Finite Group Databasebeta

← #10690701

Subgroups — #10690701

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79288C211yesmaximal, characteristic
2#73387C211yesmaximal, characteristic
3#70783C211yesmaximal, characteristic
4#70767C211yesmaximal, characteristic, complemented
5#70766C211yesmaximal, characteristic, complemented
6C3×#1274C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9103—31nomaximal, Sylow
9C2×#875C411yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#4939C411yescharacteristic
16#4935C411yescharacteristic
17#4929C411yescharacteristic
18#4925C2211yescharacteristic
19#4925C2211yescharacteristic
20#4838—22no—
21#4726—22no—
22#4688—22no—
23#4670—22no—
24#4668—22no—
25#4484—22no—
26#4468—22no—
27#4456—22no—
28#2975—31no—
29#2326—31no—
30#1479—31no—
31#1479—31no—
32#1460—31no—
33#1274—31no—
34#1274D311yescharacteristic, complemented
35C22×D12C2×C411yescharacteristic
36C22×C4×D3D412yes—
37C22×Dic6C2×C411yescharacteristic
38C2×C3×C42C2×C411yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—22no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4D412yes—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×ΓL(1,𝔽25)—24no—
50#875—22no—
51#875—22no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12—21no—
55C4×D12—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66C3×#95—21no—
67C3×#95—21no—
68C3⋊#95—22no—
69#830—21no—
70#830—21no—
71(C2×D3)⋊C8—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C2×#124—31no—
77C2×C4×Q8—31no—
78C2×C4×D4—31no—
79#436—62no—
80#436—62no—
81#431—31no—
82#431—31no—
83#431C2×D311yescharacteristic
84#431C2×D311yescharacteristic
85#407—62no—
86#391—31no—
87#390—31no—
88#388—62no—
89#385—62no—
90#384—31no—
91#376C2×D311yescharacteristic
92Q8⋊C8—62no—
93Q8⋊C8—62no—
94#323—62no—
95C23×D3C22⋊C412yes—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3C22⋊C412yes—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—82no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6M1612yescomplemented
119C2×C3×C8—22no—
120C2×C3×C8—41no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42M1612yescomplemented
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129(C2×D3)⋊C4—28no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—82no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—82no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140C4×Dic3—41no—
141C4×Dic3—41no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144ΓL(1,𝔽25)—84no—
145C22×Q8—31no—
146C22×D4—31no—
147C23×C4—32no—
148C2×M16—64no—
149#124—62no—
150#124—62no—
151C4×Q8—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×D4—61no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4—61no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—62no—
163C2×C22⋊C4—32no—
164C2×C22⋊C4—62no—
165C2×C22⋊C4—62no—
166C2×C42—61no—
167C2×C42C4×D311yescharacteristic
168C2×C42C4×D311yescharacteristic
169C2×C42C22×D311yescharacteristic
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C22⋊C8—21no—
175C22⋊C8—21no—
176C22⋊C8—64no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179C4×C8—21no—
180C4×C8—32no—
181C4×C8—61no—
182C4×C8D1212yes—
183#95—21no—
184#95—21no—
185#95—62no—
186C23×C3C2×C22⋊C411yescharacteristic
187C22×D3—22no—
188C22×D3—22no—
189C22×D3—22no—
190C22×D3—42no—
191C22×D3—42no—
192C2×C3×C4—22no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12311yescharacteristic
206C2×C3×C4C2×M1611yescharacteristic
207C2×C3×C4C2×M1611yescharacteristic
208C2×C3×C4#13111yescharacteristic
209C2×C3×C4#13111yescharacteristic
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217D12—41no—
218D12—41no—
219D12—41no—
220D12—41no—
221C4×D3—44no—
222C4×D3—82no—
223Dic6—22no—
224Dic6—22no—
225Dic6—41no—
226Dic6—41no—
227C3×C8—42no—
228C3×C8—81no—
229C3⋊C8—42no—
230C3⋊C8—81no—
231C24—32no—
232C2×Q8—32no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—32no—
247C22×C4—61no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—242no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255M16—244no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—62no—
259C2×C8—121no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—242no—
267C22⋊C4—68no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—242no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—121no—
275C42—121no—
276C42ΓL(1,𝔽25)12yescomplemented
277C42C2×D1211yescharacteristic
278C42#23511yescharacteristic
279C42D3×D411yescharacteristic
280C42#23911yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3#38611yescharacteristic
284C22×C3#40711yescharacteristic
285C22×C3#40811yescharacteristic
286C2×D3—44no—
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—42no—
290C2×D3—82no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—42no—
296C3×C4—81no—
297C3×C4#43511yescharacteristic
298C3×C4#43511yescharacteristic
299C3×C4#43611yescharacteristic
300C3×C4#43611yescharacteristic
301Dic3—42no—
302Dic3—42no—
303Dic3—81no—
304C23—62no—
305C23—62no—
306C23—62no—
307C23—122no—
308C23—122no—
309C23D3×C22⋊C411yescharacteristic
310Q8—62no—
311Q8—62no—
312Q8—121no—
313Q8—121no—
314D4—121no—
315D4—121no—
316D4—121no—
317D4—121no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—124no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—121no—
334C2×C4—242no—
335C2×C4C4×D1211yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4#89311yescharacteristic
339C2×C4#89411yescharacteristic
340C2×C4#89511yescharacteristic
341C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
342C2×C4#90111yescharacteristic
343C2×C4D3×M1611yescharacteristic
344C2×C4#90711yescharacteristic
345C8—42no—
346C8—81no—
347C8—122no—
348C8—241no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3#187011yescharacteristic
352C2×C3#190911yescharacteristic
353C2×C3#191111yescharacteristic
354D3—82no—
355C22—21no—
356C22—21no—
357C22—124no—
358C22—122no—
359C22—122no—
360C22—122no—
361C22—242no—
362C22#466511yescharacteristic
363C22#472611yescharacteristic
364C22#472711yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—81no—
371C4—122no—
372C4—122no—
373C4—241no—
374C4#468811yescharacteristic
375C4#470611yescharacteristic
376C4#483611yescharacteristic
377C4#483811yescharacteristic
378C3#910311yescharacteristic, complemented, Sylow
379C2—21no—
380C2—21no—
381C2—242no—
382C2#6866911yescharacteristic
383C2#6870511yescharacteristic
384C2#6871211yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1274D3
hypocenterC3#9103
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
4. #70767C2semidirect
5. #70766C2semidirect
6. C3×#1274C2semidirect
7. #67614C2semidirect
34. #1274D3semidirect
118. C2×Dic6M16semidirect
124. C3×C42M16semidirect
276. C42ΓL(1,𝔽25)semidirect
378. C3#9103semidirect

Maximal subgroups

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

subgroupindex
#792882
#733872
#707832
#707672
#707662
C3×#12742
#676142
#91033