Finite Group Databasebeta

← #10950935

Subgroups — #10950935

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#80758C211yesmaximal, characteristic, complemented
2C2×#4883C211yesmaximal, characteristic, complemented
3C3×#2236C211yesmaximal, characteristic, complemented
4#71085C211yesmaximal, characteristic, complemented
5#70600C211yesmaximal, characteristic, complemented
6#69327C212yesmaximal, complemented
7#69323C212yesmaximal, complemented
8#69228C212yesmaximal, complemented
9#69224C212yesmaximal, complemented
10#69108C211yesmaximal, characteristic, complemented
11D3×C22⋊C16C211yesmaximal, characteristic, complemented
12#33754—31nomaximal, Sylow
13C2×#919C2211yescharacteristic, complemented
14C22×D24C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16#5627—21no—
17C2×#895—21no—
18C3×#467C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20#5188C2211yescharacteristic, complemented
21C2×#828C2212yescomplemented
22#5115C2211yescharacteristic
23#5048—21no—
24C2×#809—21no—
25C2×D48C2212yescomplemented
26C2×D3×C16C2212yescomplemented
27#4883—21no—
28#4883—21no—
29#4883C2212yes—
30#4883C2212yes—
31#4883C2211yescharacteristic, complemented
32#4880C2211yescharacteristic, complemented
33#4807—21no—
34#4804—21no—
35C3×#366C2212yescomplemented
36C3×#358C2212yescomplemented
37C3×C22⋊C16C2211yescharacteristic
38#4549C2211yescharacteristic
39#4509C2212yescomplemented
40(C2×D3)⋊C16C2212yescomplemented
41#4491C2212yescomplemented
42#4491C2212yescomplemented
43#4490C2212yescomplemented
44#3050—31no—
45C2×#466—31no—
46#2236D311yescharacteristic, complemented
47#2234—32no—
48#2233—31no—
49#2217—32no—
50#2215—32no—
51#2187—31no—
52#2186—32no—
53#2184—31no—
54C2×C22⋊C16—31no—
55C2×#239—21no—
56C22×D12—21no—
57C22×D12—41no—
58C22×C4×D3D411yescharacteristic
59C2×C3×Q16C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C22⋊Q8—21no—
62C2×C3×C4⋊C4—21no—
63#946—21no—
64C2×#215—21no—
65#930—21no—
66C2×(C2×D3)⋊C4—41no—
67C22×C3⋊C8D411yescharacteristic
68#919—42no—
69C2×D24—21no—
70C2×D24—42no—
71C2×D24C2312yes—
72C2×D24C2311yescharacteristic
73C2×D3×C8—42no—
74C2×D3×C8D411yescharacteristic
75C2×D3×C8D411yescharacteristic
76C2×D3×C8C2311yescharacteristic
77C2×D3×C8C2311yescharacteristic
78#895—22no—
79#895—22no—
80#895—21no—
81#895—41no—
82#895—41no—
83#894—41no—
84#893—41no—
85#892—21no—
86#882—41no—
87#67⋊C4—41no—
88C2×C3×C16C2312yes—
89C3×#107—21no—
90C3×#107C2311yescharacteristic
91C3×#107C2312yes—
92C3×#103—21no—
93#835—21no—
94#828—42no—
95#821—21no—
96#818C2311yescharacteristic
97C2×C3⋊C16C2312yes—
98#809—22no—
99#809—22no—
100#809—21no—
101#808—21no—
102D48—42no—
103D3×C16—42no—
104C2×#135—31no—
105C22×D8—31no—
106C23×C8—31no—
107#525—61no—
108C2×#121—61no—
109C2×SD32—32no—
110C2×D16—32no—
111C22×C16—32no—
112#467—31no—
113#467C2×D311yescharacteristic
114#466—31no—
115#466—32no—
116#466—32no—
117#466—31no—
118#466—61no—
119#466—61no—
120#465—61no—
121#465—61no—
122#427—31no—
123C2×#107C2×D311yescharacteristic
124#416—61no—
125C2×#102—61no—
126#367—32no—
127#366C2×D312yescomplemented
128#358C2×D312yescomplemented
129#357—32no—
130#357—32no—
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic
135C23×D3—41no—
136C23×D3D811yescharacteristic
