Finite Group Databasebeta

← #33056037

Subgroups — #33056037

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619976C211yesmaximal, characteristic
2#22619081C211yesmaximal, characteristic, complemented
3#22618952C211yesmaximal, characteristic, complemented
4#22608796C211yesmaximal, characteristic, complemented
5#22608787C211yesmaximal, characteristic, complemented
6#22608764C211yesmaximal, characteristic, complemented
7#22608763C211yesmaximal, characteristic, complemented
8#22608753C211yesmaximal, characteristic, complemented
9#22608743C211yesmaximal, characteristic, complemented
10C8×#11924017C211yesmaximal, characteristic, complemented
11#22608721C211yesmaximal, characteristic, complemented
12#22608354C211yesmaximal, characteristic, complemented
13#22608336C211yesmaximal, characteristic, complemented
14#22608230C211yesmaximal, characteristic, complemented
15C8×D8×C73C211yesmaximal, characteristic, complemented
16#17989119C2211yescharacteristic
17C4×D4×D73C2211yescharacteristic
18#17988924C411yescharacteristic
19#17988904C411yescharacteristic
20#17988890C411yescharacteristic
21#17988889C411yescharacteristic
22#17988801C411yescharacteristic
23#17988800C411yescharacteristic
24#17988739C411yescharacteristic
25#17988719C411yescharacteristic
26#17988714C2211yescharacteristic, complemented
27C8×Dic146C2211yescharacteristic, complemented
28C4×#11924017C2211yescharacteristic
29C4×#11924016C2211yescharacteristic
30C4×#11924013C2211yescharacteristic
31#17988665C2211yescharacteristic, complemented
32#17988662—21no—
33#17988658—21no—
34C8×D292C2211yescharacteristic
35D4×#10596831C2211yescharacteristic
36#17988652C2211yescharacteristic
37C8×C22⋊D73—21no—
38#17988650—21no—
39#17988645—21no—
40#17988639C2211yescharacteristic
41#17988638C2211yescharacteristic
42#17988638C2211yescharacteristic
43#17988638C2211yescharacteristic
44#17988613—21no—
45#17988609C2211yescharacteristic
46C2×#14556503—21no—
47C22⋊C8×D73—21no—
48#17988339C2211yescharacteristic, complemented
49#17988338C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988319C2211yescharacteristic, complemented
52#17988317C2211yescharacteristic, complemented
53#17988316C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic, complemented
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic, complemented
58#17988309C2211yescharacteristic, complemented
59#17988309C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61C8×D4×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic, complemented
63#335×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66C82×C73C2211yescharacteristic, complemented
67C2×D4×D73C2×C411yescharacteristic
68C22×C4×D73—21no—
69C2×#11924017C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924012—21no—
73C2×#11924011—21no—
74C22×#10596831—21no—
75C2×C8×D73—21no—
76C2×C8×D73—21no—
77#14556584—21no—
78#14556582C2×C411yescharacteristic
79#14556578C2×C411yescharacteristic
80#14556573—21no—
81#14556571C2×C411yescharacteristic
82#14556562C2×C411yescharacteristic
83C4×Dic146C2311yescharacteristic
84C4×D292C2311yescharacteristic
85D4×Dic73C2×C411yescharacteristic
86#14556553C2×C411yescharacteristic
87C4×C22⋊D73—21no—
88C4×C22⋊D73—21no—
89#14556551—21no—
90#14556550—21no—
91C4⋊C4×D73C2×C411yescharacteristic
92C22⋊C4×D73—21no—
93C42×D73C2311yescharacteristic
94#14556524C2×C411yescharacteristic
95#14556523C2×C411yescharacteristic
96#14556522C2×C411yescharacteristic
97C4⋊#10596831C2311yescharacteristic
98C4⋊#10596831C2311yescharacteristic
99#14556520—21no—
100#14556520—21no—
101#14556520—21no—
102#14556520—21no—
103#14556520C2×C411yescharacteristic
104#14556520C2×C411yescharacteristic
105#14556520D411yescharacteristic
