Finite Group Databasebeta

← #10950926

Subgroups — #10950926

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
1C3×#3050C211yesmaximal, characteristic, complemented
2C2×#5115C211yesmaximal, characteristic, complemented
3#72082C212yesmaximal, complemented
4C2×#4549C211yesmaximal, characteristic, complemented
5#71085C211yesmaximal, characteristic, complemented
6#71082C212yesmaximal, complemented
7#71079C211yesmaximal, characteristic, complemented
8#71075C211yesmaximal, characteristic, complemented
9#71071C212yesmaximal, complemented
10#71069C211yesmaximal, characteristic, complemented
11#70653C212yesmaximal, complemented
12#33754—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic, complemented
14C23×C3×C8C2211yescharacteristic
15C3×#525—21no—
16C2×#930—21no—
17C22×D24C2211yescharacteristic
18C3×#467C2211yescharacteristic
19C3×#467C2211yescharacteristic, complemented
20C3×#466C2211yescharacteristic
21C3×#466C2211yescharacteristic, complemented
22C3×#465—21no—
23C3×#465—21no—
24C3×#427C2211yescharacteristic, complemented
25C3×#416—21no—
26C2×#828C2212yescomplemented
27C2×#826C2212yescomplemented
28#5115—21no—
29#5115—21no—
30#5115C2212yes—
31#5115C2212yes—
32C2×#821—21no—
33C2×#818C2211yescharacteristic
34C22×C3⋊C16C2212yescomplemented
35#4564C2212yescomplemented
36#4562C2212yescomplemented
37#4549C2211yescharacteristic
38#4549C2211yescharacteristic
39#4549C2212yescomplemented
40#4491C2212yescomplemented
41#4491C2212yescomplemented
42#4490C2212yescomplemented
43#4489C2212yescomplemented
44#3050D311yescharacteristic, complemented
45C2×#466—31no—
46#2236—31no—
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×C3×#41—21no—
56C23×C3×C4D411yescharacteristic
57C22×#67—41no—
58C22×D12—21no—
59C3×#135—42no—
60C2×C3×Q16C2311yescharacteristic
61C2×C3×SD16—21no—
62C2×C3×D8C2311yescharacteristic
63C22×C3×C8—42no—
64C22×C3×C8D411yescharacteristic
65C22×C3×C8D411yescharacteristic
66C22×C3×C8D411yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#123—41no—
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×#121—21no—
74C3×#121—21no—
75C3×#120—41no—
76C3×C4×D4—41no—
77C3×C4×D4—41no—
78C3×#117—21no—
79#930—22no—
80#930—22no—
81#930—41no—
82#930—41no—
83C2×(C2×D3)⋊C4—41no—
84C2×C4⋊Dic3—21no—
85C2×D24—21no—
86C2×D24—42no—
87C2×D24C2312yes—
88C3×#107C2311yescharacteristic
89C3×#107C2311yescharacteristic
90C3×#106—21no—
91C3×#103—21no—
92C3×#103—21no—
93C3×#102—21no—
94C3×#102—21no—
95#828—42no—
96#826—42no—
97#821—22no—
98#821—22no—
99#818—21no—
100#818C2312yes—
101C2×C3⋊C16—42no—
102C2×C3⋊C16C2312yes—
103C2×C3⋊C16C2312yes—
104C2×#135C2×D311yescharacteristic, complemented
105C22×D8—31no—
106C23×C8C2×D311yescharacteristic
107#525—21no—
108C2×#121—61no—
109C2×SD32—32no—
110C2×D16—32no—
111C22×C16—32no—
112#467C2×D311yescharacteristic
113#467C2×D311yescharacteristic, complemented
114#466—32no—
115#466—32no—
116#466—61no—
117#466—61no—
118#466C2×D311yescharacteristic
119#466C2×D311yescharacteristic, complemented
120#465—21no—
121#465—21no—
122#427C2×D311yescharacteristic, complemented
123C2×#107—31no—
124#416—21no—
125C2×#102—61no—
126#367—32no—
127#366—32no—
128#358—32no—
129#357—32no—
130#357—32no—
131#357—32no—
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16—31no—
135C24×C3D811yescharacteristic
136C23×D3—41no—
137C3×#41—42no—
138C3×#41—42no—
139C2×C3×Q8—21no—
140C2×C3×D4—21no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4D811yescharacteristic
148C22×C3×C4D811yescharacteristic
149C22×C3×C4D811yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C2×#67—42no—
154C2×#67—42no—
155C2×#67—84no—
156C22×Dic3—41no—
157C2×D12—22no—
158C2×D12—42no—
159C2×D12—42no—
160C2×D12—41no—
161C3×Q16—22no—
162C3×SD16—42no—
163C3×D8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—41no—
168C2×C3×C8—41no—
169C2×C3×C8—41no—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2411yescharacteristic
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—41no—
177C3×C4⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—41no—
182C3×C22⋊C4—41no—
183C3×C42—41no—
184(C2×D3)⋊C4—42no—
185(C2×D3)⋊C4—42no—
186C4⋊Dic3—22no—
187C4⋊Dic3—41no—
188D24—22no—
189D24—22no—
190D24—42no—
191C3⋊C16—22no—
192C3⋊C16—22no—
193C2×#41—21no—
194C22×D4—61no—
195C22×D4—121no—
196C23×C4#6711yescharacteristic
197#135—42no—
198C2×Q16C22×D311yescharacteristic
199C2×SD16—21no—
200C2×D8—32no—
201C2×D8—61no—
202C2×D8—122no—
203C2×D8C22×D311yescharacteristic
204C22×C8—42no—
205C22×C8#6711yescharacteristic
206C22×C8#6711yescharacteristic
207C22×C8#6711yescharacteristic
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211#123—41no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214#121—21no—
215#121—21no—
216#121—62no—
217#121—62no—
218#121—121no—
219#121—121no—
220#120—41no—
221C4×D4—41no—
222C4×D4—41no—
223#117—21no—
