Finite Group Databasebeta

← #10950810

Subgroups — #10950810

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81935C211yesmaximal, characteristic, complemented
2#81811C211yesmaximal, characteristic, complemented
3C3×#2243C211yesmaximal, characteristic, complemented
4#71385C211yesmaximal, characteristic, complemented
5#71320C211yesmaximal, characteristic, complemented
6#69229C211yesmaximal, characteristic, complemented
7#69228C211yesmaximal, characteristic, complemented
8#69209C211yesmaximal, characteristic, complemented
9#69206C211yesmaximal, characteristic, complemented
10#69197C211yesmaximal, characteristic
11#69196C211yesmaximal, characteristic, complemented
12#69177C211yesmaximal, characteristic, complemented
13#69175C211yesmaximal, characteristic, complemented
14#69079C211yesmaximal, characteristic, complemented
15#69072C211yesmaximal, characteristic, complemented
16#33742—31nomaximal, Sylow
17#5733—21no—
18#5682—21no—
19C3×#500C2211yescharacteristic
20C3×#495C2211yescharacteristic
21#5188C2211yescharacteristic
22Dic3×Q16C2211yescharacteristic
23C2×#829C2211yescharacteristic
24C2×#828C2211yescharacteristic
25#5174—21no—
26#5164—21no—
27#5157C2211yescharacteristic
28Dic3×D8C2211yescharacteristic
29C2×#827C2211yescharacteristic
30C2×#826C2211yescharacteristic
31#5089—21no—
32#5055—21no—
33C2×#798C2211yescharacteristic
34C2×D3×C16C2211yescharacteristic
35#4889—21no—
36#4887C2211yescharacteristic
37#4886—21no—
38#4883C2211yescharacteristic
39#4873C2211yescharacteristic
40#4872C2211yescharacteristic
41#4696C2211yescharacteristic
42#4689C2211yescharacteristic
43#4682C2211yescharacteristic
44C3×#358C2211yescharacteristic
45C3×#358C2211yescharacteristic
46C3×#357C2211yescharacteristic
47C3×#357C2211yescharacteristic
48C3×#346C2211yescharacteristic
49#4509C2211yescharacteristic
50#4509C2211yescharacteristic
51#4503C2211yescharacteristic
52#4503C2211yescharacteristic
53#4502C2211yescharacteristic
54Dic3×C16C2211yescharacteristic
55#4490C2211yescharacteristic
56#4490C2211yescharacteristic
57#4490C2211yescharacteristic
58#4490C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3175—31no—
61#3175—31no—
62#2254—31no—
63#2249—31no—
64#2243—31no—
65#2243D311yescharacteristic, complemented
66#2219—31no—
67#2218—31no—
68#2217—31no—
69#2217—31no—
70#2177—31no—
71#2176—31no—
72C4×SD32—31no—
73C4×SD32—31no—
74#2108—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×SD16—21no—
77C2×C3×D8C2311yescharacteristic
78C3×C4⋊Q8—21no—
79C3×#124—21no—
80#947—41no—
81#946—21no—
82#946—41no—
83Q8×Dic3—21no—
84#938—21no—
85D4×Dic3—21no—
86C2×Dic12C2311yescharacteristic
87C2×D24C2311yescharacteristic
88C2×#204—21no—
89C2×ΓL(1,𝔽25)D411yescharacteristic
90C2×D3×C8C2311yescharacteristic
91#898—41no—
92#897—21no—
93#895—21no—
94#894—41no—
95#893—21no—
96#887—21no—
97#886—41no—
98#885—41no—
99#879—41no—
100#878—41no—
101#875—21no—
102#872D411yescharacteristic
103#869—21no—
104C2×C3×C16C2311yescharacteristic
105C2×C3×C16C2311yescharacteristic
106C3×#107—21no—
107C3×#107C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×C4×C8C2311yescharacteristic
110#835—21no—
111#835—21no—
112#832—21no—
113#832—21no—
114#829—22no—
115#828—22no—
116#827—22no—
117#826—22no—
118#818—21no—
119#818C2311yescharacteristic
120#818C2311yescharacteristic
121#815D411yescharacteristic
122C8×Dic3C2311yescharacteristic
123C2×C3⋊C16C2311yescharacteristic
