Finite Group Databasebeta

← #10844987

Subgroups — #10844987

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79068C211yesmaximal, characteristic, complemented
2C2×#4848C211yesmaximal, characteristic
3#79008C211yesmaximal, characteristic, complemented
4#78526C211yesmaximal, characteristic
5C3×#1950C211yesmaximal, characteristic, complemented
6#70715C212yesmaximal, complemented
7#70708C212yesmaximal, complemented
8#70521C211yesmaximal, characteristic
9C4×#824C211yesmaximal, characteristic, complemented
10#68933C212yesmaximal, complemented
11#68931C212yesmaximal, complemented
12#25315—31nomaximal, Sylow
13C22×#204C2212yes—
14C2×#874C2211yescharacteristic
15#5470C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C2×C3×#108C2212yes—
18C3×#423C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C3×C4×M16C2211yescharacteristic
21C2×#824C2211yescharacteristic
22C2×#824C2211yescharacteristic
23#5131C2212yes—
24#5131C2212yes—
25C2×#823C2212yes—
26C2×#823C2212yes—
27#5121C2212yes—
28#5120C2212yes—
29C2×#821C2212yes—
30#5102C2212yes—
31#5101C2212yes—
32C4×#208C2211yescharacteristic
33#4851—21no—
34#4850—21no—
35#4848—22no—
36#4848—21no—
37#4848—21no—
38#4831—21no—
39#4830—21no—
40#4829—22no—
41#4828—22no—
42#4713—22no—
43#4694—21no—
44#4694—21no—
45C3×#341C2212yes—
46#4554C2211yescharacteristic
47#4554C2211yescharacteristic
48#3138—31no—
49#3078—31no—
50C2×#460—31no—
51#2962—31no—
52#2063—32no—
53#2046—32no—
54#1977—32no—
55#1965—31no—
56#1950D311yescharacteristic, complemented
57#1931—32no—
58C4×#101—31no—
59C22×D12—21no—
60C22×D12—21no—
61C22×D12C2312yes—
62C22×Dic6C2311yescharacteristic
63C2×C3×M16C2312yes—
64C2×C3×M16C2311yescharacteristic
65C2×C3×M16C2311yescharacteristic
66C22×C3×C8C2312yes—
67C2×C3×C42C2311yescharacteristic
68#926C2312yes—
69C2×C4⋊Dic3C2311yescharacteristic
70#924—21no—
71#924—21no—
72C2×Dic3⋊C4—21no—
73C2×C4×Dic3—21no—
74C2×#208C2311yescharacteristic
75C2×#208C2311yescharacteristic
76C2×#204—24no—
77C2×#204—22no—
78C2×#204—22no—
79C2×#204—42no—
80#874—21no—
81#874—21no—
82#874—21no—
83#874—21no—
84#870D412yes—
85#869D411yescharacteristic
86#869D411yescharacteristic
87C4×Dic6—21no—
88C4×Dic6—21no—
89C4×Dic6—21no—
90C4×Dic6—21no—
91C4×Dic6—21no—
92C4×Dic6—21no—
93C3×#108—42no—
94C3×C4⋊C8—21no—
95C3×C4⋊C8—21no—
96C3×C4⋊C8D412yes—
97C3×C4⋊C8D411yescharacteristic
98C3×C4⋊C8D411yescharacteristic
99C3×#97—21no—
100C3×C4×C8—21no—
101#824—22no—
102#824—22no—
103#823—24no—
104#823—42no—
105#821—22no—
106#821—22no—
107#821—22no—
108#821—22no—
109#821—22no—
110#821—22no—
111#818—22no—
112#817—22no—
113#816—22no—
114#816—22no—
115#815—22no—
116Dic3⋊C8—22no—
117Dic3⋊C8—22no—
118C8×Dic3—22no—
119#803—21no—
120C4×C3⋊C8—21no—
121C22×SD16—32no—
122#533—31no—
123C2×#127—31no—
124C2×C4×Q8—31no—
125#490—62no—
126#486—61no—
127#486—61no—
128#479—62no—
129#477—61no—
130#475—61no—
131#460—61no—
132#460—62no—
133#460—61no—
134#446—62no—
135#446—61no—
136C8×Q8—61no—
137C2×#108C2×D312yes—
138#428—32no—
139#424—32no—
140#423C2×D311yescharacteristic
141C2×C4⋊C8C2×D311yescharacteristic
142#419—32no—
143#418—32no—
144#418—32no—
145C2×#102—32no—
146C2×#101—31no—
147C2×#101—31no—
148C2×#100—32no—
149C2×#100—32no—
150#405—32no—
