Finite Group Databasebeta

← #10950331

Subgroups — #10950331

Order 768.

534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79728C211yesmaximal, characteristic, complemented
2#79721C211yesmaximal, characteristic
3C3×C4×Q32C211yesmaximal, characteristic, complemented
4#71503C211yesmaximal, characteristic
5#71497C211yesmaximal, characteristic, complemented
6#70913C211yesmaximal, characteristic, complemented
7#70909C211yesmaximal, characteristic, complemented
8#69334C211yesmaximal, characteristic, complemented
9#69331C211yesmaximal, characteristic, complemented
10#69234C211yesmaximal, characteristic, complemented
11#69231C211yesmaximal, characteristic, complemented
12#69226C211yesmaximal, characteristic
13#69212C211yesmaximal, characteristic, complemented
14C4×#800C211yesmaximal, characteristic, complemented
15#69053C211yesmaximal, characteristic, complemented
16#33680—31nomaximal, Sylow
17C2×#919C2211yescharacteristic
18#5583—21no—
19#5574—21no—
20C2×C3×Q32C2211yescharacteristic
21C3×C4×Q16C2211yescharacteristic
22C3×C4×Q16C2211yescharacteristic
23#5188C2211yescharacteristic
24#5183C2211yescharacteristic
25Dic3×Q16C2211yescharacteristic
26C2×#829C2211yescharacteristic
27C2×#828C2211yescharacteristic
28#5030—21no—
29#5026—21no—
30C2×#800C2211yescharacteristic
31#4887C2211yescharacteristic
32D3×#107C2211yescharacteristic
33#4878C2211yescharacteristic
34#4872C2211yescharacteristic
35#4817—21no—
36#4816—21no—
37#4803—21no—
38#4797—21no—
39C4×Dic12C2211yescharacteristic
40C4×D24C2211yescharacteristic
41#4689C2211yescharacteristic
42C3×#366C2211yescharacteristic
43C3×#358C2211yescharacteristic
44C3×#358C2211yescharacteristic
45C3×C4×C16C2211yescharacteristic
46#4564C2211yescharacteristic
47#4564C2211yescharacteristic
48#4509C2211yescharacteristic
49(C2×D3)⋊C16C2211yescharacteristic
50(C2×D3)⋊C16C2211yescharacteristic
51#4505C2211yescharacteristic
52#4503C2211yescharacteristic
53Dic3⋊C16C2211yescharacteristic
54Dic3⋊C16C2211yescharacteristic
55#4492C2211yescharacteristic
56#4491C2211yescharacteristic
57#4490C2211yescharacteristic
58#4489C2211yescharacteristic
59#4461C2211yescharacteristic
60#3177—31no—
61#2951—31no—
62#2236—31no—
63#2235—31no—
64#2229—31no—
65#2228—31no—
66#2209—31no—
67#2208—31no—
68#2189—31no—
69#2189—31no—
70#2176—31no—
71#2175—31no—
72C4×Q32D311yescharacteristic, complemented
73C4×SD32—31no—
74#2164—31no—
75C2×#239—21no—
76C2×C3×Q16C2311yescharacteristic
77C2×C3×Q16C2311yescharacteristic
78C3×C4×Q8—21no—
79C3×C4×Q8—21no—
80#946—21no—
81Q8×Dic3—21no—
82#944—21no—
83C2×#216—21no—
84C2×#215—21no—
85#919—22no—
86#919—22no—
87C2×Dic12C2311yescharacteristic
88C2×D24C2311yescharacteristic
89C2×ΓL(1,𝔽25)D411yescharacteristic
90C2×D3×C8C2311yescharacteristic
91#898—41no—
92#897—21no—
93#893—21no—
94#893—41no—
95D3×C4⋊C4—21no—
96D3×C4⋊C4—41no—
97#890—21no—
98#889—41no—
99#876—41no—
100C4×D12—21no—
101C4×D12—41no—
102#872—41no—
103#872D411yescharacteristic
104#870—41no—
105C4×Dic6—21no—
106C3×Q32—22no—
107C2×C3×C16C2311yescharacteristic
108C2×C3×C16C2311yescharacteristic
109C3×#107C2311yescharacteristic
110C3×#107C2311yescharacteristic
111C3×#103—21no—
112C3×#103—21no—
113C3×C4×C8C2311yescharacteristic
114#835—21no—
115#835—21no—
116#829—22no—
117#828—22no—
118#821—21no—
119#818C2311yescharacteristic
120#818C2311yescharacteristic
121#816—21no—
122#815D411yescharacteristic
123C8×Dic3C2311yescharacteristic
124C2×C3⋊C16C2311yescharacteristic
125C2×C3⋊C16C2311yescharacteristic
126#810—21no—
127#809—21no—
