Finite Group Databasebeta

← #10877359

Subgroups — #10877359

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79260C211yesmaximal, characteristic, complemented
2#78627C211yesmaximal, characteristic
3C3×#2091C211yesmaximal, characteristic, complemented
4#70734C212yesmaximal, complemented
5#70726C211yesmaximal, characteristic, complemented
6#70725C212yesmaximal, complemented
7#69992C212yesmaximal
8#69987C211yesmaximal, characteristic, complemented
9#69983C212yesmaximal, complemented
10#68724C211yesmaximal, characteristic, complemented
11#68718C211yesmaximal, characteristic, complemented
12#27692—31nomaximal, Sylow
13#5791C2211yescharacteristic
14C22×#208C2211yescharacteristic
15#5745C2212yes—
16C2×#907C2211yescharacteristic
17C3×#430C2211yescharacteristic
18C3×#430C2212yes—
19C3×#407C2211yescharacteristic
20C3×#407C2212yes—
21#5207C2212yes—
22#5132C2212yes—
23#5129C2211yescharacteristic
24#5125C2211yescharacteristic
25#5124C2212yes—
26C2×#822C2212yes—
27C2×#822C2212yes—
28#5122C2211yescharacteristic
29#5119C2211yescharacteristic
30#4893—22no—
31D3×#108—22no—
32#4729—22no—
33#4728—22no—
34C3×#338C2211yescharacteristic
35C3⋊#338C2211yescharacteristic
36#4559C2212yes—
37#4558C2212yes—
38#4558C2212yes—
39#4556C2212yes—
40#2930—31no—
41#2916—31no—
42#2103—31no—
43#2103—31no—
44#2091—31no—
45#2091D311yescharacteristic, complemented
46#1953—32no—
47#1891—32no—
48#1858—32no—
49#1813—31no—
50#1811—32no—
51C23×C3×C4C2311yescharacteristic
52C2×#235C2311yescharacteristic
53C2×C3×M16C2311yescharacteristic
54C2×C3×M16C2312yes—
55C2×C3×M16C2312yes—
56C2×C3×M16C2311yescharacteristic
57#953Q812yes—
58#930—21no—
59#929Q812yes—
60C4×#67—22no—
61#926C2311yescharacteristic
62#924—21no—
63C2×#208—21no—
64C2×#208—21no—
65C2×#208—24no—
66C2×#208C2312yes—
67C2×#208C2312yes—
68C22×C3⋊C8C2311yescharacteristic
69C22×C3⋊C8C2311yescharacteristic
70#907—24no—
71#907—42no—
72D3×M16—24no—
73#901—42no—
74C2×ΓL(1,𝔽25)D412yes—
75C2×D3×C8D412yes—
76C3×#108—22no—
77C3×#108—42no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8—42no—
80#830—42no—
81#824—22no—
82#823—22no—
83#822—22no—
84#822—22no—
85#822—24no—
86#822—42no—
87(C2×D3)⋊C8—22no—
88#819—22no—
89#815—21no—
90Dic3⋊C8—22no—
91C8×Dic3—21no—
92#568—32no—
93C2×#131—31no—
94C22×M16—31no—
95#525—31no—
96#436—62no—
97#435—62no—
98#430—31no—
99#430—32no—
100#430—62no—
101#430C2×D311yescharacteristic
102#430C2×D312yes—
103C2×#108—32no—
104C2×#108—32no—
105C2×#108—62no—
106#424—31no—
107#413—32no—
108#408—31no—
109#407—32no—
110#407C2×D311yescharacteristic
111#407C2×D312yes—
112#405—31no—
113#344—32no—
114#344—32no—
115#341—32no—
116#341—32no—
117#338—31no—
118#338C2×D311yescharacteristic
119C24×C3C2×Q811yescharacteristic
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4C2×Q811yescharacteristic
125C22×C3×C4C2411yescharacteristic
126C2×#67—22no—
127C2×#67C2×Q812yes—
128#235—22no—
129#235—22no—
130C2×D12#4111yescharacteristic
131C2×C4×D3C2×D412yes—
132C2×Dic6#4111yescharacteristic
133C3×M16—22no—
134C3×M16—22no—
135C3×M16—22no—
136C3×M16—22no—
137C3×M16—22no—
138C3×M16—22no—
139C2×C3×C8—21no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8C2×D412yes—
143#217—22no—
144#217C2×Q812yes—
145(C2×D3)⋊C4—22no—
146C4⋊Dic3—21no—
147Dic3⋊C4—22no—
148C4×Dic3—21no—
149#208—22no—
150#208—22no—
151#208—22no—
152#208—22no—
153#208—42no—
154#208—42no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8C2×D411yescharacteristic
160C2×C3⋊C8C2×D411yescharacteristic
161C2×C3⋊C8#4114yes—
162ΓL(1,𝔽25)—24no—
163ΓL(1,𝔽25)—44no—
164D3×C8—24no—
165D3×C8—44no—
166C2×#41—31no—
167C23×C4C22×D311yescharacteristic
168#131—64no—
169#131—122no—
170#131—122no—
171C2×M16—32no—
172C2×M16—32no—
173C2×M16—32no—
174C2×M16—64no—
175C2×M16—61no—
176C2×M16—61no—
177C2×M16—64no—
178C2×M16C22×D312yes—
