Finite Group Databasebeta

← #10950836

Subgroups — #10950836

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#80742C211yesmaximal, characteristic, complemented
2#80699C211yesmaximal, characteristic, complemented
3C3×#2217C211yesmaximal, characteristic, complemented
4#71481C211yesmaximal, characteristic, complemented
5#71474C211yesmaximal, characteristic, complemented
6#71413C211yesmaximal, characteristic, complemented
7#71399C211yesmaximal, characteristic, complemented
8#71084C211yesmaximal, characteristic, complemented
9#71073C211yesmaximal, characteristic, complemented
10#69341C211yesmaximal, characteristic, complemented
11#69339C211yesmaximal, characteristic, complemented
12#69216C211yesmaximal, characteristic, complemented
13#69206C211yesmaximal, characteristic, complemented
14#69183C211yesmaximal, characteristic, complemented
15#69174C211yesmaximal, characteristic, complemented
16#33744—31nomaximal, Sylow
17#5643—21no—
18#5619—21no—
19C2×C3×SD32C2211yescharacteristic
20C22×C3×C16C2211yescharacteristic
21C3×#467C2211yescharacteristic
22C3×#466C2211yescharacteristic
23#5191C2211yescharacteristic
24Dic3×Q16C2211yescharacteristic
25C2×#828C2211yescharacteristic
26#5151C2211yescharacteristic
27Dic3×D8C2211yescharacteristic
28C2×#826C2211yescharacteristic
29#5115C2211yescharacteristic
30#5115C2211yescharacteristic
31#5101C2211yescharacteristic
32C2×#812C2211yescharacteristic
33#5049—21no—
34#5039—21no—
35C2×D48C2211yescharacteristic
36#4878C2211yescharacteristic
37#4874C2211yescharacteristic
38#4872C2211yescharacteristic
39#4872C2211yescharacteristic
40#4817—21no—
41#4794—21no—
42#4785—21no—
43#4764—21no—
44C3×#367C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#357C2211yescharacteristic
47#4564C2211yescharacteristic
48#4562C2211yescharacteristic
49#4509C2211yescharacteristic
50#4509C2211yescharacteristic
51#4504C2211yescharacteristic
52#4502C2211yescharacteristic
53#4502C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55Dic3×C16C2211yescharacteristic
56#4491C2211yescharacteristic
57#4491C2211yescharacteristic
58#4490C2211yescharacteristic
59#4489C2211yescharacteristic
60#3176—31no—
61#3175—31no—
62#2255—31no—
63#2250—31no—
64#2242—31no—
65#2240—31no—
66#2220—31no—
67#2218—31no—
68#2217D311yescharacteristic, complemented
69#2215—31no—
70#2177—31no—
71#2175—31no—
72C4×SD32—31no—
73C4×D16—31no—
74#2108—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×D8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C3×C22⋊Q8—21no—
79C3×#121—21no—
80#947—21no—
81Q8×Dic3—21no—
82C2×#215—21no—
83#940—21no—
84#939—41no—
85#938—41no—
86D4×Dic3—21no—
87C2×#213—21no—
88#930—21no—
89#930—21no—
90#926D411yescharacteristic
91C2×#208D411yescharacteristic
92C2×D24C2311yescharacteristic
93C2×D24C2311yescharacteristic
94#898—41no—
95#897—41no—
96#893—21no—
97#893—21no—
98#890—21no—
99#888—21no—
100#886—41no—
101#885—41no—
102#883—41no—
103#883—41no—
104C3×SD32—22no—
105C2×C3×C16—22no—
106C2×C3×C16C2311yescharacteristic
107C2×C3×C16C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#107C2311yescharacteristic
110C3×#103—21no—
111C3×#102—21no—
112#835—21no—
113#832—21no—
114#828—22no—
115#826—22no—
116#821—21no—
117#821—21no—
118#818C2311yescharacteristic
119#818C2311yescharacteristic
120#815D411yescharacteristic
121#815D411yescharacteristic
122C8×Dic3C2311yescharacteristic
123C8×Dic3C2311yescharacteristic
124#812—22no—
125C2×C3⋊C16C2311yescharacteristic
