Finite Group Databasebeta

← #33144104

Subgroups — #33144104

Order 18688.

786 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614958C211yesmaximal, characteristic, complemented
2#22612879C211yesmaximal, characteristic
3#22611305C211yesmaximal, characteristic
4#22611303C211yesmaximal, characteristic
5#22611065C211yesmaximal, characteristic
6#22610880C211yesmaximal, characteristic, complemented
7#22610877C211yesmaximal, characteristic, complemented
8#22610693C211yesmaximal, characteristic, complemented
9#22610648C211yesmaximal, characteristic, complemented
10#22610517C211yesmaximal, characteristic, complemented
11#22610514C211yesmaximal, characteristic, complemented
12#22610057C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#2023C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556589—21no—
18C2×D4×Dic73C2211yescharacteristic
19C2×#14556547—21no—
20C2×#14556523C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×C8×Dic73C2211yescharacteristic
28#17988507—21no—
29#17988506—21no—
30#17988490C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988482—21no—
34#17988481—21no—
35#17988475—21no—
36#17988456C2211yescharacteristic
37#17988456C2211yescharacteristic
38#17988445—21no—
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4×#11924007C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4⋊Q8×C73C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55#390×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146—21no—
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556590—22no—
64#14556590—41no—
65#14556589—44no—
66D4×Dic73—22no—
67D4×Dic73Q814yescomplemented
68#14556547—44no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007D411yescharacteristic
81C2×#11924007D411yescharacteristic
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73C2311yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90#14556523—22no—
91#14556522—22no—
92C4⋊#10596831—41no—
93#14556520—22no—
94#14556520—41no—
95#14556512—22no—
96#14556512—41no—
97C8×Dic73—22no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105C22×Q8×C73—21no—
106C22×D4×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C4⋊Q8×C73—22no—
110C4⋊Q8×C73—41no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C42×C73C2311yescharacteristic
118#102×C73—22no—
119#102×C73—41no—
120C2×Dic146—42no—
121C2×Dic146—42no—
122C2×Dic146—81no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—24no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—42no—
129C22×Dic73—42no—
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73C2×Q812yes—
147C4⋊Dic73#4112yes—
148#11924007—22no—
149#11924007—42no—
150#11924007—42no—
151#11924007—42no—
152#11924007—42no—
153#11924007—42no—
154#11924007—81no—
155#11924007—81no—
156#11924007SD1614yes—
157C22⋊Dic73—24no—
158C22⋊Dic73—42no—
159C22⋊Dic73—81no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—42no—
163C4×Dic73—42no—
164C4×Dic73—42no—
165C4×Dic73C2×Q812yes—
166C4×Dic73#4112yes—
167C24×C73—21no—
168C2×Q8×C73—42no—
169C2×Q8×C73—42no—
170C2×Q8×C73—81no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73C2×Q812yes—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2411yescharacteristic
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—42no—
195C4⋊C4×C73#4112yes—
196C22⋊C4×C73—42no—
197C22⋊C4×C73—81no—
198C42×C73—21no—
199C42×C73—21no—
200C23×C73—22no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73C22×D411yescharacteristic
208Q8×C73—84no—
209D4×C73—22no—
210D4×C73—22no—
211D4×C73—41no—
212D4×C73—41no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—22no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73C2×SD1612yes—
228C2×C4×C73C22×Q811yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73H(5,𝔽2)11yescharacteristic
233C2×C4×C732−1+411yescharacteristic
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—22no—
239C2×Dic73—21no—
240C2×Dic73—22no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—48no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73C22⋊Q812yes—
255C2×Dic73C22⋊Q812yes—
256C2×Dic73C2×SD1612yes—
257C2×Dic73C2×SD1612yes—
258C2×Dic73AGL(1,ℤ/8)12yes—
259Dic146—84no—
260C8×C73—41no—
261C8×C73—41no—
262#10596831—41no—
263#10596831—41no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—81no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73—81no—
271C22×C73C2×C22⋊Q811yescharacteristic
272C22×C73#54611yescharacteristic
273C22×C73#54811yescharacteristic
274C22×C73C22×SD1611yescharacteristic
275C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C73#57611yescharacteristic
277C22×C73#57811yescharacteristic
278C4×C73—22no—
279C4×C73—42no—
280C4×C73—41no—
281C4×C73—81no—
282C4×C73—81no—
283C4×C73#53311yescharacteristic
284C4×C73#54411yescharacteristic
285C4×C73#55511yescharacteristic
286C4×C73#55611yescharacteristic
287Dic73—22no—
288Dic73—22no—
289Dic73—42no—
290Dic73—41no—
291Dic73—42no—
292Dic73—81no—
293Dic73—81no—
294Dic73#47614yes—
295#26024—731nomaximal, Sylow
296C2×C73—82no—
297C2×C73#243911yescharacteristic
298C2×C73C2×#47611yescharacteristic
299C2×C73#308311yescharacteristic
300C2×C73#329811yescharacteristic
301C2×C73#332511yescharacteristic
302C2×C73#336211yescharacteristic
