Finite Group Databasebeta

← #10950853

Subgroups — #10950853

Order 768.

496 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81935C211yesmaximal, characteristic
2#81910C211yesmaximal, characteristic, complemented
3C3×#2255C211yesmaximal, characteristic, complemented
4#71382C211yesmaximal, characteristic, complemented
5#71351C211yesmaximal, characteristic, complemented
6#69349C211yesmaximal, characteristic, complemented
7#69347C211yesmaximal, characteristic
8#69338C211yesmaximal, characteristic, complemented
9#69337C211yesmaximal, characteristic, complemented
10#69329C211yesmaximal, characteristic
11#69328C211yesmaximal, characteristic, complemented
12#69319C211yesmaximal, characteristic, complemented
13#69318C211yesmaximal, characteristic, complemented
14#69084C211yesmaximal, characteristic, complemented
15#69072C211yesmaximal, characteristic, complemented
16#33745—31nomaximal, Sylow
17#5733—21no—
18#5698—21no—
19C3×#500C2211yescharacteristic
20C3×#499C2211yescharacteristic
21#5089—21no—
22#5069—21no—
23C2×Dic24C2211yescharacteristic
24C2×Dic24C2211yescharacteristic
25C2×#800C2211yescharacteristic
26C2×#800C2211yescharacteristic
27C2×#798C2211yescharacteristic
28C2×D3×C16C2211yescharacteristic
29#4889—21no—
30#4887C2211yescharacteristic
31#4887C2211yescharacteristic
32#4887C2211yescharacteristic
33#4886—21no—
34#4883C2211yescharacteristic
35#4873C2211yescharacteristic
36#4873C2211yescharacteristic
37#4873C2211yescharacteristic
38#4872C2211yescharacteristic
39#4867—21no—
40#4853—21no—
41#4699C2211yescharacteristic
42#4696C2211yescharacteristic
43#4689C2211yescharacteristic
44C3×#367C2211yescharacteristic
45C3×#367C2211yescharacteristic
46C3×#366C2211yescharacteristic
47C3×#366C2211yescharacteristic
48C3×#346C2211yescharacteristic
49#4502C2211yescharacteristic
50Dic3×C16C2211yescharacteristic
51#4492C2211yescharacteristic
52#4492C2211yescharacteristic
53#4492C2211yescharacteristic
54#4492C2211yescharacteristic
55#4492C2211yescharacteristic
56#4492C2211yescharacteristic
57#4491C2211yescharacteristic
58#4491C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3177—31no—
61#3175—31no—
62#2255D311yescharacteristic, complemented
63#2251—31no—
64#2242—31no—
65#2241—31no—
66#2220—31no—
67#2219—31no—
68#2217—31no—
69#2216—31no—
70#2176—31no—
71#2175—31no—
72C4×Q32—31no—
73C4×SD32—31no—
74#2108—31no—
75C3×C4⋊Q8—21no—
76C3×#125—21no—
77#947—41no—
78#946—41no—
79C2×Dic12—21no—
80C2×Dic12C2311yescharacteristic
81C2×Dic12C2311yescharacteristic
82C2×Dic12C2311yescharacteristic
83C2×D24C2311yescharacteristic
84C2×#204—21no—
85C2×ΓL(1,𝔽25)D411yescharacteristic
86C2×D3×C8C2311yescharacteristic
87#898—41no—
88#898—41no—
89#898—41no—
90#897—21no—
91#897—21no—
92#897—21no—
93#895—21no—
94#894—41no—
95#893—21no—
96#890—41no—
97#890—41no—
98#887—21no—
99#887—21no—
100#887—21no—
101#875—21no—
102#872D411yescharacteristic
103#869—21no—
104C2×C3×C16C2311yescharacteristic
105C2×C3×C16C2311yescharacteristic
106C3×#107—21no—
107C3×#107C2311yescharacteristic
108C3×#107C2311yescharacteristic
109C3×#107C2311yescharacteristic
110C3×#107C2311yescharacteristic
111C3×#106—21no—
112C3×C4×C8C2311yescharacteristic
113#815D411yescharacteristic
114C8×Dic3C2311yescharacteristic
115C2×C3⋊C16C2311yescharacteristic
116C2×C3⋊C16C2311yescharacteristic
117#810—21no—
118#810—21no—
119#810—21no—