126C2×C3⋊C16C2311yescharacteristic
127#809—21no—
128#809—21no—
129#808—21no—
130#807—21no—
131D48—22no—
132#542—61no—
133#540—61no—
134C2×SD32—31no—
135C2×SD32C2×D311yescharacteristic
136C2×D16—31no—
137C2×D16—31no—
138C2×M32—31no—
139C22×C16C2×D311yescharacteristic
140#500—31no—
141#499—31no—
142#495—31no—
143#493—31no—
144#490—61no—
145#489—61no—
146#469—61no—
147#468—61no—
148#467C2×D311yescharacteristic
149#466—31no—
150#466—31no—
151#466C2×D311yescharacteristic
152#442—61no—
153#440—61no—
154C4×Q16—31no—
155C4×D8—31no—
156C4×D8—31no—
157C4×D8—31no—
158#405—31no—
159#367—31no—
160#367C2×D311yescharacteristic
161#366—31no—
162#366—31no—
163#358—31no—
164#358C2×D311yescharacteristic
165#357—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357C2×D311yescharacteristic
171#346—31no—
172#346—31no—
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×D4—21no—
177C2×C3×D4—41no—
178C22×C3×C4C2×D411yescharacteristic
179C2×#67—41no—
180C2×#67—41no—
181C2×#67—41no—
182C2×#67—41no—
183C22×Dic3—41no—
184C2×D12—21no—
185C2×D12—21no—
186C2×C4×D3—41no—
187C2×C4×D3—41no—
188C2×Dic6—41no—
189C3×Q16—21no—
190C3×Q16—21no—
191C3×D8—21no—
192C3×D8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2411yescharacteristic
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—41no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—41no—
202#217—41no—
203#217D811yescharacteristic
204#217D811yescharacteristic
205#215—42no—
206#213—42no—
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209(C2×D3)⋊C4—41no—
210(C2×D3)⋊C4—41no—
211(C2×D3)⋊C4—41no—
212(C2×D3)⋊C4—41no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—41no—
216C4⋊Dic3—41no—
217C4⋊Dic3D811yescharacteristic
218C4⋊Dic3D811yescharacteristic
219Dic3⋊C4—41no—
220Dic3⋊C4—41no—
221C4×Dic3—41no—
222C4×Dic3C2×D411yescharacteristic
223C4×Dic3C2×D411yescharacteristic
224#208—22no—
225C2×C3⋊C8C2×D411yescharacteristic
226C2×C3⋊C8C2×D411yescharacteristic
227D24—21no—
228D24—21no—
229D24—21no—
230D24—21no—
231C3×C16—21no—
232C3×C16#4112yes—
233C3⋊C16—21no—
234C3⋊C16#4112yes—
235C2×Q16C22×D311yescharacteristic
236C2×SD16—61no—
237C2×D8—31no—
238C2×D8—31no—
239C2×D8—61no—
240C2×D8C22×D311yescharacteristic
241C2×M16—31no—
242C22×C8C22×D311yescharacteristic
243C4⋊Q8—61no—
244#127—61no—
245#126—121no—
246#125—61no—
247#124—61no—
248#124—121no—
249#123—121no—
250#123—121no—
251#123—121no—
252C22⋊Q8—21no—
253C22⋊Q8—121no—
254#121—21no—
255#121—61no—
256#121—61no—
257#121—121no—
258#121—121no—
259C4×Q8—61no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—61no—
263#117—31no—
264SD32—22no—
265SD32—62no—
266D16—62no—
267D16—62no—
268M32—62no—
269C2×C16—22no—
270C2×C16—31no—
271C2×C16—31no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—31no—
275#107—31no—
276#107—61no—
277#107C22×D311yescharacteristic
278#107C22×D311yescharacteristic
279#106—61no—
280#103—21no—
281#103—61no—
282#102—21no—
283#102—61no—
284#102—61no—
285#102—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8—31no—
292C23×C3—41no—
293C23×C3C2×D811yescharacteristic
294C22×D3—41no—
295C22×D3—41no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—161no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4C2×D811yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308#67—161no—
309#67—161no—
310#67—161no—
311#67—161no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—161no—
315C2×Dic3C2×D811yescharacteristic
316C2×Dic3C2×D811yescharacteristic
