Finite Group Databasebeta

← #33056028

Subgroups — #33056028

Order 18688.

692 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619987C211yesmaximal, characteristic
2C8×#11924027C211yesmaximal, characteristic, complemented
3#22618640C211yesmaximal, characteristic, complemented
4#22608776C211yesmaximal, characteristic, complemented
5#22608772C211yesmaximal, characteristic, complemented
6#22608760C211yesmaximal, characteristic, complemented
7#22608755C211yesmaximal, characteristic, complemented
8#22608752C211yesmaximal, characteristic, complemented
9#22608742C211yesmaximal, characteristic, complemented
10#22608740C211yesmaximal, characteristic, complemented
11#22608721C211yesmaximal, characteristic, complemented
12#22608367C211yesmaximal, characteristic, complemented
13#22608346C211yesmaximal, characteristic, complemented
14#22608217C211yesmaximal, characteristic, complemented
15C73×#1581C211yesmaximal, characteristic, complemented
16C2×#14556599—21no—
17#17989134C2211yescharacteristic
18C4×#11924027C2211yescharacteristic
19#17988914C411yescharacteristic
20#17988910C411yescharacteristic
21#17988881C411yescharacteristic
22#17988875C411yescharacteristic
23#17988786C411yescharacteristic
24#17988780C411yescharacteristic
25#17988733C411yescharacteristic
26#17988729C411yescharacteristic
27C8×Dic146C2211yescharacteristic
28#17988711C2211yescharacteristic
29C4×#11924017C2211yescharacteristic
30C4×#11924016C2211yescharacteristic
31C4×#11924013C2211yescharacteristic
32#17988666C2211yescharacteristic
33#17988663—21no—
34D4×#10596831C2211yescharacteristic
35D4×#10596831C2211yescharacteristic
36#17988652C2211yescharacteristic
37C8×C22⋊D73—21no—
38#17988650—21no—
39#17988650—21no—
40#17988639C2211yescharacteristic
41#17988638C2211yescharacteristic
42#17988637—21no—
43#17988609C2211yescharacteristic
44#17988552—21no—
45C4×#11924011C2211yescharacteristic
46C2×C8×Dic73—21no—
47C4×C8×D73C2211yescharacteristic
48#17988339C2211yescharacteristic
49#17988338C2211yescharacteristic
50#17988337C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988317C2211yescharacteristic
53#17988316C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic
55D4⋊#10596831C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988308C2211yescharacteristic
59#17988308C2211yescharacteristic
60C4×D8×C73C2211yescharacteristic
61#436×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924028—21no—
68C2×#11924027C2×C411yescharacteristic
69C2×#11924017C2×C411yescharacteristic
70C2×#11924016C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72#14556599—44no—
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C22×#10596831—21no—
77C22×#10596831—21no—
78C2×C8×D73—21no—
79C2×C8×D73D411yescharacteristic
80C2×C8×D73D411yescharacteristic
81#14556590C2×C411yescharacteristic
82#14556587C2×C411yescharacteristic
83#14556577—21no—
84#14556566C2×C411yescharacteristic
85#14556564—21no—
86#14556562C2×C411yescharacteristic
87C4×Dic146C2311yescharacteristic
88#14556556C2×C411yescharacteristic
89C4×D292C2311yescharacteristic
90D4×Dic73C2×C411yescharacteristic
91C4×C22⋊D73—21no—
92C4×C22⋊D73—21no—
93#14556551—21no—
94#14556549C2×C411yescharacteristic
95#14556547—21no—
96C2×C4×Dic73—21no—
97C42×D73C2311yescharacteristic
98#14556524C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100#14556522C2×C411yescharacteristic
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520C2×C411yescharacteristic
107#14556520C2×C411yescharacteristic
108#14556516C2×C411yescharacteristic
109#14556514C2×C411yescharacteristic
