Finite Group Databasebeta

← #10848602

Subgroups — #10848602

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86262C211yesmaximal, characteristic
2C3×#3139C211yesmaximal, characteristic, complemented
3#83253C211yesmaximal, characteristic, complemented
4#83209C211yesmaximal, characteristic
5#71839C211yesmaximal, characteristic, complemented
6#70346C214yesmaximal
7#70345C214yesmaximal, complemented
8#69877C211yesmaximal, characteristic, complemented
9C4×#825C211yesmaximal, characteristic
10#25589—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×#944C2214yes—
14C2×#942C2212yes—
15#5728—22no—
16#5698—22no—
17#5642—24no—
18#5584—22no—
19#5570—22no—
20#5526C2212yes—
21#5516C2212yes—
22C4×#235C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C3×#491—22no—
25C3×#487—22no—
26C3×#419C2214yes—
27C3×C4×M16C2211yescharacteristic
28#5225C2212yes—
29C2×#825C2212yes—
30C2×#824C2214yes—
31#5090—22no—
32#5083—22no—
33#4975—24no—
34C42×Dic3C2211yescharacteristic
35#4924C2211yescharacteristic
36#4555C2212yes—
37#4532C2214yes—
38#3509—31no—
39#3139D311yescharacteristic, complemented
40#3039—31no—
41#2868—31no—
42#2005—34no—
43#1994—31no—
44#1966—31no—
45#1886—34no—
46C4×#104—31no—
47C22×C3×Q8C2312yes—
48C2×#235C2311yescharacteristic
49C22×Dic6C2312yes—
50C2×C3×M16C2312yes—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8D412yes—
55C3×#125D412yes—
56C3×C22⋊Q8—22no—
57C3×C4×Q8—22no—
58C3×#117C2312yes—
59C2×C3×C4⋊C4—22no—
60C2×C3×C42C2311yescharacteristic
61#946—24no—
62Q8×Dic3—24no—
63#944—28no—
64#944—44no—
65C2×#216—22no—
66#942—44no—
67C2×#215—22no—
68C4×#67—22no—
69#926C2311yescharacteristic
70C2×C4⋊Dic3D414yes—
71C2×Dic3⋊C4—28no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3C2312yes—
74C2×#208C2312yes—
75#898—28no—
76#897—24no—
77#892—24no—
78#890—28no—
79#890—24no—
80#887—24no—
81#886—24no—
82#877—24no—
83#875—22no—
84C4×D12D412yes—
85#872—21no—
86C42×D3—21no—
87#869—22no—
88#869—21no—
89#869—21no—
90#869—22no—
91C4×Dic6—24no—
92C4×Dic6D412yes—
93C3×#103—28no—
94C3×#97—21no—
95C3×C4×C8—21no—
96#835—24no—
97#825—24no—
98#824—44no—
99C4⋊(C3⋊C8)—22no—
100C2×#137—32no—
101#557—62no—
102#556—62no—
103#552—64no—
104#541—62no—
105#536—32no—
106#533—62no—
107#533C2×D311yescharacteristic
108C2×C4⋊Q8—34no—
109C2×C4⋊Q8—31no—
110C2×C4⋊Q8C2×D311yescharacteristic
111#519—32no—
112C4×#41—31no—
113#491—22no—
114#487—22no—
115#462—62no—
116#461—62no—
117#423—31no—
118C2×#104—32no—
119#419—32no—
120#419C2×D314yes—
121C2×#101—34no—
122C4×M16C2×D311yescharacteristic
123#384—64no—
124C43—31no—
125#339—32no—
126#339—34no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—24no—
131C2×C3×Q8—24no—
132C22×C3×C4—22no—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2411yescharacteristic
135C2×#67—21no—
136C22×Dic3—22no—
137C22×Dic3C2×D414yes—
138#235—22no—
139#235—22no—
140C2×D12C2×D411yescharacteristic
141C2×C4×D3—22no—
142C2×C4×D3—21no—
143C2×Dic6—22no—
144C2×Dic6—22no—
145C2×Dic6—24no—
146C2×Dic6—24no—
147C2×Dic6C2×D411yescharacteristic
148C3×M16—22no—
149C3×M16—22no—
150C2×C3×C8—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—42no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4C2×D414yes—
156C3×C22⋊C4—22no—
157C3×C42—22no—
158C3×C42C2×D412yes—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161#217—21no—
162#216—44no—
163#215—44no—
164(C2×D3)⋊C4—28no—
165(C2×D3)⋊C4—22no—
166C4⋊Dic3—28no—
167C4⋊Dic3—24no—
168C4⋊Dic3—24no—
169C4⋊Dic3C2×D412yes—
170Dic3⋊C4—28no—
171Dic3⋊C4—22no—
172Dic3⋊C4—416no—
173C4×Dic3—28no—
174C4×Dic3—28no—
175C4×Dic3—22no—
176C4×Dic3—21no—
177C4×Dic3—22no—
178C4×Dic3#4114yes—
179#208—24no—
180C2×C3⋊C8—22no—
181C2×#41—31no—
182C22×Q8—32no—
183C22×Q8C22×D312yes—
184#137—124no—
185C2×Q16—62no—
186C2×SD16—62no—
187C2×M16—32no—
188C2×M16C22×D312yes—
189C4⋊Q8—22no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—62no—
193C4⋊Q8—68no—
194C4⋊Q8—62no—
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197C4⋊Q8—124no—
