Finite Group Databasebeta

← #10877379

Subgroups — #10877379

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79260C211yesmaximal, characteristic, complemented
2#79042C211yesmaximal, characteristic
3C3×#2102C211yesmaximal, characteristic, complemented
4#70738C211yesmaximal, characteristic, complemented
5#70526C211yesmaximal, characteristic, complemented
6#69034C212yesmaximal, complemented
7#69032C212yesmaximal, complemented
8#68777C212yesmaximal
9#68748C212yesmaximal, complemented
10#68715C211yesmaximal, characteristic, complemented
11#68710C211yesmaximal, characteristic, complemented
12#27694—31nomaximal, Sylow
13#5745C2212yes—
14C2×#907C2211yescharacteristic
15C2×#907C2211yescharacteristic
16#5477C2211yescharacteristic
17C3×#430C2211yescharacteristic
18C3×#423C2211yescharacteristic
19#5139C2212yes—
20#5129C2211yescharacteristic
21#5129C2211yescharacteristic
22#5125C2211yescharacteristic
23C2×#822C2212yes—
24#5119C2212yes—
25#5006C2212yes—
26#5005C2212yes—
27#4999C2212yes—
28#4998C2212yes—
29C2×#803C2211yescharacteristic
30#4893—22no—
31D3×#108—22no—
32#4836—22no—
33#4836—22no—
34C3×#344C2212yes—
35C3×#339C2212yes—
36C3×H(3,ℤ/4)C2211yescharacteristic
37#4559C2212yes—
38#4557C2211yescharacteristic
39#4523C2212yes—
40#2932—31no—
41#2916—31no—
42#2102—31no—
43#2102D311yescharacteristic, complemented
44#2098—31no—
45#2098—31no—
46#1946—32no—
47#1884—32no—
48#1868—32no—
49#1814—32no—
50#1812—31no—
51C2×#235C2311yescharacteristic
52C2×#235C2311yescharacteristic
53C2×C3×M16C2312yes—
54C2×C3×M16C2311yescharacteristic
55C2×C3×M16C2311yescharacteristic
56C3×#117C2312yes—
57C2×C3×C42C2311yescharacteristic
58#930—22no—
59#926C2312yes—
60C2×#208C2312yes—
61C22×C3⋊C8C2311yescharacteristic
62C22×C3⋊C8C2311yescharacteristic
63#907—22no—
64#907—22no—
65#907—22no—
66#907—22no—
67#907—42no—
68D3×M16—24no—
69#901—42no—
70C2×ΓL(1,𝔽25)—21no—
71C2×ΓL(1,𝔽25)D412yes—
72C2×D3×C8—21no—
73C2×D3×C8D412yes—
74#875Q811yescharacteristic
75#875Q811yescharacteristic
76#874Q811yescharacteristic
77C4×D12—22no—
78#869Q811yescharacteristic
79C3×#108—22no—
80C3×C4⋊C8—22no—
81#825—42no—
82#822—22no—
83#822—22no—
84(C2×D3)⋊C8—22no—
85(C2×D3)⋊C8—22no—
86#819—22no—
87Dic3⋊C8—42no—
88#808—22no—
89#807—22no—
90#806—24no—
91#805—42no—
92C4⋊(C3⋊C8)—42no—
93#803—21no—
94#803—21no—
95#568—32no—
96C2×#131—31no—
97C2×#131—31no—
98#532—31no—
99#435—62no—
100#435—62no—
101#430—31no—
102#430—62no—
103#430C2×D311yescharacteristic
104C2×#108—32no—
105C2×#108—62no—
106#427—32no—
107#424—32no—
108#424—32no—
109#423C2×D311yescharacteristic
110#421—32no—
111#421—32no—
112#410—32no—
113#408—31no—
114#408—31no—
115C2×#97—31no—
116#344—32no—
117#344C2×D312yes—
118#339—32no—
119#339C2×D312yes—
120H(3,ℤ/4)—31no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4C2411yescharacteristic
124C2×#67—21no—
125C2×#67#4112yes—
126#235—22no—
127#235—22no—
128#235—22no—
129#235—22no—
130C2×D12—22no—
131C2×D12C2×Q811yescharacteristic
132C2×D12C2×Q811yescharacteristic
133C2×C4×D3—21no—
134C2×C4×D3C2×D412yes—
135C2×Dic6C2×Q811yescharacteristic
136C2×Dic6C2×Q811yescharacteristic
137C3×M16—22no—
138C3×M16—22no—
139C3×M16—22no—
140C3×M16—22no—
141C2×C3×C8—21no—
142C2×C3×C8—22no—
143C2×C3×C8C2×D412yes—
144C3×C4⋊C4—22no—
145C3×C22⋊C4—22no—
146C3×C42—21no—
147C3×C42—22no—
148C3×C42C2×Q811yescharacteristic
149C3×C42C2×Q811yescharacteristic
150#217—22no—
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4×Dic3—22no—
156#208—22no—
157#208—22no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8C2×D411yescharacteristic
162C2×C3⋊C8C2×D411yescharacteristic
163C2×C3⋊C8#4112yes—
164C2×C3⋊C8#4112yes—
165ΓL(1,𝔽25)—24no—
166ΓL(1,𝔽25)—44no—
167ΓL(1,𝔽25)—41no—
168ΓL(1,𝔽25)—41no—
169D3×C8—22no—
170D3×C8—22no—
171D3×C8—44no—
172D3×C8—42no—
173C2×#41—31no—
174C2×#41—31no—
175#131—62no—
176#131—62no—
177#131—62no—
178#131—62no—
179#131—122no—
180#131—122no—
181C2×M16—32no—
182C2×M16—32no—
183C2×M16—61no—
184C2×M16—64no—
185C2×M16C22×D311yescharacteristic
186C2×M16C22×D312yes—
187C2×M16C22×D311yescharacteristic
188C22×C8—31no—