110#14556513C2×C411yescharacteristic
111#14556512C2×C411yescharacteristic
112#14556512C2×C411yescharacteristic
113#14556511C2×C411yescharacteristic
114C22⋊#10596831—21no—
115C22⋊#10596831—21no—
116#14556503—21no—
117#14556503—21no—
118#14556503—21no—
119#14556503C2×C411yescharacteristic
120#14556503C2×C411yescharacteristic
121#14556501—21no—
122C4×#10596831C2311yescharacteristic
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73D411yescharacteristic
129C8×Dic73D411yescharacteristic
130C2×D8×C73C2×C411yescharacteristic
131C2×M16×C73—21no—
132C22×C8×C73—21no—
133C4×D4×C73C2311yescharacteristic
134C4×D4×C73C2311yescharacteristic
135#107×C73C2×C411yescharacteristic
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#102×C73C2×C411yescharacteristic
139#102×C73C2×C411yescharacteristic
140C22⋊C8×C73—21no—
141C22⋊C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145#11924028—42no—
146#11924028—42no—
147#11924027—22no—
148#11924027—22no—
149C2×Dic146—21no—
150C2×Dic146C22×C411yescharacteristic
151C2×D292—21no—
152C2×D292C22×C411yescharacteristic
153C2×C22⋊D73—21no—
154C2×C22⋊D73—21no—
155C2×C22⋊D73—21no—
156C2×C22⋊D73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C2×C4×D73—21no—
160C2×C4×D73C22×C411yescharacteristic
161C2×C4×D73C2×D411yescharacteristic
162C2×C4×D73#4111yescharacteristic
163#11924017—22no—
164#11924016—22no—
165#11924013—22no—
166#11924011—42no—
167#11924011—42no—
168#11924011—42no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—42no—
173C2×#10596831—42no—
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C22×C411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C2×#10596831C2×D411yescharacteristic
178C8×D73—22no—
179C8×D73—41no—
180C8×D73—42no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186C22⋊Dic73—21no—
187C22⋊Dic73—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73C22×C411yescharacteristic
195C4×Dic73C2×D411yescharacteristic
196C4×Dic73#4111yescharacteristic
197C2×D4×C73C22×C411yescharacteristic
198C2×D4×C73C22×C411yescharacteristic
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201D8×C73—22no—
202M16×C73—42no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—42no—
207C2×C8×C73C22×C411yescharacteristic
208C2×C8×C73C22×C411yescharacteristic
209C2×C8×C73C2×D411yescharacteristic
210C2×C8×C73C2×D411yescharacteristic
211C4⋊C4×C73C22×C411yescharacteristic
212C4⋊C4×C73C22×C411yescharacteristic
213C22⋊C4×C73—21no—
214C22⋊C4×C73—21no—
215C42×C73C2411yescharacteristic
216C23×C73—21no—
217C23×C73—21no—
218C22×D73—21no—
219C22×D73C4×D411yescharacteristic
220D4×C73—21no—
221D4×C73—21no—
222D4×C73—21no—
223D4×C73#13112yescomplemented
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73C23×C411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C22⋊D73—24no—
234C22⋊D73—42no—
235C22⋊D73—42no—
236C22⋊D73—42no—
237C2×Dic73—24no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—42no—
242C2×Dic73C4×D411yescharacteristic
243C2×Dic73C4×D411yescharacteristic
244C2×Dic73C4×D411yescharacteristic
245D292—21no—
246D292—21no—
247D292—42no—
248C4×D73—21no—
249C4×D73—21no—
250C4×D73—21no—
251C4×D73—21no—
252C4×D73—22no—
253C4×D73—41no—
254C4×D73—41no—
255Dic146—21no—
256Dic146—42no—
257Dic146#13112yescomplemented
258C8×C73—21no—
259C8×C73—21no—
260C8×C73—21no—
261C8×C73—41no—
262C8×C73—41no—
263C8×C73—41no—
264C8×C73AGL(1,ℤ/8)12yes—
265#10596831—21no—
266#10596831—21no—
267#10596831—21no—
268#10596831—21no—
269#10596831—41no—
270#10596831—41no—
271#10596831—41no—