198C4⋊Q8#6712yes—
199#126—68no—
200#125—64no—
201#125—68no—
202#125#6712yes—
203#124—62no—
204#123—64no—
205C22⋊Q8—22no—
206C22⋊Q8—64no—
207C22⋊Q8—64no—
208C4×Q8—22no—
209C4×Q8—32no—
210C4×Q8—64no—
211C4×Q8—64no—
212C4×Q8—64no—
213C4×D4—32no—
214C4×D4—62no—
215#117—31no—
216#117—64no—
217#117—64no—
218#117—61no—
219#117C22×D312yes—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—34no—
222C2×C4⋊C4—68no—
223C2×C42—32no—
224C2×C42—62no—
225C2×C42—61no—
226C2×C42C22×D311yescharacteristic
227C4⋊C8—62no—
228#104—64no—
229#103—28no—
230#103—64no—
231#101—124no—
232#97—21no—
233C4×C8—21no—
234C23×C3C22×D411yescharacteristic
235C22×D3—21no—
236C3×Q8—44no—
237C3×Q8—42no—
238C3×Q8—42no—
239C3×Q8—44no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4#12012yes—
245C2×C3×C4#12012yes—
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C22×D411yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×C3×C4H(5,𝔽2)12yes—
250#67—42no—
251C2×Dic3—28no—
252C2×Dic3—24no—
253C2×Dic3—22no—
254C2×Dic3—21no—
255C2×Dic3—24no—
256C2×Dic3—28no—
257D12—21no—
258D12—21no—
259C4×D3—42no—
260C4×D3—44no—
261Dic6—21no—
262Dic6—21no—
263Dic6—42no—
264Dic6—42no—
265Dic6—44no—
266C3×C8—42no—
267C3⋊C8—42no—
268#41—62no—
269#41—62no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—31no—
276C2×Q8—62no—
277C2×Q8—64no—
278C2×Q8—64no—
279C2×Q8—62no—
280C2×D4—31no—
281C2×D4—61no—
282C22×C4—22no—
283C22×C4—34no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4D3×D412yes—
288C22×C4C23×D311yescharacteristic
289Q16—124no—
290SD16—124no—
291M16—22no—
292M16—22no—
293M16—64no—
294C2×C8—22no—
295C2×C8—62no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—32no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—68no—
302C4⋊C4—68no—
303C4⋊C4—64no—
304C4⋊C4—64no—
305C4⋊C4—62no—
306C4⋊C4—1216no—
307C4⋊C4C2×#6714yes—
308C22⋊C4—22no—
309C22⋊C4—68no—
310C22⋊C4—61no—
311C22⋊C4—62no—
312C42—22no—
313C42—34no—
314C42—68no—
315C42—68no—
316C42—62no—
317C42—62no—
318C42—61no—
319C42D3×D412yes—
320C42C2×#6711yescharacteristic
321C42C2×#6711yescharacteristic
322C22×C3—22no—
323C22×C3C2×#12011yescharacteristic
324C22×C3#54612yes—
325C2×D3—41no—
326C2×D3—42no—
327C3×C4—22no—
328C3×C4—44no—
329C3×C4—82no—
330C3×C4#45614yes—
331C3×C4#53211yescharacteristic
332C3×C4#53511yescharacteristic
333C3×C4#55012yes—
334Dic3—28no—
335Dic3—41no—
336Dic3—44no—
337Dic3—42no—
338C23—61no—
339C23C2×D3×D411yescharacteristic
340Q8—44no—
341Q8—42no—
342Q8—42no—
343Q8—44no—
344Q8—61no—
345Q8—61no—
346Q8—122no—
347Q8—122no—
348Q8—124no—
349D4—61no—
350D4—61no—
351D4—122no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—68no—
357C2×C4—64no—
358C2×C4—68no—
359C2×C4—62no—
360C2×C4—64no—
361C2×C4—61no—
362C2×C4—124no—
363C2×C4—122no—
364C2×C4#93712yes—
365C2×C4#93712yes—
366C2×C4C2×D3×D411yescharacteristic
367C2×C4D3×#4112yes—
368C2×C4#100912yes—
369C2×C4C22×#6711yescharacteristic
370C8—42no—
371C8—122no—
372C2×C3—22no—
373C2×C3#243111yescharacteristic
374C2×C3C2×#45612yes—
375D3—81no—
376C22—22no—
377C22—121no—
378C22—122no—
379C22#567612yes—
380C22C2×#93711yescharacteristic
381C4—22no—
382C4—44no—
383C4—68no—
384C4—82no—
385C4—124no—
386C4—121no—
387C4—122no—
388C4#520514yes—
389C4#566811yescharacteristic
390C4#567411yescharacteristic
391C4#583412yes—
392C3#2558911yescharacteristic, complemented, Sylow
393C2—22no—
394C2—241no—
395C2#8295811yescharacteristic
396C2C2×#520512yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3139C2
socleC22×C3C2×#120
prime radicalC2×#235C23
hypercenter#3139D3
hypocenterC3#25589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. C3×#3139C2semidirect
3. #83253C2semidirect
5. #71839C2semidirect
7. #70345C2semidirect
8. #69877C2semidirect
39. #3139D3semidirect
392. C3#25589semidirect

Maximal subgroups

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

subgroupindex
#862622
C3×#31392
#832532
#832092
#718392
#703462
#703452
#698772
C4×#8252
#255893