Finite Group Databasebeta

← #10826081

Subgroups — #10826081

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83792C211yesmaximal, characteristic
2#78480C211yesmaximal, characteristic, complemented
3C3×#1765C211yesmaximal, characteristic, complemented
4#72011C211yesmaximal, characteristic, complemented
5#70379C211yesmaximal, characteristic
6#70368C211yesmaximal, characteristic, complemented
7#70296C211yesmaximal, characteristic
8#70294C211yesmaximal, characteristic, complemented
9#69898C211yesmaximal, characteristic, complemented
10#69895C211yesmaximal, characteristic, complemented
11#69839C211yesmaximal, characteristic, complemented
12#68936C211yesmaximal, characteristic, complemented
13#68935C211yesmaximal, characteristic, complemented
14#68779C211yesmaximal, characteristic, complemented
15#68751C211yesmaximal, characteristic, complemented
16#23834—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18#5822C2211yescharacteristic
19C2×#901C2211yescharacteristic
20#5470C2211yescharacteristic
21C2×C3×#104C2211yescharacteristic
22C3×#416C2211yescharacteristic
23C3×#405C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25#5240C2211yescharacteristic
26#5237C2211yescharacteristic
27#5236C2211yescharacteristic
28#5145—22no—
29#5144—22no—
30#5139C2211yescharacteristic
31#5132C2211yescharacteristic
32#5124C2211yescharacteristic
33#5113C2211yescharacteristic
34#5102C2211yescharacteristic
35#5102C2211yescharacteristic
36#5101C2211yescharacteristic
37#5101C2211yescharacteristic
38#5101C2211yescharacteristic
39#5006C2211yescharacteristic
40C2×#805C2211yescharacteristic
41#4924C2211yescharacteristic
42#4701—22no—
43#4701—22no—
44C3×#341C2211yescharacteristic
45C3×#339C2211yescharacteristic
46C3⋊#338C2211yescharacteristic
47#4559C2211yescharacteristic
48#4557C2211yescharacteristic
49#4556C2211yescharacteristic
50#4556C2211yescharacteristic
51#4552C2211yescharacteristic
52#4533C2211yescharacteristic
53#4531C2211yescharacteristic
54#4531C2211yescharacteristic
55#4523C2211yescharacteristic
56#2961—31no—
57#2955—31no—
58#2103—31no—
59#2102—31no—
60#2080—31no—
61#2080—31no—
62#2053—31no—
63#2050—31no—
64#2031—31no—
65#2031—31no—
66#1765—31no—
67#1765D311yescharacteristic, complemented
68#1757—31no—
69#1757—31no—
70#1747—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C3×#131—21no—
74C3×#131—21no—
75C3×#131—21no—
76C3×#131—21no—
77C2×C3×M16—21no—
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2311yescharacteristic
80C22×C3×C8—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#117C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85#951—21no—
86#950—22no—
87#949—21no—
88#942—21no—
89#932—21no—
90#926C2311yescharacteristic
91#926C2311yescharacteristic
92#926C2311yescharacteristic
93#924—22no—
94C2×#208C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C2×#208C2311yescharacteristic
97C2×#208C2311yescharacteristic
98#901—21no—
99#901—21no—
100#901—21no—
101#901—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×D3×C8—21no—
104#870—21no—
105#869—21no—
106C4×Dic6—22no—
107C3×#104—21no—
108C3×#104—21no—
109C3×#103—21no—
110C3×#102—21no—
111C3×#97—21no—
112C3×#97—21no—
113C3×#97—21no—
114C3×C4×C8—21no—
115#837—22no—
116#836—21no—
117#834—22no—
118#833—21no—
119#825—22no—
120#824—21no—
121#823—21no—
122#822—22no—
123#821—21no—
124#816—21no—
125#815—21no—
126#815—21no—
127#815D412yes—
128Dic3⋊C8—22no—
129Dic3⋊C8—22no—
130C8×Dic3—21no—
131C8×Dic3—21no—
132C8×Dic3D412yes—
133#806—22no—
134#805—21no—
135#805—21no—
136C4⋊(C3⋊C8)—22no—
137#575—31no—
138C2×#131—31no—
139C2×#131C2×D311yescharacteristic
140#533—31no—
141#446—62no—
142#446—62no—
143#444—62no—
144#443—62no—
145#430—31no—
146#424—31no—
147#424—31no—
148#423—31no—
149#421—31no—
150#421—31no—