272C22×C73—22no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73C2×C4×D411yescharacteristic
279C2×D73—21no—
280C2×D73—21no—
281C2×D73—21no—
282C2×D73—41no—
283C2×D73—41no—
284C2×D73—41no—
285C4×C73—21no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73C2×#13111yescharacteristic
289C4×C73C2×#13111yescharacteristic
290C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C73#57511yescharacteristic
292Dic73—22no—
293Dic73—21no—
294Dic73—21no—
295Dic73—41no—
296Dic73—41no—
297Dic73#44412yes—
298#17917—731nomaximal, Sylow
299C2×C73—42no—
300C2×C73—81no—
301C2×C73#292911yescharacteristic
302C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C73C2×#44411yescharacteristic
304D73—41no—
305D73—81no—
306#3106—731no—
307C8×#41—731no—
308#2931—731no—
309#1583—731no—
310#1583—731no—
311#1581—731no—
312#1581D7311yescharacteristic, complemented
313#1580—731no—
314#1580—731no—
315SD16⋊C8—731no—
316SD16⋊C8—731no—
317#1574—731no—
318#1489—731no—
319#1484—731no—
320#1451—731no—
321C73#1791711yescharacteristic, complemented, Sylow
322C2×#131—1461no—
323#532—731no—
324C4×#41—731no—
325#482—731no—
326#482—731no—
327#477—731no—
328#476—731no—
329#460—731no—
330#459—731no—
331#449—731no—
332#449—731no—
333C8×Q8—731no—
334C8×Q8—731no—
335C4×SD16—731no—
336C4×SD16—731no—
337C4×D8—731no—
338C4×D8C2×D7311yescharacteristic
339#436—731no—
340#436—1461no—
341#436C2×D7311yescharacteristic
342C8×D4—731no—
343C8×D4—731no—
344C8×D4—731no—
345C8×D4—1461no—
346C8×D4—1461no—
347C8×D4—1461no—
348C8×D4C2×D7311yescharacteristic
349#431—731no—
350#431—731no—
351#431—1461no—
352#423—731no—
353#408—1461no—
354C4×M16—731no—
355C2×C4×C8—731no—
356C2×C4×C8—1461no—
357#335—731no—
358#335C2×D7311yescharacteristic
359#334—731no—
360#334—731no—
361Q8⋊C8—731no—
362Q8⋊C8—731no—
363D4⋊C8—731no—
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8—731no—
367D4⋊C8C2×D7311yescharacteristic
368D4⋊C8C2×D7311yescharacteristic
369#322—731no—
370#322—731no—
371#322—731no—
372#322C2×D7311yescharacteristic
373C2×#41—731no—
374C2×#41—1461no—
375C2×SD16—731no—
376C2×SD16—731no—
377C2×D8—731no—
378C2×D8C4×D7311yescharacteristic
379#131—2924no—
380C2×M16—21no—
381C2×M16—1461no—
382C2×M16—1461no—
383C2×M16—1461no—
384C22×C8—21no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—1461no—
388C22×C8—1461no—
389C22×C8—1461no—
390C4⋊Q8—731no—
391#127—731no—
392#124—1461no—
393#121—731no—
394#121—731no—
395#121—1461no—
396C4×Q8—731no—
397C4×Q8—731no—
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—1461no—
401C4×D4—1461no—
402C4×D4—1461no—
403C4×D4C22×D7311yescharacteristic
404C4×D4C22×D7311yescharacteristic
405#117—731no—
406#117—1461no—
407C2×C42—731no—
408C2×C42—1461no—
409#107—731no—
410#107C4×D7311yescharacteristic
411#106—731no—
412#106—731no—
413C4⋊C8—731no—
414C4⋊C8—731no—
415C4⋊C8—731no—
416C4⋊C8—731no—
417C4⋊C8—1461no—
418C4⋊C8—1461no—
419C4⋊C8—1461no—
420C4⋊C8C22×D7311yescharacteristic
421C4⋊C8C22×D7311yescharacteristic
422#103—731no—
423#103—731no—
424#102—731no—
425#102—731no—
426#102—731no—
427#102—731no—
428#102C4×D7311yescharacteristic
429#102C4×D7311yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—731no—
433C22⋊C8—731no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438C22⋊C8—1461no—
439#97—1461no—
440C4×C8—731no—
441C4×C8—731no—
442C4×C8—731no—
443C4×C8—731no—
444C4×C8—731no—
445C4×C8—1461no—
446C4×C8—1461no—
447C4×C8—1461no—
448C4×C8C22×D7311yescharacteristic
449C4×C8C22×D7311yescharacteristic
450C4×C8C22×D7311yescharacteristic
451#41—2922no—
452#41—2922no—
453#41—1462no—
454#41—1462no—