137C2×C3×Q8—21no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C2×#67—41no—
141C22×Dic3D811yescharacteristic
142#239—42no—
143#239—42no—
144C2×D12—22no—
145C2×D12—21no—
146C2×D12—42no—
147C2×D12—42no—
148C2×D12—41no—
149C2×D12—41no—
150C2×D12—42no—
151C2×D12—42no—
152C2×D12—41no—
153C2×D12—84no—
154C2×C4×D3—41no—
155C2×C4×D3—41no—
156C2×C4×D3—41no—
157C2×C4×D3D811yescharacteristic
158C2×C4×D3D811yescharacteristic
159C2×C4×D3C2×D411yescharacteristic
160C2×C4×D3C2×D411yescharacteristic
161C3×Q16—22no—
162C2×C3×C8—22no—
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8C2411yescharacteristic
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—41no—
169C3×C22⋊C4—41no—
170#215—42no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—41no—
175(C2×D3)⋊C4—42no—
176(C2×D3)⋊C4—42no—
177C4⋊Dic3—21no—
178Dic3⋊C4—41no—
179C4×Dic3—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8C2×D411yescharacteristic
182C2×C3⋊C8C2×D411yescharacteristic
183D24—22no—
184D24—22no—
185D24—22no—
186D24—42no—
187D3×C8—22no—
188D3×C8—22no—
189D3×C8—41no—
190D3×C8—41no—
191C3×C16—22no—
192C3⋊C16—22no—
193C2×#41—61no—
194C22×D4—61no—
195C22×D4—121no—
196C23×C4—31no—
197#135—122no—
198C2×Q16C22×D311yescharacteristic
199C2×SD16—61no—
200C2×D8—32no—
201C2×D8—31no—
202C2×D8—61no—
203C2×D8—122no—
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8—122no—
210C22×C8C22×D311yescharacteristic
211#123—121no—
212C22⋊Q8—21no—
213C22⋊Q8—61no—
214#121—62no—
215#121—62no—
216#121—61no—
217#121—61no—
218#121—121no—
219#121—121no—
220#120—121no—
221C4×D4—121no—
222C4×D4—121no—
223#117—61no—
224C2×C4⋊C4—21no—
225C2×C22⋊C4—121no—
226SD32—122no—
227D16—122no—
228C2×C16—32no—
229C2×C16—122no—
230C2×C16C22×D312yes—
231#107—21no—
232#107—31no—
233#107C22×D311yescharacteristic
234#107C22×D312yes—
235#106—61no—
236#103—21no—
237#103—61no—
238#102—62no—
239#102—62no—
240#102—61no—
241#102—61no—
242C23×C3C2×D811yescharacteristic
243C22×D3—22no—
244C22×D3—22no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—42no—
248C22×D3—82no—
249C22×D3—82no—
250C22×D3—82no—
251C22×D3C2×D811yescharacteristic
252C22×D3C2×D811yescharacteristic
253C3×Q8—41no—
254C3×Q8—41no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—82no—
261C2×C3×C4C2×D811yescharacteristic
262C2×C3×C4C22×D411yescharacteristic
263#67—82no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3C2×D811yescharacteristic
267C2×Dic3C2×D811yescharacteristic
268D12—42no—
269D12—41no—
270D12—42no—
271D12—41no—
272D12—82no—
273D12—84no—
274D12—82no—
275D12—82no—
276D12—84no—
277C4×D3—21no—
278C4×D3—21no—
279C4×D3—21no—
280C4×D3—21no—
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—82no—
284C3×C8—21no—
285C3×C8H(5,𝔽2)12yes—
286C3⋊C8—21no—
287C3⋊C8—21no—
288C24—31no—
289C24—121no—
290#41—122no—
291#41—122no—
292C2×Q8—21no—
293C2×D4—62no—
294C2×D4—61no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—122no—
300C2×D4—122no—
301C2×D4—121no—
302C2×D4—121no—
303C2×D4—244no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—41no—
310C22×C4—121no—
311C22×C4—121no—
312C22×C4—121no—
313C22×C4D3×D411yescharacteristic
314Q16—22no—
315SD16—122no—
316D8—62no—
317D8—62no—
318D8—62no—
319D8—122no—
320C2×C8—22no—
321C2×C8—31no—
322C2×C8—31no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—121no—