189C22×C8—32no—
190C22×C8—31no—
191C22×C8—61no—
192C4⋊Q8—31no—
193#127—31no—
194#124—31no—
195#124—31no—
196#121—62no—
197C4×D4—62no—
198#117—32no—
199#117C22×D312yes—
200C2×C42C22×D311yescharacteristic
201#108—22no—
202#108—62no—
203#108—62no—
204#108—62no—
205#107—62no—
206#106—62no—
207C4⋊C8—22no—
208C4⋊C8—122no—
209C4⋊C8—122no—
210#104—122no—
211#104—122no—
212#99—64no—
213C22⋊C8—62no—
214C22⋊C8—62no—
215#97—61no—
216#97—61no—
217C23×C3C2×#4111yescharacteristic
218C22×D3—21no—
219C22×D3C22⋊Q812yes—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—42no—
223C2×C3×C4—81no—
224C2×C3×C4C22×D411yescharacteristic
225C2×C3×C4C22×Q811yescharacteristic
226C2×C3×C4C2×#4112yes—
227C2×C3×C4H(5,𝔽2)12yes—
228#67—24no—
229#67—42no—
230C2×Dic3—21no—
231C2×Dic3—42no—
232C2×Dic3C22⋊Q812yes—
233D12—22no—
234D12—22no—
235D12—82no—
236C4×D3—22no—
237C4×D3—22no—
238C4×D3—41no—
239C4×D3—41no—
240Dic6—22no—
241Dic6—22no—
242C3×C8—22no—
243C3×C8—41no—
244C3×C8—42no—
245C3⋊C8—42no—
246C3⋊C8—41no—
247C3⋊C8#12311yescharacteristic
248C3⋊C8#12311yescharacteristic
249C3⋊C8#12312yes—
250#41—62no—
251#41—62no—
252#41—62no—
253#41—62no—
254C2×Q8—31no—
255C2×Q8—31no—
256C2×D4—32no—
257C2×D4—31no—
258C2×D4—31no—
259C2×D4—62no—
260C2×D4—61no—
261C22×C4—21no—
262C22×C4—32no—
263C22×C4—61no—
264C22×C4C23×D311yescharacteristic
265M16—22no—
266M16—22no—
267M16—22no—
268M16—22no—
269M16—64no—
270M16—62no—
271M16—62no—
272M16—121no—
273M16—121no—
274M16—124no—
275C2×C8—21no—
276C2×C8—22no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—32no—
280C2×C8—32no—
281C2×C8—62no—
282C2×C8—62no—
283C2×C8—61no—
284C2×C8—62no—
285C2×C8—62no—
286C2×C8—122no—
287C2×C8—124no—
288C2×C8D3×D412yes—
289C4⋊C4—22no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C22⋊C4—22no—
293C22⋊C4—62no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C42—21no—
297C42—22no—
298C42—62no—
299C42D3×Q811yescharacteristic
300C42D3×Q811yescharacteristic
301C22×C3—41no—
302C22×C3C2×C22⋊Q811yescharacteristic
303C22×C3#55312yes—
304C2×D3—22no—
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—41no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—82no—
311C3×C4C2×#12311yescharacteristic
312C3×C4#53511yescharacteristic
313C3×C4#55412yes—
314Dic3—22no—
315Dic3—41no—
316Dic3—82no—
317C23—32no—
318C23—61no—
319C23D3×#4111yescharacteristic
320Q8—62no—
321Q8—62no—
322D4—64no—
323D4—62no—
324D4—62no—
325D4—122no—
326D4—242no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—32no—
330C2×C4—42no—
331C2×C4—62no—
332C2×C4—62no—
333C2×C4—61no—
334C2×C4—81no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4C2×D3×D411yescharacteristic
339C2×C4C2×D3×Q811yescharacteristic
340C2×C4D3×#4112yes—
341C2×C4#100912yes—
342C8—22no—
343C8—31no—
344C8—31no—
345C8—32no—
346C8—41no—
347C8—42no—
348C8—121no—
349C8—122no—
350C2×C3—22no—
351C2×C3—21no—
352C2×C3#248711yescharacteristic
353D3—42no—
354D3—81no—
355C22—41no—
356C22—62no—
357C22—122no—
358C22—122no—
359C22—121no—
360C22D3×C22⋊Q811yescharacteristic
361C22#570912yes—
362C4—41no—
363C4—41no—
364C4—62no—
365C4—82no—
366C4—121no—
367C4—242no—
368C4D3×#12311yescharacteristic
369C4#567411yescharacteristic
370C4#569012yes—
371C3#2769411yescharacteristic, complemented, Sylow
372C2—21no—
373C2—22no—
374C2—122no—
375C2—241no—
376C2#7814611yescharacteristic
subgroupG/H
centerC4#5674
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2102C2
socleC2×C3#2487
prime radical#5477C22
hypercenter#2102D3
hypocenterC3#27694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2102

Complements

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

NH ≅ G/Ntype
1. #79260C2semidirect
3. C3×#2102C2semidirect
4. #70738C2semidirect
5. #70526C2semidirect
6. #69034C2semidirect
7. #69032C2semidirect
9. #68748C2semidirect
10. #68715C2semidirect
11. #68710C2semidirect
43. #2102D3semidirect
371. C3#27694semidirect

Maximal subgroups

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

subgroupindex
#792602
#790422
C3×#21022
#707382
#705262
#690342
#690322
#687772
#687482
#687152
#687102
#276943