455C2×Q8—731no—
456C2×Q8—1461no—
457C2×D4—731no—
458C2×D4—1461no—
459C2×D4—1461no—
460C2×D4—1461no—
461C2×D4—1461no—
462C2×D4—1461no—
463C2×D4C2×C4×D7311yescharacteristic
464C2×D4C2×C4×D7311yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—731no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4—1461no—
473SD16—1462no—
474SD16—1462no—
475D8—22no—
476D8—1462no—
477M16—42no—
478M16—2922no—
479M16—2922no—
480M16—2922no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—2921no—
486C2×C8—2922no—
487C2×C8—2922no—
488C2×C8—2922no—
489C2×C8—731no—
490C2×C8—731no—
491C2×C8—731no—
492C2×C8—731no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1462no—
497C2×C8C2×C4×D7311yescharacteristic
498C2×C8C2×C4×D7311yescharacteristic
499C2×C8D4×D7311yescharacteristic
500C2×C8D4×D7311yescharacteristic
501C4⋊C4—731no—
502C4⋊C4—731no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4C2×C4×D7311yescharacteristic
507C4⋊C4C2×C4×D7311yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C22⋊C4—1461no—
513C22⋊C4—1461no—
514C22⋊C4—1461no—
515C42—731no—
516C42—731no—
517C42—731no—
518C42—1461no—
519C42—1461no—
520C42—1461no—
521C42C23×D7311yescharacteristic
522C23—21no—
523C23—21no—
524C23—731no—
525C23—1461no—
526Q8—2922no—
527Q8—732no—
528Q8—1461no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—2922no—
533D4—2922no—
534D4—2922no—
535D4—2922no—
536D4—1464no—
537D4—1461no—
538D4—1461no—
539D4#1455659912yescomplemented
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—2922no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—731no—
550C2×C4—731no—
551C2×C4—731no—
552C2×C4—1462no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1464no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4C22×C4×D7311yescharacteristic
562C2×C4C2×D4×D7311yescharacteristic
563C2×C4#41×D7311yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—41no—
569C8—41no—
570C8—2921no—
571C8—2921no—
572C8—2921no—
573C8—1461no—
574C8—1461no—
575C8—1461no—
576C8—1461no—
577C8#1455661512yes—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—2921no—
585C22—2921no—
586C22—2921no—
587C22—1461no—
588C22—1461no—
589C22—1461no—
590C22C4×D4×D7311yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—2921no—
595C4—2921no—
596C4—732no—
597C4—1461no—
598C4—1461no—
599C4—1462no—
600C4#131×D7311yescharacteristic
601C4C2×#1455659911yescharacteristic
602C4C2×#1455661511yescharacteristic
603C4#1798938411yescharacteristic
604C2—42no—
605C2—81no—
606C2—2921no—
607C2—5841no—
608C2#2261862211yescharacteristic
609C2C4×#1455661511yescharacteristic
610C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1581C2
socleC22×C73C2×C4×D4
prime radical#22619987C2
cosocleC42×C73C24
hypercenter#1581D73
hypocenterC73#17917
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. C8×#11924027C2semidirect
3. #22618640C2semidirect
4. #22608776C2semidirect
5. #22608772C2semidirect
6. #22608760C2semidirect
7. #22608755C2semidirect
8. #22608752C2semidirect
9. #22608742C2semidirect
10. #22608740C2semidirect
11. #22608721C2semidirect
12. #22608367C2semidirect
13. #22608346C2semidirect
14. #22608217C2semidirect
15. C73×#1581C2semidirect
223. D4×C73#131semidirect
257. Dic146#131semidirect
312. #1581D73semidirect
321. C73#17917semidirect
539. D4#14556599semidirect

Maximal subgroups

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

subgroupindex
#226199872
C8×#119240272
#226186402
#226087762
#226087722
#226087602
#226087552
#226087522
#226087422
#226087402
#226087212
#226083672
#226083462
#226082172
C73×#15812
#1791773