224C2×C4⋊C4—61no—
225C2×C22⋊C4—121no—
226SD32—122no—
227D16—122no—
228C2×C16—32no—
229C2×C16—32no—
230C2×C16—122no—
231#107—32no—
232#107—61no—
233#107C22×D311yescharacteristic
234#107C22×D311yescharacteristic
235#106—21no—
236#103—21no—
237#103—21no—
238#102—21no—
239#102—21no—
240#102—62no—
241#102—62no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3C2×D811yescharacteristic
247C23×C3C2×D811yescharacteristic
248C23×C3C2×D811yescharacteristic
249C22×D3—42no—
250C22×D3—82no—
251C22×D3—82no—
252C22×D3—82no—
253C3×Q8—41no—
254C3×Q8—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—82no—
258C3×D4—82no—
259C3×D4—82no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—82no—
273C2×C3×C4C2×D811yescharacteristic
274C2×C3×C4C2×D811yescharacteristic
275C2×C3×C4C2×D811yescharacteristic
276C2×C3×C4C22×D411yescharacteristic
277#67—84no—
278#67—84no—
279C2×Dic3—42no—
280C2×Dic3—82no—
281D12—42no—
282D12—42no—
283D12—82no—
284C3×C8—21no—
285C3×C8—21no—
286C3×C8—21no—
287C3×C8H(5,𝔽2)12yes—
288C24—121no—
289C24#21311yescharacteristic
290#41—42no—
291#41—42no—
292C2×Q8—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—62no—
298C2×D4—122no—
299C2×D4—122no—
300C2×D4—122no—
301C2×D4—122no—
302C2×D4—121no—
303C2×D4—244no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—121no—
308C22×C4#21311yescharacteristic
309C22×C4#21311yescharacteristic
310C22×C4#21311yescharacteristic
311C22×C4C2×#6711yescharacteristic
312C22×C4C2×#6711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314Q16—22no—
315SD16—42no—
316D8—22no—
317D8—62no—
318D8—62no—
319D8—122no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8C2×#6711yescharacteristic
327C2×C8C2×#6711yescharacteristic
328C2×C8C2×#6711yescharacteristic
329C2×C8C23×D311yescharacteristic
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—62no—
335C4⋊C4—121no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—122no—
342C22⋊C4—122no—
343C42—41no—
344C16—62no—
345C16—62no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—44no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3#50812yes—
358C22×C3#50912yes—
359C22×C3C22×D811yescharacteristic
360C2×D3—82no—
361C2×D3—82no—
362C2×D3—82no—
363C2×D3—164no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4—81no—
370C3×C4#53411yescharacteristic
371C3×C4#57511yescharacteristic
372Dic3—82no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—41no—
377C23—122no—
378C23—242no—
379C23—242no—
380C23—242no—
381C23C2×#21311yescharacteristic
382C23C2×#21311yescharacteristic
383C23C2×#21311yescharacteristic
384Q8—41no—
385Q8—41no—
386D4—41no—
387D4—41no—
388D4—82no—
389D4—82no—
390D4—82no—
391D4—122no—
392D4—122no—
393D4—242no—
394D4—244no—
395D4—244no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—82no—
409C2×C4—122no—
410C2×C4—242no—
411C2×C4C2×#21311yescharacteristic
412C2×C4C2×#21311yescharacteristic
413C2×C4C2×#21311yescharacteristic
414C2×C4C22×#6711yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8#100412yes—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—41no—
422C2×C3—161no—
423C2×C3#318611yescharacteristic
424C2×C3C2×#50811yescharacteristic
425C2×C3C2×#50911yescharacteristic
426D3—162no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—44no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—242no—
439C22—242no—
440C22—242no—
441C22—484no—
442C22#519312yes—
443C22#519412yes—
444C22C22×#21311yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—242no—
452C4#580411yescharacteristic
453C4#582211yescharacteristic
454C3#3375411yescharacteristic, complemented, Sylow
455C2—22no—
456C2—22no—
457C2—41no—
458C2—161no—
459C2—482no—
460C2#8262511yescharacteristic
461C2C2×#519411yescharacteristic
462C2C2×#519311yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3050C2
socleC22×C3C22×D8
hypercenter#3050D3
hypocenterC3#33754
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C3×#3050C2semidirect
2. C2×#5115C2semidirect
3. #72082C2semidirect
4. C2×#4549C2semidirect
5. #71085C2semidirect
6. #71082C2semidirect
7. #71079C2semidirect
8. #71075C2semidirect
9. #71071C2semidirect
10. #71069C2semidirect
11. #70653C2semidirect
13. C2×C3×#135C22semidirect
19. C3×#467C22semidirect
21. C3×#466C22semidirect
24. C3×#427C22semidirect
26. C2×#828C22semidirect
27. C2×#826C22semidirect
34. C22×C3⋊C16C22semidirect
35. #4564C22semidirect
36. #4562C22semidirect
39. #4549C22semidirect
40. #4491C22semidirect
41. #4491C22semidirect
42. #4490C22semidirect
43. #4489C22semidirect
44. #3050D3semidirect
104. C2×#135C2×D3semidirect
113. #467C2×D3semidirect
119. #466C2×D3semidirect
122. #427C2×D3semidirect
454. C3#33754semidirect

Maximal subgroups

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

subgroupindex
C3×#30502
C2×#51152
#720822
C2×#45492
#710852
#710822
#710792
#710752
#710712
#710692
#706532
#337543