Finite Group Databasebeta

← #10826084

Subgroups — #10826084

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83792C211yesmaximal, characteristic
2#78543C211yesmaximal, characteristic, complemented
3C3×#1757C211yesmaximal, characteristic, complemented
4#72009C211yesmaximal, characteristic, complemented
5#70144C211yesmaximal, characteristic
6#70132C211yesmaximal, characteristic, complemented
7#70043C211yesmaximal, characteristic
8#70038C211yesmaximal, characteristic, complemented
9#69898C211yesmaximal, characteristic, complemented
10#69895C211yesmaximal, characteristic, complemented
11#69888C211yesmaximal, characteristic, complemented
12#69043C211yesmaximal, characteristic, complemented
13#69042C211yesmaximal, characteristic, complemented
14#69024C211yesmaximal, characteristic, complemented
15#69016C211yesmaximal, characteristic, complemented
16#23834—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18#5822C2211yescharacteristic
19C2×#901C2211yescharacteristic
20#5517C2211yescharacteristic
21C3×#413C2211yescharacteristic
22C3×#410C2211yescharacteristic
23C3×#405C2211yescharacteristic
24C3×#405C2211yescharacteristic
25#5236C2211yescharacteristic
26C2×#837C2211yescharacteristic
27#5233C2211yescharacteristic
28#5145—22no—
29#5144—22no—
30#5139C2211yescharacteristic
31C2×#825C2211yescharacteristic
32#5132C2211yescharacteristic
33#5124C2211yescharacteristic
34#5119C2211yescharacteristic
35#5118C2211yescharacteristic
36#5101C2211yescharacteristic
37#5101C2211yescharacteristic
38C2×#815C2211yescharacteristic
39#5006C2211yescharacteristic
40#5002C2211yescharacteristic
41#4999C2211yescharacteristic
42#4701—22no—
43#4679—22no—
44C3×#344C2211yescharacteristic
45C3×#344C2211yescharacteristic
46C3⋊#338C2211yescharacteristic
47#4559C2211yescharacteristic
48#4558C2211yescharacteristic
49#4558C2211yescharacteristic
50#4557C2211yescharacteristic
51#4555C2211yescharacteristic
52#4533C2211yescharacteristic
53#4532C2211yescharacteristic
54#4531C2211yescharacteristic
55#4531C2211yescharacteristic
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#1765—31no—
68#1757—31no—
69#1757D311yescharacteristic, complemented
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
80C2×C3×M16C2311yescharacteristic
81C22×C3×C8—21no—
82C22×C3×C8C2311yescharacteristic
83C3×#117C2311yescharacteristic
84C3×#117C2311yescharacteristic
85#951—21no—
86#950—22no—
87#949—21no—
88#942—21no—
89#932—21no—
90#926C2311yescharacteristic
91#926C2311yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C2×#208C2311yescharacteristic
94C2×#208C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97#901—21no—
98#901—21no—
99#901—21no—
100#901—21no—
101C2×ΓL(1,𝔽25)—21no—
102C2×D3×C8—21no—
103#890—21no—
104#888—21no—
105#878—22no—
106C4×Dic6—22no—
107C3×#101—21no—
108C3×#100—21no—
109C3×#99—22no—
110C3×#97—21no—
111C3×#97—21no—
112C3×C4×C8—21no—
113C3×C4×C8—21no—
114#837—22no—
115#837—21no—
116#837—21no—
117#835—21no—
118#832—21no—
119#825—22no—
120#825—21no—
121#825—21no—
122#824—21no—
123#823—21no—
124#822—22no—
125#815—21no—
126#815—21no—
127#815—21no—
128#815D412yes—
129Dic3⋊C8—22no—
130Dic3⋊C8—22no—
131C8×Dic3—21no—
132C8×Dic3D412yes—
133#810—21no—
134#809—21no—
135#806—22no—
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×#104—31no—
153#416—31no—
154#416—31no—
155#413—31no—
156#413C2×D311yescharacteristic
157#410—31no—
158#410C2×D311yescharacteristic
159#405—31no—