326C2×C8—121no—
327C2×C8—121no—
328C2×C8D3×D411yescharacteristic
329C2×C8C23×D311yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—61no—
335C4⋊C4—121no—
336C22⋊C4—41no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C42—121no—
344C16—22no—
345C16—62no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3#50812yes—
349C22×C3C22×D811yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—21no—
355C2×D3—44no—
356C2×D3—81no—
357C2×D3—82no—
358C2×D3—82no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—82no—
362C2×D3—164no—
363C2×D3#50912yes—
364C3×C4—21no—
365C3×C4—82no—
366C3×C4—81no—
367C3×C4—81no—
368C3×C4#53411yescharacteristic
369C3×C4#57511yescharacteristic
370Dic3—21no—
371Dic3—21no—
372Dic3—81no—
373C23—31no—
374C23—31no—
375C23—62no—
376C23—62no—
377C23—121no—
378C23—121no—
379C23—122no—
380C23—242no—
381C23—242no—
382C23—242no—
383C23D3×D811yescharacteristic
384Q8—41no—
385Q8—41no—
386D4—122no—
387D4—121no—
388D4—122no—
389D4—121no—
390D4—242no—
391D4—244no—
392D4—242no—
393D4—244no—
394D4—242no—
395D4—242no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—82no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—242no—
413C2×C4D3×D811yescharacteristic
414C2×C4C2×D3×D411yescharacteristic
415C8—21no—
416C8—61no—
417C8—61no—
418C8#100912yes—
419C2×C3—22no—
420C2×C3#318611yescharacteristic
421C2×C3C2×#50811yescharacteristic
422C2×C3C2×#50911yescharacteristic
423D3—22no—
424D3—41no—
425D3—161no—
426D3—162no—
427C22—21no—
428C22—21no—
429C22—32no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—124no—
436C22—241no—
437C22—242no—
438C22—242no—
439C22—241no—
440C22—241no—
441C22—242no—
442C22—484no—
443C22#491912yes—
444C22C2×D3×D811yescharacteristic
445C4—21no—
446C4—61no—
447C4—61no—
448C4—82no—
449C4—81no—
450C4—81no—
451C4—241no—
452C4#565311yescharacteristic
453C4#577711yescharacteristic
454C3#3375411yescharacteristic, complemented, Sylow
455C2—22no—
456C2—62no—
457C2—121no—
458C2—482no—
459C2—481no—
460C2#8129911yescharacteristic
461C2C2×#491911yescharacteristic
462C2D3×#50911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2236C2
socleC22×C3C22×D8
hypercenter#2236D3
hypocenterC3#33754
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2236

Complements

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

NH ≅ G/Ntype
1. #80758C2semidirect
2. C2×#4883C2semidirect
3. C3×#2236C2semidirect
4. #71085C2semidirect
5. #70600C2semidirect
6. #69327C2semidirect
7. #69323C2semidirect
8. #69228C2semidirect
9. #69224C2semidirect
10. #69108C2semidirect
11. D3×C22⋊C16C2semidirect
13. C2×#919C22semidirect
20. #5188C22semidirect
21. C2×#828C22semidirect
25. C2×D48C22semidirect
26. C2×D3×C16C22semidirect
31. #4883C22semidirect
32. #4880C22semidirect
35. C3×#366C22semidirect
36. C3×#358C22semidirect
39. #4509C22semidirect
40. (C2×D3)⋊C16C22semidirect
41. #4491C22semidirect
42. #4491C22semidirect
43. #4490C22semidirect
46. #2236D3semidirect
127. #366C2×D3semidirect
128. #358C2×D3semidirect
454. C3#33754semidirect

Maximal subgroups

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

subgroupindex
#807582
C2×#48832
C3×#22362
#710852
#706002
#693272
#693232
#692282
#692242
#691082
D3×C22⋊C162
#337543