128#808—21no—
129#807—21no—
130#803D411yescharacteristic
131#800—22no—
132C2×#135—31no—
133#541—61no—
134#520—61no—
135C2×Q32—31no—
136C2×Q32C2×D311yescharacteristic
137C2×SD32—31no—
138C2×SD32—31no—
139#499—31no—
140#494—31no—
141#491—61no—
142#470—61no—
143#467—31no—
144#467—31no—
145#441—61no—
146#440—61no—
147#440—61no—
148C4×Q16—31no—
149C4×Q16—31no—
150C4×Q16C2×D311yescharacteristic
151C4×Q16C2×D311yescharacteristic
152C4×D8—31no—
153C4×D8—31no—
154C2×#107—31no—
155#417—61no—
156#405—31no—
157#367—31no—
158#367—31no—
159#366—31no—
160#366C2×D311yescharacteristic
161C4⋊C16—31no—
162C4⋊C16—31no—
163#358—31no—
164#358—31no—
165#358—31no—
166#358—31no—
167#358C2×D311yescharacteristic
168#358C2×D311yescharacteristic
169#357—31no—
170#357—31no—
171C22⋊C16—31no—
172C22⋊C16—31no—
173#346—31no—
174C4×C16C2×D311yescharacteristic
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177#239—24no—
178#239—42no—
179C2×D12—21no—
180C2×D12—41no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×C4×D3C2×D411yescharacteristic
186C2×Dic6—21no—
187C3×Q16—21no—
188C3×Q16—21no—
189C3×Q16—21no—
190C3×Q16#4112yes—
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2411yescharacteristic
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—41no—
196C3×C4⋊C4—41no—
197C3×C42—41no—
198C3×C42—41no—
199C3×C42C2×D411yescharacteristic
200#216—42no—
201#215—24no—
202(C2×D3)⋊C4—41no—
203(C2×D3)⋊C4—41no—
204(C2×D3)⋊C4—41no—
205(C2×D3)⋊C4—41no—
206(C2×D3)⋊C4D811yescharacteristic
207(C2×D3)⋊C4D811yescharacteristic
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—41no—
211C4⋊Dic3—41no—
212C4⋊Dic3—41no—
213Dic3⋊C4—41no—
214Dic3⋊C4—41no—
215Dic3⋊C4—41no—
216Dic3⋊C4—41no—
217Dic3⋊C4—41no—
218Dic3⋊C4—41no—
219Dic3⋊C4D811yescharacteristic
220Dic3⋊C4D811yescharacteristic
221C4×Dic3—41no—
222C4×Dic3—41no—
223C4×Dic3C2×D411yescharacteristic
224C2×C3⋊C8C2×D411yescharacteristic
225C2×C3⋊C8C2×D411yescharacteristic
226Dic12—21no—
227Dic12—21no—
228D24—21no—
229D24#4112yes—
230ΓL(1,𝔽25)—41no—
231D3×C8—21no—
232D3×C8—21no—
233C3×C16—21no—
234C3×C16—21no—
235C3⋊C16—21no—
236C3⋊C16—21no—
237C2×#41—61no—
238#135—62no—
239#135—62no—
240C2×Q16—31no—
241C2×Q16—61no—
242C2×Q16C22×D311yescharacteristic
243C2×Q16C22×D311yescharacteristic
244C2×SD16—61no—
245C2×D8—31no—
246C2×M16—31no—
247C22×C8—31no—
248C4⋊Q8—61no—
249#126—121no—
250#126—121no—
251#125—61no—
252#125—121no—
253#125—121no—
254C22⋊Q8—61no—
255C22⋊Q8—61no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—61no—
259C4×Q8—61no—
260C4×D4—61no—
261C4×D4—61no—
262C4×D4—121no—
263C4×D4—121no—
264#117—31no—
265#117—121no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—121no—
268Q32—22no—
269Q32—62no—
270SD32—62no—
271SD32—62no—
272C2×C16—31no—
273C2×C16—31no—
274C2×C16C22×D311yescharacteristic
275C2×C16C22×D311yescharacteristic
276#107—31no—
277#107—31no—
278#107—61no—
279#107C22×D311yescharacteristic
280#107C22×D311yescharacteristic
281#106—61no—
282#103—21no—
283#103—21no—
284#103—61no—
285#103—61no—
286#103—61no—
287#103—61no—
288#102—61no—
289#102—61no—
290#97—31no—
291#97—31no—
292C4×C8—31no—
293C4×C8C22×D311yescharacteristic
294C22×D3—41no—
295C22×D3C2×D811yescharacteristic
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×Q8—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4C2×D811yescharacteristic
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3C2×D811yescharacteristic
311C2×Dic3C2×D811yescharacteristic
312C2×Dic3C2×D811yescharacteristic
313D12—22no—
314D12—44no—