160#405—31no—
161#405C2×D311yescharacteristic
162#405C2×D311yescharacteristic
163C2×#97—31no—
164#344—31no—
165#344—31no—
166#344C2×D311yescharacteristic
167#344C2×D311yescharacteristic
168#341—31no—
169#341—31no—
170#339—31no—
171#339—31no—
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×C4C2411yescharacteristic
185C2×#67—21no—
186C22×Dic3—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—21no—
200C3×M16—21no—
201C3×M16—41no—
202C3×M16—41no—
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×C4⋊C4—21no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4#4112yes—
218C3×C42—21no—
219C3×C42—21no—
220#217—21no—
221#217—21no—
222#216—22no—
223#215—22no—
224#214—22no—
225#213—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229Dic3⋊C4—22no—
230Dic3⋊C4—22no—
231C4×Dic3—21no—
232C4×Dic3—21no—
233C4×Dic3C2×D412yes—
234C4×Dic3#4111yescharacteristic
235C4×Dic3#4111yescharacteristic
236#208—21no—
237#208—21no—
238#208—21no—
239#208—21no—
240#208—21no—
241#208—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8C2×D412yes—
248ΓL(1,𝔽25)—41no—
249ΓL(1,𝔽25)—41no—
250D3×C8—41no—
251D3×C8—41no—
252C2×#41—31no—
253C2×#41C22×D311yescharacteristic
254#137—61no—
255#137—61no—
256AGL(1,ℤ/8)—61no—
257AGL(1,ℤ/8)—61no—
258#135—62no—
259#131—21no—
260#131—21no—
261#131—21no—
262#131—21no—
263#131—61no—
264#131—61no—
265#131—61no—
266#131—61no—
267C2×M16—21no—
268C2×M16—31no—
269C2×M16—31no—
270C2×M16—31no—
271C2×M16—61no—
272C2×M16C22×D311yescharacteristic
273C2×M16C22×D311yescharacteristic
274C2×M16C22×D311yescharacteristic
275C22×C8—21no—
276C22×C8—31no—
277C22×C8—61no—
278C22×C8C22×D311yescharacteristic
279C4⋊Q8—61no—
280#125—61no—
281C22⋊Q8—62no—
282C4×Q8—62no—
283#117—31no—
284#117—31no—
285#117C22×D311yescharacteristic
286#117C22×D311yescharacteristic
287C2×C42—31no—
288#108—62no—
289C4⋊C8—62no—
290C4⋊C8—62no—
291C4⋊C8—62no—
292#104—61no—
293#104—61no—
294#104—62no—
295#104—61no—
296#104—61no—
297#104—62no—
298#103—61no—
299#103—61no—
300#102—61no—
301#102—61no—
302#101—21no—
303#101—61no—
304#100—21no—
305#100—61no—
306#99—22no—
307#99—62no—
308#97—21no—
309#97—21no—
310#97—32no—
311#97—61no—
312#97—61no—
313#97—61no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—32no—
317C4×C8—61no—
318C23×C3—21no—
319C23×C3C2×#4111yescharacteristic
320C22×D3—21no—
321C3×Q8—21no—
322C3×Q8—21no—
323C3×D4—21no—
324C3×D4—21no—
325C3×D4—41no—
326C3×D4—41no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338#67—41no—
339#67—41no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—22no—
343C2×Dic3—21no—
344C2×Dic3—41no—
345C2×Dic3—81no—
346D12—21no—
347D12—21no—
348C4×D3—41no—
349C4×D3—41no—
350Dic6—21no—
351Dic6—21no—
352Dic6—82no—
353C3×C8—41no—
354C3×C8—41no—
355C3×C8—41no—
356C3×C8#12411yescharacteristic
357C3×C8#12411yescharacteristic
358C3×C8#12411yescharacteristic
359C3×C8#12411yescharacteristic
360C3⋊C8—22no—
361C3⋊C8—41no—
362C3⋊C8—41no—
363C3⋊C8—41no—
364#41—21no—
365#41—21no—
366#41—21no—
367#41—21no—
368#41—61no—
369#41—61no—
370#41—61no—
371#41—61no—
372C2×Q8—31no—
373C2×Q8—62no—
374C2×Q8#23511yescharacteristic
375C2×D4—21no—
376C2×D4—31no—
377C2×D4—61no—
378C2×D4#23511yescharacteristic
379C22×C4—21no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4C23×D311yescharacteristic
383Q16—62no—
384SD16—62no—
385SD16—62no—
386D8—62no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—21no—
391M16—21no—
392M16—21no—
393M16—41no—
394M16—41no—
