Finite Group Databasebeta

← #10777935

Subgroups — #10777935

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79301C212yesmaximal, complemented
2#78470C211yesmaximal, characteristic, complemented
3#78447C211yesmaximal, characteristic, complemented
4C3×#1564C211yesmaximal, characteristic, complemented
5#70773C212yesmaximal, complemented
6#68594C212yesmaximal, complemented
7#68584C212yesmaximal, complemented
8#68553C211yesmaximal, characteristic, complemented
9#68550C211yesmaximal, characteristic, complemented
10#68543C211yesmaximal, characteristic, complemented
11#68540C211yesmaximal, characteristic, complemented
12#19160—31nomaximal, Sylow
13#5477C2212yes—
14C4×#235C2211yescharacteristic
15C3×#431C2212yes—
16C2×C3×C4×C8C2211yescharacteristic
17#5115—21no—
18#5115—21no—
19#5111—21no—
20#5111—21no—
21#4925C2211yescharacteristic
22#4850C2212yes—
23#4848C2212yes—
24#4843C2212yes—
25#4842C2212yes—
26#4737—22no—
27#4734—22no—
28#4695C2211yescharacteristic
29#4694C2212yes—
30C4×D24—22no—
31C4×D24—21no—
32C4×D24—21no—
33C4×#204—22no—
34#4688C2211yescharacteristic
35#4688C2211yescharacteristic
36C4×ΓL(1,𝔽25)—21no—
37#4685—21no—
38#4682C2211yescharacteristic
39#4679C2211yescharacteristic
40#4679C2211yescharacteristic
41C3×#335C2212yes—
42C3×#334C2211yescharacteristic
43C3×#334C2211yescharacteristic
44#4485C2212yes—
45#4485C2212yes—
46#4481C2212yes—
47#4481C2212yes—
48#3169—31no—
49#3106—32no—
50#2965—31no—
51#1698—32no—
52#1697—32no—
53#1696—31no—
54#1695—31no—
55#1688—31no—
56#1687—31no—
57#1564D311yescharacteristic, complemented
58#1543—32no—
59C2×#235—22no—
60C2×#235D411yescharacteristic
61C22×C3×C8D411yescharacteristic
62C22×C3×C8D411yescharacteristic
63C2×C3×C42C2311yescharacteristic
64#930—22no—
65#930—22no—
66C4×#67—21no—
67C4×#67—21no—
68#926D411yescharacteristic
69C2×D24—22no—
70C2×D24—21no—
71C2×D24—21no—
72C2×#204—22no—
73C2×ΓL(1,𝔽25)—21no—
74C2×ΓL(1,𝔽25)—21no—
75#875—22no—
76#874C2312yes—
77C4×D12—22no—
78C4×D12—22no—
79C4×D12D411yescharacteristic
80C4×D12C2311yescharacteristic
81#872—21no—
82C42×D3—21no—
83#869C2312yes—
84C4×Dic6D411yescharacteristic
85C4×Dic6C2311yescharacteristic
86C3×C4⋊C8C2312yes—
87C3×C4⋊C8C2312yes—
88C3×C22⋊C8—22no—
89C3×C4×C8—22no—
90C3×C4×C8D411yescharacteristic
91C3×C4×C8D411yescharacteristic
92C3×C4×C8C2311yescharacteristic
93C3×C4×C8C2311yescharacteristic
94#830—21no—
95#821—22no—
96#821—22no—
97#821—22no—
98#821—22no—
99#821—22no—
100(C2×D3)⋊C8—21no—
101(C2×D3)⋊C8—21no—
102#818—21no—
103#818—21no—
104#818—22no—
105#817—22no—
106#816—22no—
107#815—21no—
108#815—21no—
109Dic3⋊C8—21no—
110Dic3⋊C8—21no—
111C4⋊(C3⋊C8)C2311yescharacteristic
112C4⋊(C3⋊C8)C2311yescharacteristic
113C4×C3⋊C8—21no—
114#532—32no—
115C4×#41—31no—
116#500—31no—
117#493—31no—
118#486—32no—
119#482—62no—
120#477—32no—
121#476—32no—
122#466—61no—
123#466—61no—
124#460—32no—
125#459—32no—
126#449—62no—
127#446—31no—
128#446—31no—
129C4×SD16—62no—
130C4×D8—62no—
131C4×D8—61no—
132C4×D8—61no—
133#436—31no—
134#436—31no—
135#435—61no—
136#435—61no—
137#431—31no—
138#431—61no—
139#431C2×D312yes—
140C4×M16—61no—
141C2×C4×C8C2×D311yescharacteristic
142#335C2×D312yes—
143#334C2×D311yescharacteristic
144#334C2×D311yescharacteristic
145#333—32no—
146#333—32no—
147#331—32no—
148#331—32no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C2×#67—21no—
153C2×#67—42no—
154#235—42no—
155#235—84no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12C2×D411yescharacteristic
160C2×C4×D3—21no—
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—42no—
164C2×Dic6—22no—
165C2×Dic6C2×D411yescharacteristic
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—41no—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2×D411yescharacteristic
174C3×C42C2×D411yescharacteristic
175C3×C42C2×D411yescharacteristic
176C3×C42#4111yescharacteristic
177C3×C42C2411yescharacteristic
178#217—21no—
179(C2×D3)⋊C4—21no—
180(C2×D3)⋊C4—21no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—42no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3C2×D411yescharacteristic
186C4⋊Dic3C2×D411yescharacteristic
187Dic3⋊C4—21no—
188Dic3⋊C4—21no—
189C4×Dic3—21no—
190C4×Dic3—21no—
191C4×Dic3—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194D24—42no—
195D24—44no—
196D24—42no—
197#204—44no—
198ΓL(1,𝔽25)—42no—
199ΓL(1,𝔽25)—42no—
200C2×#41—31no—
201C2×#41—62no—
202C2×SD16—62no—
203C2×D8—62no—
204C2×D8—61no—
205C2×D8—61no—