151C4×M16—31no—
152C4×M16C2×D311yescharacteristic
153#343—31no—
154#343—31no—
155#341C2×D312yes—
156C23×D3—22no—
157C22×C3×C4#4111yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—21no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—41no—
169C2×D12—41no—
170C2×D12—41no—
171C2×D12—41no—
172C2×D12—82no—
173C2×D12—82no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6#4112yes—
182C3×M16—22no—
183C3×M16—21no—
184C3×M16—21no—
185C3×M16—22no—
186C3×M16—21no—
187C3×M16—21no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8C2×D412yes—
194C2×C3×C8C2×D412yes—
195C3×C42C2×D411yescharacteristic
196C3×C42C2×D411yescharacteristic
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D411yescharacteristic
199#217—22no—
200C4⋊Dic3—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3C2×D412yes—
205C4⋊Dic3C2×D412yes—
206Dic3⋊C4—22no—
207Dic3⋊C4—22no—
208Dic3⋊C4—41no—
209Dic3⋊C4—41no—
210C4×Dic3—22no—
211C4×Dic3—41no—
212C4×Dic3—41no—
213#208—21no—
214#208—21no—
215#208—21no—
216#208#4112yes—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219#204—44no—
220#204—44no—
221C22×Q8—31no—
222C22×D4—32no—
223C22×D4—61no—
224C22×D4—61no—
225C2×SD16—64no—
226C2×SD16—62no—
227C2×SD16—62no—
228C2×SD16—122no—
229C2×M16—31no—
230C2×M16—31no—
231C2×M16C22×D311yescharacteristic
232C2×M16C22×D312yes—
233C2×M16C22×D311yescharacteristic
234C22×C8C22×D312yes—
235C4⋊Q8—31no—
236C4⋊Q8—31no—
237#127—61no—
238#127—61no—
239#127—61no—
240#127—61no—
241#125—32no—
242C22⋊Q8—61no—
243C22⋊Q8—61no—
244C4×Q8—61no—
245C4×Q8—61no—
246C4×Q8—61no—
247C4×Q8—61no—
248C4×Q8—61no—
249C4×Q8—61no—
250#117—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—61no—
253C2×C42—61no—
254C2×C42C22×D311yescharacteristic
255#108—42no—
256#107—62no—
257#106—62no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—62no—
261C4⋊C8—62no—
262C4⋊C8D1211yescharacteristic
263C4⋊C8D1211yescharacteristic
264C4⋊C8D1212yes—
265#103—62no—
266#103—62no—
267#102—62no—
268#102—62no—
269#102—62no—
270#102—62no—
271#102—62no—
272#102—62no—
273#101—62no—
274#101—62no—
275#100—64no—
276#100—122no—
277#97—21no—
278#97—62no—
279#97—61no—
280C4×C8—21no—
281C4×C8—62no—
282C4×C8—61no—
283C23×C3C2×#4111yescharacteristic
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—82no—
288C22×D3—82no—
289C2×C3×C4—22no—
290C2×C3×C4—41no—
291C2×C3×C4#12111yescharacteristic
292C2×C3×C4#12111yescharacteristic
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12111yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4H(5,𝔽2)11yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—81no—
308D12—42no—
309D12—42no—
310D12—42no—
311D12—42no—
312D12—82no—
313D12—82no—
314D12—82no—
315D12—82no—
316Dic6—22no—
317Dic6—22no—
318Dic6—41no—
319Dic6—41no—
320Dic6—42no—
321Dic6—81no—
322C3×C8—22no—
323C3×C8—22no—
324C3×C8—41no—
325C3×C8—42no—
326C3×C8—41no—
327C3⋊C8—22no—
328C3⋊C8—41no—
329C24—62no—
330C2×Q8—32no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—61no—
338C2×D4—62no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—121no—
345C2×D4—121no—
346C2×D4—121no—
347C2×D4—121no—
348C2×D4—242no—
349C2×D4—242no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4#23911yescharacteristic
353C22×C4#23911yescharacteristic