179C2×M16C22×D312yes—
180C2×M16C22×D311yescharacteristic
181C2×M16C22×D311yescharacteristic
182C22×C8—31no—
183C22×C8—32no—
184C22×C8—31no—
185#123—32no—
186C22⋊Q8—61no—
187#121—61no—
188#120—32no—
189C4×D4—62no—
190#117—31no—
191#108—22no—
192#108—42no—
193#108—64no—
194#108—62no—
195#108—62no—
196#108—62no—
197#108—122no—
198C4⋊C8—62no—
199#101—62no—
200#100—62no—
201C22⋊C8—22no—
202C22⋊C8—42no—
203C22⋊C8—62no—
204C22⋊C8—122no—
205#97—61no—
206C4×C8—61no—
207C23×C3—21no—
208C23×C3—22no—
209C23×C3—81no—
210C23×C3C22×Q811yescharacteristic
211C22×D3C22⋊Q812yes—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4C22×D411yescharacteristic
218C2×C3×C4C2×#4112yes—
219C2×C3×C4C2×#4111yescharacteristic
220C2×C3×C4H(5,𝔽2)12yes—
221#67—24no—
222#67—82no—
223C2×Dic3—22no—
224C2×Dic3C22⋊Q812yes—
225D12—22no—
226C4×D3—22no—
227C4×D3—22no—
228Dic6—22no—
229C3×C8—22no—
230C3×C8—42no—
231C3×C8—41no—
232C3×C8—42no—
233C3⋊C8—41no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C3⋊C8#12312yes—
237C3⋊C8#12312yes—
238C24D3×Q811yescharacteristic
239#41—62no—
240#41—62no—
241C2×Q8—31no—
242C2×D4—32no—
243C2×D4—31no—
244C2×D4—62no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—32no—
250C22×C4D3×Q811yescharacteristic
251C22×C4C23×D311yescharacteristic
252M16—22no—
253M16—22no—
254M16—22no—
255M16—22no—
256M16—22no—
257M16—22no—
258M16—64no—
259M16—62no—
260M16—62no—
261M16—62no—
262M16—62no—
263M16—122no—
264M16—122no—
265M16—124no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—31no—
270C2×C8—31no—
271C2×C8—34no—
272C2×C8—64no—
273C2×C8—61no—
274C2×C8—62no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8—124no—
278C2×C8D3×D412yes—
279C4⋊C4—62no—
280C4⋊C4—61no—
281C22⋊C4—32no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C42—61no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—44no—
289C22×C3—41no—
290C22×C3C2×C22⋊Q811yescharacteristic
291C22×C3#55412yes—
292C2×D3—22no—
293C2×D3—42no—
294C3×C4—41no—
295C3×C4C2×#12311yescharacteristic
296C3×C4#53511yescharacteristic
297C3×C4#55312yes—
298Dic3—22no—
299Dic3—42no—
300C23—22no—
301C23—21no—
302C23—32no—
303C23—81no—
304C23C2×D3×Q811yescharacteristic
305Q8—62no—
306D4—64no—
307D4—62no—
308D4—242no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—32no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4C2×D3×D411yescharacteristic
319C2×C4#100412yes—
320C2×C4D3×#4112yes—
321C2×C4D3×#4111yescharacteristic
322C8—22no—
323C8—32no—
324C8—32no—
325C8—42no—
326C8—42no—
327C8—41no—
328C8—121no—
329C8—122no—
330C8—122no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—81no—
334C2×C3#248711yescharacteristic
335D3—42no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—44no—
340C22—41no—
341C22—62no—
342C22—122no—
343C22D3×C22⋊Q811yescharacteristic
344C22#563612yes—
345C4—41no—
346C4—62no—
347C4—122no—
348C4D3×#12311yescharacteristic
349C4#566112yes—
350C4#571511yescharacteristic
351C3#2769211yescharacteristic, complemented, Sylow
352C2—21no—
353C2—22no—
354C2—81no—
355C2—122no—
356C2#7802911yescharacteristic
subgroupG/H
centerC4#5715
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2091C2
socleC2×C3#2487
prime radical#5791C22
hypercenter#2091D3
hypocenterC3#27692
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2091

Complements

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

NH ≅ G/Ntype
1. #79260C2semidirect
3. C3×#2091C2semidirect
4. #70734C2semidirect
5. #70726C2semidirect
6. #70725C2semidirect
8. #69987C2semidirect
9. #69983C2semidirect
10. #68724C2semidirect
11. #68718C2semidirect
45. #2091D3semidirect
351. C3#27692semidirect

Maximal subgroups

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

subgroupindex
#792602
#786272
C3×#20912
#707342
#707262
#707252
#699922
#699872
#699832
#687242
#687182
#276923