Finite Group Databasebeta

← #10802129

Subgroups — #10802129

Order 768.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83928C211yesmaximal, characteristic
2C2×#5196C211yesmaximal, characteristic, complemented
3#78989C212yesmaximal, complemented
4#78549C211yesmaximal, characteristic, complemented
5C3×#1976C211yesmaximal, characteristic, complemented
6#70665C211yesmaximal, characteristic, complemented
7#70659C211yesmaximal, characteristic, complemented
8#70401C212yesmaximal, complemented
9#70055C212yesmaximal, complemented
10#70040C212yesmaximal, complemented
11#69901C211yesmaximal, characteristic, complemented
12#21558—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14#5822C2212yes—
15C2×#948C2212yes—
16C2×#907C2211yescharacteristic
17C2×#901C2211yescharacteristic
18#5522C2211yescharacteristic
19C3×#427C2211yescharacteristic
20C3×#421C2212yes—
21C3×#410C2212yes—
22C3×#405C2211yescharacteristic
23#5237C2212yes—
24C2×#837C2212yes—
25#5196—21no—
26#5196—21no—
27#5196—21no—
28#5196—21no—
29#5192C412yescomplemented
30#5177C414yescomplemented
31#5160C412yescomplemented
32#5124C2212yes—
33Dic3×M16C2211yescharacteristic
34C2×#819C2211yescharacteristic
35C2×#819C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#5005C2212yes—
38#4997C2212yes—
39#4997C2211yescharacteristic
40#4824—24no—
41#4823—24no—
42#4708—21no—
43#4707—21no—
44#4689—21no—
45C4×D3×C8—21no—
46#4533C2212yes—
47#4533C2212yes—
48#4532C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#443—31no—
52C4×#131—31no—
53#1976D311yescharacteristic, complemented
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1772—31no—
59#1757—32no—
60C2×C3×#41C2312yes—
61C2×#235C2311yescharacteristic
62C3×#135—21no—
63C3×#135—21no—
64C3×#135—21no—
65C3×#135—21no—
66C2×C3×Q16C2×C411yescharacteristic, complemented
67C2×C3×SD16C2×C412yescomplemented
68C2×C3×D8C2×C411yescharacteristic, complemented
69C2×C3×M16C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×#117C2312yes—
72C3×#117C2311yescharacteristic
73#951—22no—
74#950—24no—
75#949—22no—
76#948—22no—
77#948—22no—
78#948—22no—
79#948—22no—
80#942C2×C414yes—
81#932C2×C414yes—
82C2×C4×Dic3C2311yescharacteristic
83C2×#208—22no—
84C2×#208C2312yes—
85C2×#208C2312yes—
86C2×#208C2311yescharacteristic
87C22×C3⋊C8—22no—
88C22×C3⋊C8C2311yescharacteristic
89#907—21no—
90#907—21no—
91#907D412yes—
92#907D412yes—
93#901—21no—
94#901—21no—
95#901C2×C412yes—
96#901C2×C412yes—
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100C2×D3×C8—21no—
101#893—21no—
102#887—21no—
103#67⋊C4—22no—
104#872—21no—
105C42×D3—21no—
106C3×#107—21no—
107C3×#106—21no—
108C3×#104—24no—
109C3×#99—24no—
110C3×#97—21no—
111C3×C4×C8—21no—
112#837—22no—
113#837—22no—
114#836C2×C414yescomplemented
115#833C2×C414yescomplemented
116#822—24no—
117#819—21no—
118#819—21no—
119#819C2×C412yescomplemented
120#819C2×C412yescomplemented
121#815—21no—
122C8×Dic3—21no—
123C8×Dic3D411yescharacteristic
124C8×Dic3D411yescharacteristic
125C8×Dic3D411yescharacteristic
126C8×Dic3D411yescharacteristic
127#805—24no—
128#803—21no—
129#803—22no—
130C4×C3⋊C8—21no—
131C4×C3⋊C8—22no—
132#575—32no—
133C2×#135C2×D311yescharacteristic
134C2×#131—32no—
135C2×#131—31no—
136C2×#131—31no—
137C4×#41—31no—
138#473—32no—
139#472—32no—
140#471—34no—
141#444—64no—
142#443—61no—
143#443—61no—
144#443—61no—
145#443—61no—
146#443—64no—
147#430—32no—
148C2×#108—31no—
149C2×#108—31no—
150#427C2×D311yescharacteristic
151#421—32no—
152#421C2×D312yes—
153C2×#104—32no—
154#413—32no—
155#410C2×D312yes—
156#405—31no—
157#405—32no—
158#405—61no—
159#405—61no—
160#405C2×D311yescharacteristic
161C4×M16—31no—
162C4×M16—61no—
163C2×C4×C8—31no—
164C2×C4×C8—61no—
165#344—32no—
166#344—32no—
167#339—32no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C2×C3×Q8C22×C412yes—