124C2×C3⋊C16C2311yescharacteristic
125#810—21no—
126#810—21no—
127#809—21no—
128#809—21no—
129#803D411yescharacteristic
130#798—22no—
131D3×C16—22no—
132#542—61no—
133#542—61no—
134C2×Q32—31no—
135C2×SD32—31no—
136C2×SD32—31no—
137C2×D16—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500—31no—
141#500C2×D311yescharacteristic
142#495—31no—
143#495C2×D311yescharacteristic
144#489—61no—
145#489—61no—
146#468—61no—
147#468—61no—
148#467—31no—
149#467—31no—
150#466—31no—
151#466—31no—
152#440—61no—
153#440—61no—
154C4×Q16—31no—
155C4×Q16—31no—
156C4×D8—31no—
157C4×D8—31no—
158#405—31no—
159#367—31no—
160#367—31no—
161#367—31no—
162#367—31no—
163#358—31no—
164#358—31no—
165#358C2×D311yescharacteristic
166#358C2×D311yescharacteristic
167#357—31no—
168#357—31no—
169#357C2×D311yescharacteristic
170#357C2×D311yescharacteristic
171#346—31no—
172#346C2×D311yescharacteristic
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×Q8—41no—
177C2×C3×D4—21no—
178C2×#67—41no—
179C22×Dic3—41no—
180C2×D12—21no—
181C2×D12—41no—
182C2×C4×D3—41no—
183C2×C4×D3C2×D411yescharacteristic
184C2×Dic6—21no—
185C2×Dic6—41no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×SD16—42no—
189C3×D8—21no—
190C3×D8—21no—
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2411yescharacteristic
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—41no—
196C3×C22⋊C4—41no—
197C3×C22⋊C4—41no—
198C3×C42C2×D411yescharacteristic
199#217—41no—
200#217—41no—
201(C2×D3)⋊C4—41no—
202(C2×D3)⋊C4—41no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4—41no—
207(C2×D3)⋊C4D811yescharacteristic
208(C2×D3)⋊C4D811yescharacteristic
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—41no—
212C4⋊Dic3—41no—
213C4⋊Dic3—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4D811yescharacteristic
217Dic3⋊C4D811yescharacteristic
218C4×Dic3—41no—
219C4×Dic3—41no—
220C4×Dic3C2×D411yescharacteristic
221C2×C3⋊C8C2×D411yescharacteristic
222C2×C3⋊C8C2×D411yescharacteristic
223Dic12—21no—
224Dic12—21no—
225D24—21no—
226D24—21no—
227#204—42no—
228ΓL(1,𝔽25)—22no—
229D3×C8—21no—
230D3×C8—21no—
231C3×C16—21no—
232C3×C16—21no—
233C3⋊C16#4112yes—
234C3⋊C16#4112yes—
235C2×Q16—31no—
236C2×Q16C22×D311yescharacteristic
237C2×SD16—21no—
238C2×SD16—61no—
239C2×D8—31no—
240C2×D8C22×D311yescharacteristic
241C2×M16—31no—
242C22×C8—31no—
243C4⋊Q8—21no—
244C4⋊Q8—61no—
245#126—121no—
246#126—121no—
247#124—21no—
248#124—61no—
249#124—121no—
250#124—121no—
251#123—121no—
252#123—121no—
253C22⋊Q8—61no—
254C22⋊Q8—61no—
255C22⋊Q8—121no—
256C22⋊Q8—121no—
257#121—61no—
258#121—61no—
259C4×Q8—61no—
260C4×Q8—61no—
261C4×D4—61no—
262C4×D4—61no—
263#117—31no—
264Q32—62no—
265SD32—62no—
266SD32—62no—
267D16—62no—
268M32—62no—
269C2×C16—31no—
270C2×C16—31no—
271C2×C16—62no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—21no—
275#107—31no—
276#107—31no—
277#107—61no—
278#107C22×D311yescharacteristic
279#107C22×D311yescharacteristic
280#103—61no—
281#103—61no—
282#103—61no—
283#103—61no—
284#102—61no—
285#102—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C22×D3—41no—
294C22×D3C2×D811yescharacteristic
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—161no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4C2×D811yescharacteristic
304C2×C3×C4C2×D811yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306#67—161no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—161no—
311C2×Dic3C2×D811yescharacteristic