317C2×Dic3C2×D811yescharacteristic
318C2×Dic3C2×D811yescharacteristic
319D12—41no—
320D12—41no—
321D12—41no—
322D12—41no—
323C4×D3—161no—
324C4×D3—161no—
325Dic6—161no—
326C3×C8—21no—
327C3×C8C2×#4111yescharacteristic
328C3×C8C2×#4111yescharacteristic
329C3⋊C8—21no—
330C3⋊C8#13512yes—
331C2×Q8—21no—
332C2×Q8—121no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—121no—
338C2×D4—121no—
339C2×D4—121no—
340C2×D4—121no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4D3×D411yescharacteristic
345Q16—21no—
346Q16—21no—
347SD16—122no—
348D8—21no—
349D8—21no—
350D8—61no—
351D8—61no—
352D8—61no—
353D8—61no—
354D8—122no—
355M16—62no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—31no—
359C2×C8—31no—
360C2×C8D3×D411yescharacteristic
361C2×C8C23×D311yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—31no—
365C4⋊C4—31no—
366C4⋊C4—41no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C4⋊C4—121no—
373C22⋊C4—31no—
374C22⋊C4—31no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C22⋊C4—121no—
382C22⋊C4—121no—
383C22⋊C4—121no—
384C42—31no—
385C42—31no—
386C42—121no—
387C16—21no—
388C16—32no—
389C16—61no—
390C16#23712yes—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3C22×D811yescharacteristic
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—21no—
405C3×C4—81no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4#53211yescharacteristic
409C3×C4C2×#13511yescharacteristic
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—21no—
414Dic3—81no—
415Dic3—81no—
416C23—41no—
417C23—121no—
418C23—121no—
419C23D3×D811yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—481no—
423D4—41no—
424D4—41no—
425D4—121no—
426D4—121no—
427D4—121no—
428D4—121no—
429D4—161no—
430D4—481no—
431D4—481no—
432D4—481no—
433D4—481no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—481no—
446C2×C4—481no—
447C2×C4—481no—
448C2×C4D3×D811yescharacteristic
449C2×C4C2×D3×D411yescharacteristic
450C8—21no—
451C8—32no—
452C8—61no—
453C8C2×#23711yescharacteristic
454C8D3×#4111yescharacteristic
455C2×C3—41no—
456C2×C3—161no—
457C2×C3#316911yescharacteristic
458C2×C3#341511yescharacteristic
459C2×C3#341611yescharacteristic
460D3—161no—
461D3—161no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22—241no—
473C22—241no—
474C22C2×D3×D811yescharacteristic
475C4—21no—
476C4—61no—
477C4—61no—
478C4—61no—
479C4—61no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—241no—
484C4—241no—
485C4#559611yescharacteristic
486C4D3×#13511yescharacteristic
487C3#3374411yescharacteristic, complemented, Sylow
488C2—41no—
489C2—161no—
490C2—481no—
491C2—481no—
492C2#8069711yescharacteristic
493C2#8202411yescharacteristic
494C2#8203111yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2217C2
socleC22×C3C22×D8
hypercenter#2217D3
hypocenterC3#33744
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #80742C2semidirect
2. #80699C2semidirect
3. C3×#2217C2semidirect
4. #71481C2semidirect
5. #71474C2semidirect
6. #71413C2semidirect
7. #71399C2semidirect
8. #71084C2semidirect
9. #71073C2semidirect
10. #69341C2semidirect
11. #69339C2semidirect
12. #69216C2semidirect
13. #69206C2semidirect
14. #69183C2semidirect
15. #69174C2semidirect
68. #2217D3semidirect
487. C3#33744semidirect

Maximal subgroups

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

subgroupindex
#807422
#806992
C3×#22172
#714812
#714742
#714132
#713992
#710842
#710732
#693412
#693392
#692162
#692062
#691832
#691742
#337443