395M16—61no—
396M16—61no—
397M16—61no—
398M16—61no—
399M16—61no—
400M16—61no—
401M16—121no—
402M16—121no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—32no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—121no—
415C2×C8—121no—
416C2×C8#23511yescharacteristic
417C2×C8#23511yescharacteristic
418C2×C8D3×D411yescharacteristic
419C2×C8D3×D411yescharacteristic
420C2×C8#23711yescharacteristic
421C2×C8#23711yescharacteristic
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—62no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C22⋊C4—21no—
430C22⋊C4—61no—
431C22⋊C4—61no—
432C22⋊C4#23512yes—
433C42—21no—
434C42—21no—
435C42—32no—
436C42—31no—
437C42—31no—
438C42—61no—
439C42—61no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—81no—
443C22×C3#52511yescharacteristic
444C22×C3#52911yescharacteristic
445C22×C3#52911yescharacteristic
446C2×D3—41no—
447C2×D3—81no—
448C3×C4—42no—
449C3×C4—41no—
450C3×C4—81no—
451C3×C4#52511yescharacteristic
452C3×C4C2×#12411yescharacteristic
453C3×C4#52911yescharacteristic
454C3×C4#52911yescharacteristic
455Dic3—41no—
456Dic3—42no—
457Dic3—41no—
458Dic3—41no—
459Dic3—81no—
460C23—21no—
461C23—61no—
462C23C2×#23511yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—61no—
466Q8—61no—
467Q8—242no—
468D4—21no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—61no—
473D4—61no—
474D4—121no—
475D4—121no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—62no—
484C2×C4—61no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—241no—
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×#23511yescharacteristic
491C2×C4C2×D3×D411yescharacteristic
492C2×C4C2×#23711yescharacteristic
493C2×C4D3×#4111yescharacteristic
494C2×C4D3×#4111yescharacteristic
495C8—41no—
496C8—41no—
497C8—41no—
498C8—62no—
499C8—121no—
500C8—121no—
501C8—121no—
502C8#88511yescharacteristic
503C8#88511yescharacteristic
504C8#88511yescharacteristic
505C8#88511yescharacteristic
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—81no—
510C2×C3#239811yescharacteristic
511D3—81no—
512C22—41no—
513C22—41no—
514C22—81no—
515C22—121no—
516C22—241no—
517C22#547911yescharacteristic
518C22#554711yescharacteristic
519C22#555711yescharacteristic
520C4—42no—
521C4—41no—
522C4—81no—
523C4—121no—
524C4—121no—
525C4—122no—
526C4—121no—
527C4—241no—
528C4C2×#88511yescharacteristic
529C4#552811yescharacteristic
530C4#553811yescharacteristic
531C4#554711yescharacteristic
532C3#2383411yescharacteristic, complemented, Sylow
533C2—21no—
534C2—21no—
535C2—21no—
536C2—81no—
537C2—241no—
538C2#7348111yescharacteristic
subgroupG/H
centerC4C2×#885
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1757C2
socleC2×C3#2398
prime radical#5822C22
hypercenter#1757D3
hypocenterC3#23834
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1757

Complements

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

NH ≅ G/Ntype
2. #78543C2semidirect
3. C3×#1757C2semidirect
4. #72009C2semidirect
6. #70132C2semidirect
8. #70038C2semidirect
9. #69898C2semidirect
10. #69895C2semidirect
11. #69888C2semidirect
12. #69043C2semidirect
13. #69042C2semidirect
14. #69024C2semidirect
15. #69016C2semidirect
69. #1757D3semidirect
532. C3#23834semidirect

Maximal subgroups

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

subgroupindex
#837922
#785432
C3×#17572
#720092
#701442
#701322
#700432
#700382
#698982
#698952
#698882
#690432
#690422
#690242
#690162
#238343