303C2×C73#336711yescharacteristic
304#2551—731no—
305#2354—731no—
306#2065—731no—
307#2065—731no—
308#2023—731no—
309#2023D7311yescharacteristic
310#1987—731no—
311#1987—731no—
312#1948—731no—
313#1939—731no—
314#1920—731no—
315#1920—731no—
316#1834—731no—
317#1802—731no—
318C4×#102—731no—
319C73#2602411yescharacteristic, complemented, Sylow
320C2×C4⋊Q8—731no—
321C2×C4⋊Q8—1461no—
322C2×C4⋊Q8C2×D7311yescharacteristic
323C2×C4×D4—731no—
324C2×#117—1461no—
325C2×#106—731no—
326C2×#106—731no—
327C2×C4⋊C8—731no—
328C2×C4⋊C8—731no—
329C2×C4⋊C8—731no—
330C2×#102—731no—
331C2×#102—731no—
332C2×#102C2×D7311yescharacteristic
333C2×#102C2×D7311yescharacteristic
334C2×C4×C8—731no—
335#395—1461no—
336#395—1461no—
337#390—731no—
338#390C2×D7311yescharacteristic
339#389—731no—
340#389—731no—
341#389—1461no—
342#388—1461no—
343#388—1461no—
344#384—731no—
345#384—731no—
346#380—1461no—
347C4×C4⋊C4—731no—
348C4×C4⋊C4—731no—
349C4×C4⋊C4—731no—
350C4×C4⋊C4—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340C2×D7311yescharacteristic
360#340C2×D7311yescharacteristic
361#336—731no—
362#336C2×D7311yescharacteristic
363C22×Q8—21no—
364C22×Q8—1461no—
365C22×D4C22×D7311yescharacteristic
366C23×C4—1461no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8C22×D7311yescharacteristic
370C22×C8C22×D7311yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—41no—
373C4⋊Q8—2924no—
374C4⋊Q8—2921no—
375C4⋊Q8—1462no—
376C4×D4—734no—
377C4×D4—1462no—
378#117—2924no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4C22×D7311yescharacteristic
394C2×C4⋊C4C22×D7311yescharacteristic
395C2×C4⋊C4C22×D7311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—1461no—
404C2×C42—1461no—
405C2×C42—1461no—
406C2×C42C22×D7311yescharacteristic
407#106—1462no—
408#106—1462no—
409C4⋊C8—2921no—
410C4⋊C8—2921no—
411C4⋊C8—1462no—
412#102—22no—
413#102—41no—
414#102—2921no—
415#102—1462no—
416C4×C8—1462no—
417#95—731no—
418#95—731no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424C24—21no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—2922no—
429C2×Q8—2922no—
430C2×Q8—5841no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D7312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—2922no—
441C22×C4—2922no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1464no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4D4×D7311yescharacteristic
452C22×C4D4×D7311yescharacteristic
453C22×C4C23×D7311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—2922no—
473C4⋊C4—2922no—
474C4⋊C4—2922no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—5841no—
478C4⋊C4—5841no—
479C4⋊C4—734no—
480C4⋊C4—732no—
481C4⋊C4—732no—
482C4⋊C4—1462no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4#1192402712yes—
490C22⋊C4—42no—
491C22⋊C4—81no—
492C22⋊C4—2922no—
493C22⋊C4—5841no—
494C22⋊C4—1464no—
495C42—21no—
496C42—21no—
497C42—2922no—
498C42—2922no—
499C42—2922no—
500C42—732no—
501C42—732no—
502C42—1461no—
503C42—1461no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D7311yescharacteristic
512Q8—84no—
513Q8—5844no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2928no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—732no—
544C2×C4—732no—
545C2×C4—732no—
546C2×C4—732no—
547C2×C4—732no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4SD16×D7312yes—
558C2×C4C2×Q8×D7311yescharacteristic
559C2×C4#41×D7311yescharacteristic
560C2×C4#41×D7311yescharacteristic
561C2×C4C2×#1192402711yescharacteristic
562C2×C4#1455663611yescharacteristic
563C2×C4#1455663811yescharacteristic
564C8—41no—
565C8—41no—
566C8—2921no—
567C8—2921no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C22⋊Q8×D7311yescharacteristic
576C22#1798922411yescharacteristic
577C22#1798924511yescharacteristic
578C22C2×SD16×D7311yescharacteristic
579C22AGL(1,ℤ/8)×D7311yescharacteristic
580C22#1798938811yescharacteristic
581C22#1798940311yescharacteristic
582C4—22no—
583C4—41no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—2922no—
588C4—2922no—
589C4—2921no—
590C4—5841no—
591C4—5841no—
592C4—734no—
593C4—1462no—
594C4—1462no—
595C4#1798913911yescharacteristic
596C4#1798921611yescharacteristic
597C4#1798929911yescharacteristic
598C4#1798930611yescharacteristic
599C2—82no—
600C2#2261361511yescharacteristic
601C2#476×D7311yescharacteristic
602C2#2261977711yescharacteristic
603C2#2262178911yescharacteristic
604C2#2262216611yescharacteristic
605C2#2262259911yescharacteristic
606C2#2262264611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2023C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2023D73
hypocenterC73#26024
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #22614958C2semidirect
6. #22610880C2semidirect
7. #22610877C2semidirect
8. #22610693C2semidirect
9. #22610648C2semidirect
10. #22610517C2semidirect
11. #22610514C2semidirect
12. #22610057C2semidirect
67. D4×Dic73Q8semidirect
319. C73#26024semidirect

Maximal subgroups

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

subgroupindex
#226149582
#226128792
#226113052
#226113032
#226110652
#226108802
#226108772
#226106932
#226106482
#226105172
#226105142
#226100572
#226099292
#102×Dic732
C73×#20232
#2602473