206C2×M16—61no—
207C2×M16—61no—
208C22×C8D1211yescharacteristic
209C22×C8D1211yescharacteristic
210C4⋊Q8—32no—
211#127—32no—
212#124—62no—
213#121—62no—
214#121—62no—
215C4×Q8—31no—
216C4×Q8—31no—
217C4×D4—31no—
218C4×D4—31no—
219C4×D4—61no—
220C4×D4—61no—
221C4×D4—62no—
222C4×D4—62no—
223#117—31no—
224#117—61no—
225C2×C42—61no—
226C2×C42C22×D311yescharacteristic
227#107—62no—
228#107—61no—
229#107—61no—
230#106—62no—
231C4⋊C8—31no—
232C4⋊C8—31no—
233C4⋊C8—61no—
234C4⋊C8—61no—
235C4⋊C8C22×D312yes—
236C4⋊C8C22×D312yes—
237#103—62no—
238#102—62no—
239#102—62no—
240#102—62no—
241#102—62no—
242#102—62no—
243C22⋊C8—22no—
244C22⋊C8—61no—
245C22⋊C8—61no—
246C22⋊C8—61no—
247#97—61no—
248#97—61no—
249C4×C8—22no—
250C4×C8—61no—
251C4×C8D1211yescharacteristic
252C4×C8D1211yescharacteristic
253C4×C8C22×D311yescharacteristic
254C4×C8C22×D311yescharacteristic
255C23×C3#12111yescharacteristic
256C22×D3—21no—
257C22×D3—42no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C22×D411yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268#67—42no—
269#67—84no—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—42no—
274D12—22no—
275D12—42no—
276D12—42no—
277D12—42no—
278D12—42no—
279D12—82no—
280C4×D3—42no—
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—42no—
284C4×D3—84no—
285Dic6—22no—
286Dic6—82no—
287C3×C8—21no—
288C3×C8—21no—
289C3×C8—21no—
290C3×C8—42no—
291C3×C8—42no—
292C3×C8#13512yes—
293C3⋊C8—41no—
294C3⋊C8—41no—
295#41—122no—
296#41—244no—
297C2×Q8—31no—
298C2×Q8—62no—
299C2×D4—31no—
300C2×D4—61no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—122no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—122no—
309C22×C4C2×D1211yescharacteristic
310C22×C4D3×D411yescharacteristic
311C22×C4#23911yescharacteristic
312SD16—124no—
313D8—122no—
314D8—124no—
315D8—122no—
316M16—122no—
317M16—122no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8C2×D1211yescharacteristic
325C2×C8C2×D1211yescharacteristic
326C2×C8C2×D1211yescharacteristic
327C2×C8C2×D1211yescharacteristic
328C4⋊C4—31no—
329C4⋊C4—31no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C42—61no—
340C42—61no—
341C42—61no—
342C42C2×D1211yescharacteristic
343C42D3×D411yescharacteristic
344C42#23911yescharacteristic
345C42C23×D311yescharacteristic
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3C2×#12111yescharacteristic
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4#47212yes—
360C3×C4C2×#13511yescharacteristic
361C3×C4#57511yescharacteristic
362C3×C4#57512yes—
363Dic3—41no—
364Dic3—41no—
365Dic3—41no—
366Dic3—82no—
367C23—61no—
368C23—122no—
369C23#89511yescharacteristic
370Q8—62no—
371Q8—242no—
372D4—62no—
373D4—122no—
374D4—122no—
375D4—122no—
376D4—122no—
377D4—122no—
378D4—242no—
379D4—244no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—122no—
392C2×C4—244no—
393C2×C4#89511yescharacteristic
394C2×C4#89511yescharacteristic
395C2×C4#89511yescharacteristic
396C2×C4C22×D1211yescharacteristic
397C2×C4C2×D3×D411yescharacteristic
398C2×C4C2×#23911yescharacteristic
399C8—21no—
400C8—21no—
401C8—21no—
402C8—42no—
403C8—42no—
404C8—121no—
405C8—121no—
406C8#90412yes—
407C2×C3—41no—
408C2×C3#303511yescharacteristic
409C2×C3#305411yescharacteristic
410C2×C3C2×#47211yescharacteristic
411D3—81no—
412D3—162no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—242no—
420C22—242no—
421C22—242no—
422C22C2×#89511yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—121no—
427C4—121no—
428C4—121no—
429C4—242no—
430C4#489712yes—
431C4C2×#90411yescharacteristic
432C4#574811yescharacteristic
433C4#576812yes—
434C3#1916011yescharacteristic, complemented, Sylow
435C2—41no—
436C2—241no—
437C2—482no—
438C2#7902111yescharacteristic
439C2#7918111yescharacteristic
440C2C2×#489711yescharacteristic
subgroupG/H
centerC2×C4#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1564C2
socleC22×C3C2×#121
hypercenter#1564D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #79301C2semidirect
2. #78470C2semidirect
3. #78447C2semidirect
4. C3×#1564C2semidirect
5. #70773C2semidirect
6. #68594C2semidirect
7. #68584C2semidirect
8. #68553C2semidirect
9. #68550C2semidirect
10. #68543C2semidirect
11. #68540C2semidirect
57. #1564D3semidirect
434. C3#19160semidirect

Maximal subgroups

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

subgroupindex
#793012
#784702
#784472
C3×#15642
#707732
#685942
#685842
#685532
#685502
#685432
#685402
#191603