173C2×C3×D4—22no—
174C2×C3×D4C22×C412yes—
175C22×C3×C4—22no—
176C22×C3×C4C2411yescharacteristic
177C2×#67—21no—
178C22×Dic3—21no—
179#235—21no—
180#235—21no—
181#235C22×C412yes—
182#235C2×D412yes—
183C2×D12#4111yescharacteristic
184C2×C4×D3—21no—
185C2×C4×D3—22no—
186C2×Dic6#4111yescharacteristic
187C3×Q16—21no—
188C3×Q16—21no—
189C3×SD16—22no—
190C3×SD16—22no—
191C3×D8—21no—
192C3×D8—21no—
193C3×M16—21no—
194C3×M16C2×D412yes—
195C2×C3×C8—21no—
196C2×C3×C8C22×C411yescharacteristic
197C2×C3×C8C22×C411yescharacteristic
198C2×C3×C8C2×D411yescharacteristic
199C2×C3×C8C2×D411yescharacteristic
200C2×C3×C8#4111yescharacteristic
201C2×C3×C8#4111yescharacteristic
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4#4112yes—
206C3×C42—21no—
207C3×C42—22no—
208#216—24no—
209#215—24no—
210#214—24no—
211#213—24no—
212(C2×D3)⋊C4—22no—
213Dic3⋊C4—22no—
214C4×Dic3—22no—
215C4×Dic3—21no—
216C4×Dic3C2×D411yescharacteristic
217C4×Dic3C2×D411yescharacteristic
218#208—21no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208—42no—
223#208—42no—
224#208C22×C414yes—
225#208C22×C412yes—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—42no—
230C2×C3⋊C8—42no—
231C2×C3⋊C8C2×D411yescharacteristic
232C2×C3⋊C8C2×D411yescharacteristic
233C2×C3⋊C8C2×D411yescharacteristic
234C2×C3⋊C8C2×D411yescharacteristic
235C2×C3⋊C8#4111yescharacteristic
236C2×C3⋊C8#4111yescharacteristic
237ΓL(1,𝔽25)—22no—
238ΓL(1,𝔽25)—22no—
239ΓL(1,𝔽25)—22no—
240ΓL(1,𝔽25)—22no—
241D3×C8—22no—
242D3×C8—22no—
243D3×C8—22no—
244D3×C8—22no—
245C2×#41—31no—
246C2×#41C22×D312yes—
247#137—34no—
248#137—62no—
249AGL(1,ℤ/8)—34no—
250AGL(1,ℤ/8)—62no—
251#135—21no—
252#135—21no—
253#135—21no—
254#135—21no—
255#135—64no—
256C2×Q16C4×D311yescharacteristic, complemented
257C2×SD16C4×D312yescomplemented
258C2×D8C4×D311yescharacteristic, complemented
259#131—32no—
260#131—32no—
261#131—32no—
262#131—32no—
263#131—61no—
264#131—61no—
265#131—62no—
266#131—62no—
267#131—62no—
268#131—62no—
269#131—61no—
270#131—61no—
271C2×M16—31no—
272C2×M16—32no—
273C2×M16—32no—
274C2×M16—61no—
275C2×M16—62no—
276C2×M16—61no—
277C2×M16C22×D311yescharacteristic
278C22×C8—31no—
279C22×C8—61no—
280C22×C8—62no—
281C22×C8—61no—
282C22×C8C22×D311yescharacteristic
283C4×Q8—61no—
284C4×D4—61no—
285C4×D4—62no—
286#117—61no—
287#117C22×D312yes—
288#117C22×D311yescharacteristic
289C2×C42—31no—
290C2×C42—61no—
291#108—32no—
292#108—32no—
293#108—64no—
294#108—61no—
295#108—61no—
296#107—21no—
297#106—21no—
298#104—24no—
299#104—64no—
300#104—62no—
301#104—62no—
302#101—34no—
303#100—34no—
304#99—24no—
305#97—21no—
306#97—61no—
307#97—62no—
308#97—61no—
309C4×C8—21no—
310C4×C8—31no—
311C4×C8—31no—
312C4×C8—31no—
313C4×C8—31no—
314C4×C8—61no—
315C4×C8—62no—
316C4×C8—61no—
317C23×C3—22no—
318C23×C3C2×#4111yescharacteristic
319C22×D3—21no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—42no—
325C3×D4—42no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4C23×C411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337#67—22no—
338#67—22no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—42no—
343D12—21no—
344D12C4×D412yes—
345C4×D3—22no—
346C4×D3—22no—
347C4×D3—44no—
348Dic6—21no—
349Dic6C4×D412yes—
350C3×C8—22no—
351C3×C8C4×D411yescharacteristic
352C3×C8C4×D411yescharacteristic
353C3×C8C4×D411yescharacteristic
354C3×C8C4×D411yescharacteristic
355C3⋊C8—22no—
356C3⋊C8—24no—
357C3⋊C8—42no—
358C3⋊C8#12111yescharacteristic
359C3⋊C8#12111yescharacteristic
360C3⋊C8#12111yescharacteristic
361C3⋊C8#12111yescharacteristic
362#41—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41—32no—
367#41—32no—
368#41—61no—
369#41—61no—
370C2×Q8—31no—
371C2×Q8C2×C4×D312yes—
372C2×D4—22no—
373C2×D4—31no—
374C2×D4—61no—
375C2×D4C2×C4×D312yes—