151C2×#104—31no—
152C2×#104C2×D311yescharacteristic
153#416—31no—
154#416C2×D311yescharacteristic
155#413—31no—
156#413—31no—
157#410—31no—
158#410—31no—
159#405—31no—
160#405—31no—
161#405—31no—
162#405C2×D311yescharacteristic
163C2×#97C2×D311yescharacteristic
164#344—31no—
165#344—31no—
166#344—31no—
167#344—31no—
168#341—31no—
169#341C2×D311yescharacteristic
170#339—31no—
171#339C2×D311yescharacteristic
172#338—31no—
173H(3,ℤ/4)—31no—
174H(3,ℤ/4)—31no—
175H(3,ℤ/4)—31no—
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C2×C3×Q8#4111yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4#4111yescharacteristic
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2411yescharacteristic
186C2×#67—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190#235—21no—
191C2×D12#4111yescharacteristic
192C2×C4×D3—21no—
193C2×Dic6—22no—
194C2×Dic6#4111yescharacteristic
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—21no—
199C3×M16—41no—
200C3×M16—41no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—41no—
207C2×C3×C8—41no—
208C2×C3×C8C2×D411yescharacteristic
209C2×C3×C8C2×D411yescharacteristic
210C2×C3×C8#4111yescharacteristic
211C2×C3×C8#4111yescharacteristic
212C2×C3×C8#4111yescharacteristic
213C2×C3×C8#4111yescharacteristic
214C3×C4⋊C4—21no—
215C3×C22⋊C4—21no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42#4111yescharacteristic
219C3×C42#4111yescharacteristic
220#217—21no—
221#217—21no—
222#217#4112yes—
223#216—22no—
224#215—22no—
225#214—22no—
226#213—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231Dic3⋊C4—24no—
232C4×Dic3—21no—
233C4×Dic3—21no—
234C4×Dic3C2×D412yes—
235#208—21no—
236#208—21no—
237#208—21no—
238#208—21no—
239#208—21no—
240#208—21no—
241#208—21no—
242#208—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8C2×D412yes—
247ΓL(1,𝔽25)—41no—
248ΓL(1,𝔽25)—41no—
249D3×C8—41no—
250D3×C8—41no—
251C2×#41—31no—
252C2×#41C22×D311yescharacteristic
253#137—61no—
254#137—61no—
255AGL(1,ℤ/8)—61no—
256AGL(1,ℤ/8)—61no—
257#135—62no—
258#131—21no—
259#131—21no—
260#131—21no—
261#131—21no—
262#131—61no—
263#131—61no—
264#131—61no—
265#131—61no—
266C2×M16—21no—
267C2×M16—31no—
268C2×M16—31no—
269C2×M16—31no—
270C2×M16—31no—
271C2×M16—61no—
272C2×M16C22×D311yescharacteristic
273C2×M16C22×D311yescharacteristic
274C22×C8—21no—
275C22×C8—61no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4⋊Q8—61no—
279#125—61no—
280C22⋊Q8—62no—
281C4×Q8—62no—
282#117—31no—
283#117—31no—
284#117—31no—
285#117C22×D311yescharacteristic
286C2×C42C22×D311yescharacteristic
287#108—62no—
288C4⋊C8—62no—
289C4⋊C8—62no—
290C4⋊C8—62no—
291#104—21no—
292#104—21no—
293#104—62no—
294#104—62no—
295#104—61no—
296#104—61no—
297#103—21no—
298#103—61no—
299#102—21no—
300#102—61no—
301#101—61no—
302#101—61no—
303#100—61no—
304#100—61no—
305#99—62no—
306#99—62no—
307#97—21no—
308#97—21no—
309#97—21no—
310#97—32no—
311#97—61no—
312#97—61no—
313C4×C8—21no—
314C4×C8—32no—
315C4×C8—61no—
316C4×C8—61no—
317C23×C3—21no—
318C23×C3C2×#4111yescharacteristic
319C22×D3—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×D4—21no—
323C3×D4—21no—
324C3×D4—41no—
325C3×D4—41no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—81no—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339#67—41no—
340#67—41no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—21no—
344C2×Dic3—41no—
345D12—21no—
346D12—21no—
347C4×D3—41no—
348C4×D3—41no—
349Dic6—21no—
350Dic6—21no—
351Dic6—82no—
352C3×C8—41no—
353C3×C8—41no—
354C3×C8—41no—
355C3×C8#12411yescharacteristic
356C3×C8#12411yescharacteristic
357C3×C8#12411yescharacteristic
358C3×C8#12411yescharacteristic
359C3⋊C8—22no—
360C3⋊C8—41no—
361C3⋊C8—41no—
362C3⋊C8—41no—
363#41—21no—
364#41—21no—
365#41—21no—
366#41—21no—
367#41—61no—