120#810—21no—
121#810—21no—
122#810—21no—
123#809—21no—
124#809—21no—
125#803D411yescharacteristic
126Dic24—22no—
127Dic24—22no—
128#800—22no—
129#800—22no—
130#798—22no—
131D3×C16—22no—
132#542—61no—
133#541—61no—
134C2×Q32—31no—
135C2×Q32—31no—
136C2×SD32—31no—
137C2×SD32—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500C2×D311yescharacteristic
141#499C2×D311yescharacteristic
142#495—31no—
143#494—31no—
144#491—61no—
145#489—61no—
146#470—61no—
147#468—61no—
148#467—31no—
149#467—31no—
150#467—31no—
151#466—31no—
152#441—61no—
153#440—61no—
154C4×Q16—31no—
155C4×Q16—31no—
156C4×Q16—31no—
157C4×D8—31no—
158#405—31no—
159#367C2×D311yescharacteristic
160#367C2×D311yescharacteristic
161#366C2×D311yescharacteristic
162#366C2×D311yescharacteristic
163#358—31no—
164#358—31no—
165#358—31no—
166#358—31no—
167#358—31no—
168#358—31no—
169#357—31no—
170#357—31no—
171#346—31no—
172#346C2×D311yescharacteristic
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—41no—
176C2×D12—21no—
177C2×D12—41no—
178C2×C4×D3—41no—
179C2×C4×D3C2×D411yescharacteristic
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8C2411yescharacteristic
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—41no—
190C3×C4⋊C4—41no—
191C3×C4⋊C4—41no—
192C3×C42C2×D411yescharacteristic
193(C2×D3)⋊C4—41no—
194(C2×D3)⋊C4—41no—
195(C2×D3)⋊C4—41no—
196(C2×D3)⋊C4—41no—
197(C2×D3)⋊C4—41no—
198(C2×D3)⋊C4—41no—
199(C2×D3)⋊C4—41no—
200(C2×D3)⋊C4D811yescharacteristic
201(C2×D3)⋊C4D811yescharacteristic
202C4⋊Dic3—41no—
203C4⋊Dic3—41no—
204C4⋊Dic3—41no—
205C4⋊Dic3—41no—
206C4⋊Dic3—41no—
207Dic3⋊C4—41no—
208Dic3⋊C4—41no—
209Dic3⋊C4—41no—
210Dic3⋊C4D811yescharacteristic
211Dic3⋊C4D811yescharacteristic
212C4×Dic3—41no—
213C4×Dic3—41no—
214C4×Dic3—41no—
215C4×Dic3C2×D411yescharacteristic
216C2×C3⋊C8C2×D411yescharacteristic
217C2×C3⋊C8C2×D411yescharacteristic
218Dic12—21no—
219Dic12—21no—
220Dic12—21no—
221Dic12—21no—
222Dic12—21no—
223Dic12—21no—
224Dic12—42no—
225D24—21no—
226D24—21no—
227#204—42no—
228ΓL(1,𝔽25)—22no—
229D3×C8—21no—
230D3×C8—21no—
231C3×C16#4112yes—
232C3×C16#4112yes—
233C3⋊C16—21no—
234C3⋊C16—21no—
235C2×Q16—31no—
236C2×Q16—31no—
237C2×Q16—31no—
238C2×Q16—61no—
239C2×SD16—61no—
240C2×D8—31no—
241C2×M16—31no—
242C22×C8—31no—
243C4⋊Q8—21no—
244C4⋊Q8—61no—
245#126—121no—
246#126—121no—
247#126—121no—
248#125—21no—
249#125—121no—
250#125—121no—
251#124—61no—
252#124—121no—
253#123—121no—
254C22⋊Q8—61no—
255C22⋊Q8—61no—
256C22⋊Q8—61no—
257C22⋊Q8—121no—
258#121—61no—
259C4×Q8—61no—
260C4×Q8—61no—
261C4×Q8—61no—
262C4×D4—61no—
263#117—31no—
264Q32—62no—
265Q32—62no—
266SD32—62no—
267SD32—62no—
268M32—62no—
269C2×C16—31no—
270C2×C16—31no—
271C2×C16—62no—
272C2×C16C22×D311yescharacteristic
273C2×C16C22×D311yescharacteristic
274#107—21no—
275#107C22×D311yescharacteristic
276#107C22×D311yescharacteristic
277#107C22×D311yescharacteristic
278#107C22×D311yescharacteristic
279#106—21no—
280#103—61no—
281#103—61no—
282#103—61no—
283#103—61no—
284#103—61no—
285#103—61no—
286#102—61no—
287#102—61no—
288#97—31no—
289#97—31no—
290C4×C8—31no—
291C4×C8C22×D311yescharacteristic
292C22×D3—41no—
293C22×D3C2×D811yescharacteristic
294C3×Q8—161no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4C2×D811yescharacteristic
300C2×C3×C4C2×D811yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3C2×D811yescharacteristic