106#14556520D411yescharacteristic
107#14556516C2×C411yescharacteristic
108#14556514C2×C411yescharacteristic
109#14556513C2×C411yescharacteristic
110#14556512C2×C411yescharacteristic
111#14556512C2×C411yescharacteristic
112#14556511C2×C411yescharacteristic
113C22⋊#10596831—21no—
114C22⋊#10596831—21no—
115#14556503—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503C2×C411yescharacteristic
120#14556503C2×C411yescharacteristic
121#14556503D411yescharacteristic
122#14556503D411yescharacteristic
123#14556501—21no—
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C4×#10596831C2311yescharacteristic
127C8×Dic73—21no—
128C8×Dic73—21no—
129C2×D8×C73C2×C411yescharacteristic
130C22×C8×C73—21no—
131C22×C8×C73—21no—
132C4×D4×C73C2311yescharacteristic
133C4×D4×C73C2311yescharacteristic
134#107×C73C2×C411yescharacteristic
135C4⋊C8×C73C2311yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137#102×C73C2×C411yescharacteristic
138#102×C73C2×C411yescharacteristic
139C22⋊C8×C73—21no—
140C22⋊C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C23×D73—21no—
145C2×Dic146C22×C411yescharacteristic
146C2×D292C22×C411yescharacteristic
147C2×C22⋊D73—21no—
148C2×C22⋊D73—21no—
149D4×D73—22no—
150D4×D73—22no—
151C22×Dic73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—81no—
156C2×C4×D73—81no—
157C2×C4×D73C22×C411yescharacteristic
158C2×C4×D73C2×D411yescharacteristic
159C2×C4×D73#4111yescharacteristic
160#11924017—22no—
161#11924016—22no—
162#11924013—22no—
163#11924012—81no—
164#11924011—81no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—81no—
169C2×#10596831C22×C411yescharacteristic
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C2×D411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C8×D73—81no—
174C8×D73—81no—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73C22×C411yescharacteristic
178C4⋊Dic73C22×C411yescharacteristic
179#11924007—21no—
180#11924007—21no—
181#11924007—21no—
182C22⋊Dic73—21no—
183C22⋊Dic73—21no—
184C22⋊Dic73—21no—
185C22⋊Dic73—21no—
186#11924005—21no—
187#11924005—21no—
188#11924005—21no—
189C4×Dic73—21no—
190C4×Dic73C22×C411yescharacteristic
191C4×Dic73C2×D411yescharacteristic
192C4×Dic73#4111yescharacteristic
193C2×D4×C73C22×C411yescharacteristic
194C2×D4×C73C22×C411yescharacteristic
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197D8×C73—22no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—81no—
202C2×C8×C73—81no—
203C2×C8×C73C22×C411yescharacteristic
204C2×C8×C73C22×C411yescharacteristic
205C2×C8×C73C2×D411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C4⋊C4×C73C22×C411yescharacteristic
208C4⋊C4×C73C22×C411yescharacteristic
209C22⋊C4×C73—21no—
210C22⋊C4×C73—21no—
211C42×C73C2411yescharacteristic
212C23×C73—21no—
213C23×C73—21no—
214C22×D73—21no—
215C22×D73—24no—
216C22×D73—81no—
217C22×D73—81no—
218C22×D73C4×D411yescharacteristic
219D4×C73—21no—
220D4×C73—21no—
221D4×C73—21no—
222D4×C73#13112yes—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73C23×C411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C22⋊D73—24no—
233C22⋊D73—81no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—81no—
238C2×Dic73C4×D411yescharacteristic
239C2×Dic73C4×D411yescharacteristic
240C2×Dic73C4×D411yescharacteristic
241D292—21no—
242D292#13112yes—
243C4×D73—21no—
244C4×D73—21no—
245C4×D73—21no—
246C4×D73—21no—
247C4×D73—41no—
248C4×D73—41no—
249C4×D73—41no—
250C4×D73—81no—
251C4×D73—81no—
252Dic146—21no—
253Dic146—21no—
254C8×C73—21no—
255C8×C73—21no—
256C8×C73—21no—
257C8×C73—21no—
258C8×C73—41no—
259C8×C73—41no—
260C8×C73—41no—
261#10596831—21no—
262#10596831—21no—
263#10596831—21no—
264#10596831—21no—