376C22×C4—22no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4C23×D311yescharacteristic
381Q16—21no—
382Q16—21no—
383Q16—64no—
384SD16—22no—
385SD16—22no—
386SD16—64no—
387SD16—64no—
388D8—21no—
389D8—21no—
390D8—64no—
391M16—21no—
392M16—32no—
393M16—34no—
394M16—62no—
395M16—62no—
396M16—61no—
397M16—62no—
398M16—62no—
399M16—62no—
400M16—62no—
401M16—62no—
402M16—122no—
403M16—122no—
404M16D3×D412yes—
405C2×C8—21no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—31no—
411C2×C8—31no—
412C2×C8—62no—
413C2×C8—62no—
414C2×C8—61no—
415C2×C8—62no—
416C2×C8—62no—
417C2×C8—62no—
418C2×C8—62no—
419C2×C8—122no—
420C2×C8—122no—
421C2×C8C2×C4×D311yescharacteristic
422C2×C8C2×C4×D311yescharacteristic
423C2×C8D3×D411yescharacteristic
424C2×C8D3×D411yescharacteristic
425C2×C8#23711yescharacteristic
426C2×C8#23711yescharacteristic
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—62no—
430C22⋊C4—22no—
431C22⋊C4—62no—
432C22⋊C4#23712yes—
433C42—21no—
434C42—22no—
435C42—31no—
436C42—31no—
437C42—62no—
438C42—61no—
439C22×C3—41no—
440C22×C3—42no—
441C22×C3—82no—
442C22×C3C4×#4111yescharacteristic
443C22×C3#52511yescharacteristic
444C22×C3#53211yescharacteristic
445C2×D3—22no—
446C2×D3—42no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4C2×C4×D411yescharacteristic
451C3×C4C2×C4×D411yescharacteristic
452C3×C4C2×#12111yescharacteristic
453C3×C4#52911yescharacteristic
454Dic3—22no—
455Dic3—22no—
456Dic3—41no—
457C23—22no—
458C23—61no—
459C23C2×#23711yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—32no—
463Q8—61no—
464D4—22no—
465D4—22no—
466D4—32no—
467D4—42no—
468D4—42no—
469D4—62no—
470D4—62no—
471D4—61no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—61no—
482C2×C4—124no—
483C2×C4—122no—
484C2×C4C22×C4×D311yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×D3×D411yescharacteristic
487C2×C4C2×#23711yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4D3×#4111yescharacteristic
490C8—22no—
491C8—31no—
492C8—31no—
493C8—31no—
494C8—31no—
495C8—62no—
496C8—64no—
497C8—122no—
498C8#67⋊C411yescharacteristic
499C8#67⋊C411yescharacteristic
500C8#67⋊C411yescharacteristic
501C8#67⋊C411yescharacteristic
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—82no—
506C2×C3C4×#12111yescharacteristic
507D3—42no—
508C22—41no—
509C22—42no—
510C22—62no—
511C22—82no—
512C22—122no—
513C22C4×#23711yescharacteristic
514C22#560611yescharacteristic
515C22#571611yescharacteristic
516C4—42no—
517C4—42no—
518C4—62no—
519C4—62no—
520C4—82no—
521C4—121no—
522C4C2×#67⋊C411yescharacteristic
523C4C4×D3×D411yescharacteristic
524C4D3×#12111yescharacteristic
525C4#568011yescharacteristic
526C3#2155811yescharacteristic, complemented, Sylow
527C2—21no—
528C2—21no—
529C2—21no—
530C2—82no—
531C2—122no—
532C2#7629111yescharacteristic
subgroupG/H
centerC4D3×#121
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1976C2
socleC2×C3C4×#121
prime radical#83928C2
cosocleC22×C3×C4C24
hypercenter#1976D3
hypocenterC3#21558
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. C2×#5196C2semidirect
3. #78989C2semidirect
4. #78549C2semidirect
5. C3×#1976C2semidirect
6. #70665C2semidirect
7. #70659C2semidirect
8. #70401C2semidirect
9. #70055C2semidirect
10. #70040C2semidirect
11. #69901C2semidirect
29. #5192C4semidirect
30. #5177C4semidirect
31. #5160C4semidirect
53. #1976D3semidirect
66. C2×C3×Q16C2×C4semidirect
67. C2×C3×SD16C2×C4semidirect
68. C2×C3×D8C2×C4semidirect
114. #836C2×C4semidirect
115. #833C2×C4semidirect
119. #819C2×C4semidirect
120. #819C2×C4semidirect
256. C2×Q16C4×D3semidirect
257. C2×SD16C4×D3semidirect
258. C2×D8C4×D3semidirect
526. C3#21558semidirect

Maximal subgroups

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

subgroupindex
#839282
C2×#51962
#789892
#785492
C3×#19762
#706652
#706592
#704012
#700552
#700402
#699012
#215583