306C2×Dic3C2×D811yescharacteristic
307C2×Dic3C2×D811yescharacteristic
308D12—41no—
309D12—41no—
310D12—161no—
311C4×D3—21no—
312C4×D3—21no—
313C4×D3—161no—
314Dic6—41no—
315Dic6—41no—
316Dic6—41no—
317Dic6—41no—
318Dic6—41no—
319Dic6—41no—
320C3×C8#13512yes—
321C3×C8C2×#4111yescharacteristic
322C3×C8C2×#4111yescharacteristic
323C3⋊C8—21no—
324C3⋊C8—21no—
325C2×Q8—41no—
326C2×Q8—61no—
327C2×Q8—61no—
328C2×Q8—61no—
329C2×D4—61no—
330C2×D4—121no—
331C22×C4—31no—
332C22×C4—121no—
333Q16—61no—
334Q16—61no—
335Q16—61no—
336Q16—61no—
337Q16—61no—
338Q16—61no—
339Q16—122no—
340SD16—122no—
341D8—61no—
342D8—61no—
343M16—62no—
344C2×C8—31no—
345C2×C8—31no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8D3×D411yescharacteristic
349C2×C8C23×D311yescharacteristic
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—31no—
355C4⋊C4—31no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—121no—
360C4⋊C4—121no—
361C4⋊C4—121no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C22⋊C4—31no—
368C22⋊C4—31no—
369C22⋊C4—121no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C42—31no—
377C42—121no—
378C42—121no—
379C42—121no—
380C42D3×D411yescharacteristic
381C16—61no—
382C16—61no—
383C16#23912yes—
384C16#23912yes—
385C22×C3C22×D811yescharacteristic
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—81no—
390C2×D3—81no—
391C2×D3—81no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4#53211yescharacteristic
399C3×C4C2×#13511yescharacteristic
400Dic3—21no—
401Dic3—21no—
402Dic3—21no—
403Dic3—81no—
404Dic3—81no—
405Dic3—81no—
406C23—31no—
407C23—121no—
408Q8—121no—
409Q8—121no—
410Q8—121no—
411Q8—121no—
412Q8—121no—
413Q8—121no—
414Q8—161no—
415D4—121no—
416D4—121no—
417D4—481no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—481no—
431C2×C4D3×D811yescharacteristic
432C2×C4D3×D811yescharacteristic
433C2×C4C2×D3×D411yescharacteristic
434C8—61no—
435C8—61no—
436C8#91912yes—
437C8C2×#23911yescharacteristic
438C8C2×#23911yescharacteristic
439C2×C3#316911yescharacteristic
440C2×C3#341611yescharacteristic
441C2×C3#341711yescharacteristic
442D3—41no—
443D3—161no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—241no—
448C22—241no—
449C22—241no—
450C22C2×D3×D811yescharacteristic
451C4—21no—
452C4—21no—
453C4—61no—
454C4—61no—
455C4—61no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—241no—
461C4—241no—
462C4—241no—
463C4#572511yescharacteristic
464C4C2×#91911yescharacteristic
465C3#3374511yescharacteristic, complemented, Sylow
466C2—121no—
467C2—481no—
468C2#8193311yescharacteristic
469C2#8203711yescharacteristic
470C2#8204111yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2255C2
socleC22×C3C22×D8
prime radical#4883C22
hypercenter#2255D3
hypocenterC3#33745
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
2. #81910C2semidirect
3. C3×#2255C2semidirect
4. #71382C2semidirect
5. #71351C2semidirect
6. #69349C2semidirect
8. #69338C2semidirect
9. #69337C2semidirect
11. #69328C2semidirect
12. #69319C2semidirect
13. #69318C2semidirect
14. #69084C2semidirect
15. #69072C2semidirect
62. #2255D3semidirect
465. C3#33745semidirect

Maximal subgroups

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

subgroupindex
#819352
#819102
C3×#22552
#713822
#713512
#693492
#693472
#693382
#693372
#693292
#693282
#693192
#693182
#690842
#690722
#337453