265#10596831—41no—
266#10596831—41no—
267#10596831—41no—
268C22×C73—22no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73C2×C4×D411yescharacteristic
275C2×D73—21no—
276C2×D73—22no—
277C2×D73—21no—
278C2×D73—21no—
279C2×D73—41no—
280C2×D73—41no—
281C2×D73—84no—
282C4×C73—21no—
283C4×C73—41no—
284C4×C73—41no—
285C4×C73C2×#13111yescharacteristic
286C4×C73C2×#13111yescharacteristic
287C4×C73#57611yescharacteristic
288C4×C73#57711yescharacteristic
289Dic73—21no—
290Dic73—21no—
291Dic73—21no—
292Dic73—41no—
293Dic73—41no—
294Dic73—41no—
295#17918—731nomaximal, Sylow
296C2×C73—42no—
297C2×C73—81no—
298C2×C73#292911yescharacteristic
299C2×C73#294611yescharacteristic
300C2×C73#295711yescharacteristic
301D73—41no—
302D73—42no—
303#3105—731no—
304#2993—731no—
305#2972—731no—
306#1586—731no—
307#1585—731no—
308#1582—731no—
309#1581—731no—
310#1580—731no—
311SD16⋊C8—731no—
312C8×SD16—731no—
313C8×D8D7311yescharacteristic, complemented
314#1574—731no—
315#1486—731no—
316#1482—731no—
317#1454—731no—
318C73#1791811yescharacteristic, complemented, Sylow
319#529—731no—
320C2×C4×D4—731no—
321#484—731no—
322#481—731no—
323#478—731no—
324#478—731no—
325#463—731no—
326#463—731no—
327#450—731no—
328#447—731no—
329#446—731no—
330C8×Q8—731no—
331C4×SD16—731no—
332C4×SD16—731no—
333C4×D8—731no—
334C4×D8C2×D7311yescharacteristic
335#436—731no—
336#436—1461no—
337#435—1461no—
338C8×D4—731no—
339C8×D4—731no—
340C8×D4—731no—
341C8×D4—1461no—
342C8×D4—1461no—
343C8×D4C2×D7311yescharacteristic
344C8×D4C2×D7311yescharacteristic, complemented
345#433—1461no—
346#431—731no—
347#431—731no—
348#431—731no—
349#431—731no—
350#424—1461no—
351#423—731no—
352C2×C22⋊C8—1461no—
353C2×C22⋊C8—1461no—
354#335—731no—
355#335C2×D7311yescharacteristic, complemented
356#334—731no—
357#334—731no—
358Q8⋊C8—731no—
359Q8⋊C8—731no—
360D4⋊C8—731no—
361D4⋊C8—731no—
362D4⋊C8—731no—
363D4⋊C8—731no—
364D4⋊C8C2×D7311yescharacteristic
365D4⋊C8C2×D7311yescharacteristic, complemented
366#322—731no—
367#322—731no—
368#322—731no—
369C82C2×D7311yescharacteristic, complemented
370C22×D4—731no—
371C23×C4—1461no—
372C2×SD16—731no—
373C2×SD16—731no—
374C2×D8—731no—
375C2×D8C4×D7311yescharacteristic
376C2×M16—1461no—
377C2×M16—1461no—
378C22×C8—21no—
379C22×C8—21no—
380C22×C8—1461no—
381C22×C8—1461no—
382C22×C8—1461no—
383#126—1461no—
384#125—731no—
385#124—731no—
386#123—1461no—
387C22⋊Q8—731no—
388#121—731no—
389C4×Q8—731no—
390C4×D4—731no—
391C4×D4—731no—
392C4×D4—731no—
393C4×D4—1461no—
394C4×D4—1461no—
395C4×D4—1461no—
396C4×D4C22×D7311yescharacteristic
397C4×D4C22×D7311yescharacteristic
398#117—1461no—
399C2×C4⋊C4—731no—
400C2×C22⋊C4—1461no—
401C2×C42—731no—
402#107—731no—
403#107C4×D7311yescharacteristic
404#106—731no—
405#106—731no—
406C4⋊C8—731no—
407C4⋊C8—731no—
408C4⋊C8—731no—
409C4⋊C8—731no—
410C4⋊C8—731no—
411C4⋊C8—731no—
412C4⋊C8—1461no—
413C4⋊C8—1461no—
414C4⋊C8—1461no—
415C4⋊C8—1461no—
416C4⋊C8C22×D7311yescharacteristic
417C4⋊C8C22×D7311yescharacteristic
418#103—731no—
419#103—731no—
420#102—731no—
421#102—731no—
422#102—731no—
423#102—731no—
424#102C4×D7311yescharacteristic
425#102C4×D7311yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—731no—
429C22⋊C8—731no—
430C22⋊C8—731no—
431C22⋊C8—731no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438#97—1461no—
439C4×C8—731no—
440C4×C8—731no—
441C4×C8—731no—
442C4×C8—1461no—
443C4×C8—1461no—
444C4×C8C22×D7311yescharacteristic
445C4×C8C22×D7311yescharacteristic
446C4×C8C22×D7311yescharacteristic
447C24—1461no—
448C2×Q8—731no—
449C2×D4—731no—
450C2×D4—1461no—
451C2×D4—1462no—
452C2×D4—1461no—
453C2×D4—1462no—
454C2×D4C2×C4×D7311yescharacteristic
455C2×D4C2×C4×D7311yescharacteristic