368#41—61no—
369#41—61no—
370#41—61no—
371C2×Q8—31no—
372C2×Q8—62no—
373C2×Q8#23511yescharacteristic
374C2×D4—21no—
375C2×D4—31no—
376C2×D4—61no—
377C2×D4#23511yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—61no—
381C22×C4C23×D311yescharacteristic
382Q16—62no—
383SD16—62no—
384SD16—62no—
385D8—62no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—41no—
391M16—41no—
392M16—61no—
393M16—61no—
394M16—61no—
395M16—61no—
396M16—61no—
397M16—61no—
398M16—61no—
399M16—61no—
400M16—121no—
401M16—121no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—32no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—121no—
414C2×C8—121no—
415C2×C8#23511yescharacteristic
416C2×C8#23511yescharacteristic
417C2×C8D3×D411yescharacteristic
418C2×C8D3×D411yescharacteristic
419C2×C8#23711yescharacteristic
420C2×C8#23711yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—64no—
423C4⋊C4—62no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—61no—
427C22⋊C4—21no—
428C22⋊C4—32no—
429C22⋊C4—61no—
430C22⋊C4—61no—
431C42—21no—
432C42—21no—
433C42—32no—
434C42—61no—
435C42—61no—
436C42#23511yescharacteristic
437C42#23511yescharacteristic
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—81no—
441C22×C3#52511yescharacteristic
442C22×C3#52911yescharacteristic
443C22×C3#52911yescharacteristic
444C2×D3—41no—
445C2×D3—81no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—81no—
450C3×C4#52511yescharacteristic
451C3×C4C2×#12411yescharacteristic
452C3×C4#52911yescharacteristic
453C3×C4#52911yescharacteristic
454Dic3—42no—
455Dic3—42no—
456Dic3—41no—
457Dic3—81no—
458C23—21no—
459C23—61no—
460C23D3×#4111yescharacteristic
461Q8—21no—
462Q8—21no—
463Q8—61no—
464Q8—61no—
465Q8—242no—
466D4—21no—
467D4—21no—
468D4—41no—
469D4—41no—
470D4—61no—
471D4—61no—
472D4—121no—
473D4—121no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—61no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—81no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4C2×#23511yescharacteristic
488C2×C4C2×#23511yescharacteristic
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×#23711yescharacteristic
492C2×C4D3×#4111yescharacteristic
493C8—41no—
494C8—41no—
495C8—41no—
496C8—62no—
497C8—121no—
498C8—121no—
499C8—121no—
500C8#88511yescharacteristic
501C8#88511yescharacteristic
502C8#88511yescharacteristic
503C8#88511yescharacteristic
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—81no—
508C2×C3#239811yescharacteristic
509D3—81no—
510C22—41no—
511C22—41no—
512C22—81no—
513C22—121no—
514C22—241no—
515C22#552811yescharacteristic
516C22#553811yescharacteristic
517C22#554711yescharacteristic
518C4—41no—
519C4—41no—
520C4—41no—
521C4—81no—
522C4—122no—
523C4—122no—
524C4—121no—
525C4—241no—
526C4#547911yescharacteristic
527C4C2×#88511yescharacteristic
528C4#554711yescharacteristic
529C4#555711yescharacteristic
530C3#2383411yescharacteristic, complemented, Sylow
531C2—21no—
532C2—21no—
533C2—21no—
534C2—81no—
535C2—241no—
536C2#7348111yescharacteristic
subgroupG/H
centerC4C2×#885
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1765C2
socleC2×C3#2398
prime radical#5822C22
hypercenter#1765D3
hypocenterC3#23834
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #78480C2semidirect
3. C3×#1765C2semidirect
4. #72011C2semidirect
6. #70368C2semidirect
8. #70294C2semidirect
9. #69898C2semidirect
10. #69895C2semidirect
11. #69839C2semidirect
12. #68936C2semidirect
13. #68935C2semidirect
14. #68779C2semidirect
15. #68751C2semidirect
67. #1765D3semidirect
530. C3#23834semidirect

Maximal subgroups

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

subgroupindex
#837922
#784802
C3×#17652
#720112
#703792
#703682
#702962
#702942
#698982
#698952
#698392
#689362
#689352
#687792
#687512
#238343