354C22×C4C23×D311yescharacteristic
355SD16—124no—
356SD16—124no—
357M16—22no—
358M16—21no—
359M16—21no—
360M16—22no—
361M16—21no—
362M16—21no—
363M16—32no—
364M16—61no—
365M16—61no—
366M16—61no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—61no—
373C2×C8—61no—
374C2×C8C2×D1212yes—
375C2×C8C2×D1212yes—
376C4⋊C4—32no—
377C4⋊C4—32no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—121no—
385C4⋊C4—121no—
386C22⋊C4—62no—
387C42—62no—
388C42—121no—
389C42—121no—
390C42C2×D1211yescharacteristic
391C42C2×D1211yescharacteristic
392C42D3×D411yescharacteristic
393C42D3×D411yescharacteristic
394C22×C3—21no—
395C22×C3—21no—
396C22×C3C2×#12111yescharacteristic
397C22×C3#52911yescharacteristic
398C22×C3#54211yescharacteristic
399C2×D3—82no—
400C2×D3—82no—
401C2×D3—82no—
402C2×D3—162no—
403C2×D3—162no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4#47112yes—
408C3×C4#54611yescharacteristic
409C3×C4#54811yescharacteristic
410C3×C4#54811yescharacteristic
411C3×C4#55011yescharacteristic
412Dic3—42no—
413Dic3—42no—
414Dic3—42no—
415Dic3—81no—
416C23—122no—
417C23—122no—
418C23—122no—
419C23—242no—
420C23—242no—
421C23C2×#23911yescharacteristic
422Q8—62no—
423Q8—62no—
424Q8—122no—
425Q8—121no—
426Q8—121no—
427Q8—241no—
428D4—122no—
429D4—122no—
430D4—122no—
431D4—122no—
432D4—242no—
433D4—242no—
434D4—242no—
435D4—242no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—241no—
445C2×C4#89511yescharacteristic
446C2×C4#89511yescharacteristic
447C2×C4#89511yescharacteristic
448C2×C4#89511yescharacteristic
449C2×C4C22×D1211yescharacteristic
450C2×C4C2×D3×D411yescharacteristic
451C2×C4#100411yescharacteristic
452C2×C4#100711yescharacteristic
453C2×C4D3×#4111yescharacteristic
454C2×C4D3×#4111yescharacteristic
455C8—22no—
456C8—22no—
457C8—41no—
458C8—41no—
459C8—42no—
460C8—62no—
461C8—121no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3#242611yescharacteristic
465C2×C3C2×#47111yescharacteristic
466C2×C3#306811yescharacteristic
467D3—162no—
468C22—21no—
469C22—21no—
470C22—242no—
471C22—242no—
472C22—242no—
473C22—482no—
474C22—482no—
475C22C2×#89511yescharacteristic
476C22#569111yescharacteristic
477C22#573311yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—122no—
482C4—122no—
483C4—122no—
484C4—241no—
485C4#489812yes—
486C4#555411yescharacteristic
487C4#557611yescharacteristic
488C4#559911yescharacteristic
489C4#563311yescharacteristic
490C3#2531511yescharacteristic, complemented, Sylow
491C2—21no—
492C2—21no—
493C2—482no—
494C2#7621811yescharacteristic
495C2C2×#489811yescharacteristic
496C2#7926111yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1950C2
socleC22×C3C2×#121
prime radicalC2×#874C22
hypercenter#1950D3
hypocenterC3#25315
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
1. #79068C2semidirect
3. #79008C2semidirect
5. C3×#1950C2semidirect
6. #70715C2semidirect
7. #70708C2semidirect
9. C4×#824C2semidirect
10. #68933C2semidirect
11. #68931C2semidirect
56. #1950D3semidirect
490. C3#25315semidirect

Maximal subgroups

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

subgroupindex
#790682
C2×#48482
#790082
#785262
C3×#19502
#707152
#707082
#705212
C4×#8242
#689332
#689312
#253153