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—5841no—
459C22×C4—5841no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467SD16—1462no—
468SD16—1462no—
469D8—22no—
470D8—1462no—
471M16—5841no—
472M16—5841no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—81no—
477C2×C8—81no—
478C2×C8—5841no—
479C2×C8—5841no—
480C2×C8—5841no—
481C2×C8—731no—
482C2×C8—731no—
483C2×C8—731no—
484C2×C8—731no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8C2×C4×D7311yescharacteristic
489C2×C8C2×C4×D7311yescharacteristic
490C2×C8D4×D7311yescharacteristic
491C2×C8D4×D7311yescharacteristic
492C4⋊C4—731no—
493C4⋊C4—731no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4C2×C4×D7311yescharacteristic
500C4⋊C4C2×C4×D7311yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—1461no—
504C22⋊C4—1461no—
505C22⋊C4—1461no—
506C22⋊C4—1461no—
507C22⋊C4—1461no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C42—731no—
511C42—731no—
512C42—731no—
513C42—1461no—
514C42C23×D7311yescharacteristic
515C23—21no—
516C23—21no—
517C23—5841no—
518C23—5841no—
519C23—731no—
520C23—1464no—
521C23—1461no—
522Q8—1461no—
523Q8—1461no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—5841no—
528D4—732no—
529D4—1464no—
530D4—1461no—
531D4#1455659712yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—5841no—
542C2×C4—5841no—
543C2×C4—5841no—
544C2×C4—731no—
545C2×C4—731no—
546C2×C4—731no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4C22×C4×D7311yescharacteristic
555C2×C4C2×D4×D7311yescharacteristic
556C2×C4#41×D7311yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—2921no—
565C8—2921no—
566C8—2921no—
567C8—1461no—
568C8—1461no—
569C8—1461no—
570C8—1461no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—2921no—
578C22—2921no—
579C22—5844no—
580C22—1461no—
581C22—1461no—
582C22—1461no—
583C22—1462no—
584C22C4×D4×D7311yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—2921no—
589C4—2921no—
590C4—2921no—
591C4—1461no—
592C4—1461no—
593C4—1461no—
594C4#131×D7311yescharacteristic
595C4C2×#1455659711yescharacteristic
596C4#1798938811yescharacteristic
597C4#1798939411yescharacteristic
598C2—42no—
599C2—81no—
600C2—2922no—
601C2—2921no—
602C2#2261862011yescharacteristic
603C2#2261875611yescharacteristic
604C2#2261884311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC8×D8×C73C2
socleC22×C73C2×C4×D4
prime radical#17988719C4
cosocleC42×C73C24
hypercenterC8×D8D73
hypocenterC73#17918
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22619081C2semidirect
3. #22618952C2semidirect
4. #22608796C2semidirect
5. #22608787C2semidirect
6. #22608764C2semidirect
7. #22608763C2semidirect
8. #22608753C2semidirect
9. #22608743C2semidirect
10. C8×#11924017C2semidirect
11. #22608721C2semidirect
12. #22608354C2semidirect
13. #22608336C2semidirect
14. #22608230C2semidirect
15. C8×D8×C73C2semidirect
26. #17988714C22semidirect
27. C8×Dic146C22semidirect
31. #17988665C22semidirect
48. #17988339C22semidirect
49. #17988338C22semidirect
50. #17988337C22semidirect
51. #17988319C22semidirect
52. #17988317C22semidirect
55. D4⋊#10596831C22semidirect
57. #17988309C22semidirect
58. #17988309C22semidirect
59. #17988309C22semidirect
62. C8×D4×C73C22semidirect
63. #335×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. C82×C73C22semidirect
313. C8×D8D73semidirect
318. C73#17918semidirect
344. C8×D4C2×D73semidirect
355. #335C2×D73semidirect
365. D4⋊C8C2×D73semidirect
369. C82C2×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226199762
#226190812
#226189522
#226087962
#226087872
#226087642
#226087632
#226087532
#226087432
C8×#119240172
#226087212
#226083542
#226083362
#226082302
C8×D8×C732
#1791873