315D12—41no—
316C4×D3—21no—
317C4×D3—21no—
318C4×D3—44no—
319C4×D3—161no—
320C4×D3—161no—
321C4×D3—161no—
322Dic6—41no—
323Dic6—41no—
324C3×C8—21no—
325C3×C8C2×#4111yescharacteristic
326C3×C8C2×#4111yescharacteristic
327C3⋊C8—21no—
328C3⋊C8—21no—
329#41—64no—
330#41—122no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—61no—
334C2×D4—61no—
335C2×D4—121no—
336C22×C4—31no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4—121no—
340C22×C4—121no—
341Q16—21no—
342Q16—21no—
343Q16—21no—
344Q16—61no—
345Q16—61no—
346Q16—122no—
347Q16#23512yes—
348SD16—64no—
349D8—32no—
350D8—61no—
351M16—121no—
352C2×C8—31no—
353C2×C8—31no—
354C2×C8—61no—
355C2×C8—61no—
356C2×C8D3×D411yescharacteristic
357C2×C8C23×D311yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—31no—
361C4⋊C4—31no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—121no—
367C4⋊C4—121no—
368C4⋊C4—121no—
369C4⋊C4—121no—
370C4⋊C4—121no—
371C4⋊C4—121no—
372C4⋊C4—121no—
373C4⋊C4—121no—
374C4⋊C4—121no—
375C22⋊C4—31no—
376C22⋊C4—31no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C42—31no—
382C42—41no—
383C42—41no—
384C42—121no—
385C42—121no—
386C42D3×D411yescharacteristic
387C16—21no—
388C16—21no—
389C16—61no—
390C16—61no—
391C22×C3C22×D811yescharacteristic
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—42no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—42no—
401C3×C4—81no—
402C3×C4—81no—
403C3×C4—81no—
404C3×C4#52511yescharacteristic
405C3×C4#57511yescharacteristic
406Dic3—21no—
407Dic3—21no—
408Dic3—21no—
409Dic3—81no—
410Dic3—81no—
411Dic3—81no—
412C23—31no—
413C23—121no—
414Q8—22no—
415Q8—41no—
416Q8—41no—
417Q8—41no—
418Q8—121no—
419Q8—121no—
420D4—62no—
421D4—124no—
422D4—121no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—124no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—481no—
437C2×C4—481no—
438C2×C4—481no—
439C2×C4D3×D811yescharacteristic
440C2×C4D3×D811yescharacteristic
441C2×C4C2×D3×D411yescharacteristic
442C8—21no—
443C8—61no—
444C8—61no—
445C8C2×#23511yescharacteristic
446C8D3×#4111yescharacteristic
447C2×C3#310011yescharacteristic
448C2×C3#341611yescharacteristic
449C2×C3#341711yescharacteristic
450D3—41no—
451D3—82no—
452C22—61no—
453C22—61no—
454C22—61no—
455C22—122no—
456C22—241no—
457C22—241no—
458C22C2×D3×D811yescharacteristic
459C4—21no—
460C4—21no—
461C4—42no—
462C4—61no—
463C4—61no—
464C4—61no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—241no—
469C4—241no—
470C4—241no—
471C4#554711yescharacteristic
472C4#576811yescharacteristic
473C3#3368011yescharacteristic, complemented, Sylow
474C2—121no—
475C2—242no—
476C2#7959011yescharacteristic
477C2#8202811yescharacteristic
478C2#8203211yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×C4×Q32C2
socleC22×C3C22×D8
prime radicalC2×#919C22
hypercenterC4×Q32D3
hypocenterC3#33680
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×Q32

Complements

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

NH ≅ G/Ntype
1. #79728C2semidirect
3. C3×C4×Q32C2semidirect
5. #71497C2semidirect
6. #70913C2semidirect
7. #70909C2semidirect
8. #69334C2semidirect
9. #69331C2semidirect
10. #69234C2semidirect
11. #69231C2semidirect
13. #69212C2semidirect
14. C4×#800C2semidirect
15. #69053C2semidirect
72. C4×Q32D3semidirect
473. C3#33680semidirect

Maximal subgroups

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

subgroupindex
#797282
#797212
C3×C4×Q322
#715032
#714972
#709132
#709092
#693342
#693312
#692342
#692312
#692262
#692122
C4×#8002
#690532
#336803