312C2×Dic3C2×D811yescharacteristic
313C2×Dic3C2×D811yescharacteristic
314D12—41no—
315D12—41no—
316D12—161no—
317C4×D3—21no—
318C4×D3—21no—
319C4×D3—161no—
320Dic6—41no—
321Dic6—41no—
322Dic6—161no—
323C3×C8#13512yes—
324C3×C8C2×#4111yescharacteristic
325C3×C8C2×#4111yescharacteristic
326C3⋊C8—21no—
327C3⋊C8—21no—
328C2×Q8—21no—
329C2×Q8—41no—
330C2×Q8—61no—
331C2×Q8—121no—
332C2×D4—21no—
333C2×D4—61no—
334C2×D4—121no—
335C2×D4—121no—
336C22×C4—31no—
337C22×C4—121no—
338C22×C4—121no—
339Q16—21no—
340Q16—21no—
341Q16—61no—
342Q16—61no—
343SD16—42no—
344SD16—122no—
345D8—21no—
346D8—21no—
347D8—61no—
348D8—61no—
349M16—62no—
350C2×C8—31no—
351C2×C8—31no—
352C2×C8—61no—
353C2×C8—61no—
354C2×C8D3×D411yescharacteristic
355C2×C8C23×D311yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—41no—
361C4⋊C4—61no—
362C4⋊C4—61no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C22⋊C4—31no—
369C22⋊C4—31no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C42—31no—
381C42—121no—
382C42—121no—
383C42D3×D411yescharacteristic
384C16—21no—
385C16—21no—
386C16—32no—
387C16—32no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3C22×D811yescharacteristic
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—81no—
401C3×C4—81no—
402C3×C4—81no—
403C3×C4#53211yescharacteristic
404C3×C4C2×#13511yescharacteristic
405Dic3—21no—
406Dic3—21no—
407Dic3—21no—
408Dic3—81no—
409Dic3—81no—
410Dic3—81no—
411C23—31no—
412C23—41no—
413C23—121no—
414Q8—41no—
415Q8—41no—
416Q8—121no—
417Q8—121no—
418Q8—161no—
419Q8—481no—
420D4—41no—
421D4—41no—
422D4—121no—
423D4—121no—
424D4—481no—
425D4—481no—
426C2×C4—31no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—481no—
438C2×C4—481no—
439C2×C4D3×D811yescharacteristic
440C2×C4D3×D811yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C8—61no—
443C8—61no—
444C8#91612yes—
445C8D3×#4111yescharacteristic
446C8D3×#4111yescharacteristic
447C2×C3—161no—
448C2×C3#316911yescharacteristic
449C2×C3#341611yescharacteristic
450C2×C3#341611yescharacteristic
451D3—41no—
452D3—161no—
453C22—61no—
454C22—61no—
455C22—61no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22C2×D3×D811yescharacteristic
463C4—21no—
464C4—21no—
465C4—61no—
466C4—61no—
467C4—61no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—241no—
472C4—241no—
473C4—241no—
474C4#566811yescharacteristic
475C4C2×#91611yescharacteristic
476C3#3374211yescharacteristic, complemented, Sylow
477C2—121no—
478C2—161no—
479C2—481no—
480C2#8155711yescharacteristic
481C2#8203711yescharacteristic
482C2#8204011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2243C2
socleC22×C3C22×D8
prime radical#69197C2
hypercenter#2243D3
hypocenterC3#33742
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2243

Complements

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

NH ≅ G/Ntype
1. #81935C2semidirect
2. #81811C2semidirect
3. C3×#2243C2semidirect
4. #71385C2semidirect
5. #71320C2semidirect
6. #69229C2semidirect
7. #69228C2semidirect
8. #69209C2semidirect
9. #69206C2semidirect
11. #69196C2semidirect
12. #69177C2semidirect
13. #69175C2semidirect
14. #69079C2semidirect
15. #69072C2semidirect
65. #2243D3semidirect
476. C3#33742semidirect

Maximal subgroups

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

subgroupindex
#819352
#818112
C3×#22432
#713852
#713202
#692292
#692282
#692092
#692062
#691972
#691962
#691772
#691752
#690792
#690722
#337423