Finite Group Databasebeta

← #10848599

Subgroups — #10848599

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#85830C211yesmaximal, characteristic
2#82533C211yesmaximal, characteristic
3#78537C211yesmaximal, characteristic, complemented
4#73311C211yesmaximal, characteristic, complemented
5C3×C4×#104C211yesmaximal, characteristic
6#70389C214yesmaximal
7#69877C211yesmaximal, characteristic, complemented
8#69396C211yesmaximal, characteristic, complemented
9#68765C214yesmaximal, complemented
10#25589—31nomaximal, Sylow
11C3×C4×#41C2211yescharacteristic
12#5825C2212yes—
13C2×#951C2212yes—
14#5567—22no—
15#5542—22no—
16#5541—24no—
17#5518—22no—
18#5516C2212yes—
19#5470—22no—
20#5470C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×#869C2214yes—
24C2×C3×#104C2212yes—
25C3×C4×M16C2211yescharacteristic
26C3×C43C2211yescharacteristic
27C2×#824C2214yes—
28#5120C2214yes—
29#5031—22no—
30#5019—22no—
31#5004C2212yes—
32#4935—24no—
33#4924C2211yescharacteristic
34#4703—22no—
35#4681—22no—
36C3×#339C2212yes—
37#4523C2214yes—
38#3509—31no—
39#3139—31no—
40#3039—31no—
41#2868—31no—
42#2005—34no—
43#1994—31no—
44#1966—31no—
45#1886—34no—
46C4×#104D311yescharacteristic
47C2×C3×#41C2311yescharacteristic
48C22×Dic6C2312yes—
49C22×Dic6C2312yes—
50C2×C3×M16C2312yes—
51C3×C4×Q8D412yes—
52C3×C4×D4—22no—
53C3×C4×D4D412yes—
54C3×#117—21no—
55C3×#117C2311yescharacteristic
56C2×C3×C42—22no—
57C2×C3×C42—21no—
58C2×C3×C42C2312yes—
59C2×C3×C42C2311yescharacteristic
60#951—44no—
61#944—22no—
62C2×#216—22no—
63#936—22no—
64#935—24no—
65C2×#214—22no—
66#926—24no—
67#926C2312yes—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—28no—
70C2×C4⋊Dic3D414yes—
71#924—24no—
72#924—22no—
73#924—24no—
74C2×#208C2312yes—
75#889—28no—
76#887—24no—
77#879—28no—
78#877—24no—
79#870—24no—
80#870D412yes—
81#869—21no—
82#869—21no—
83#869—22no—
84#869—21no—
85#869—21no—
86#869—22no—
87#869—28no—
88#869—44no—
89#869D412yes—
90C4×Dic6—24no—
91C4×Dic6—22no—
92C4×Dic6—24no—
93C3×#104—24no—
94C3×#97—21no—
95C3×C4×C8—21no—
96#824—44no—
97#816—28no—
98#810—24no—
99C4⋊(C3⋊C8)—22no—
100C2×#137—32no—
101#557—62no—
102#556—62no—
103#552—64no—
104#541—62no—
105#536—32no—
106#533—31no—
107#533—62no—
108C2×C4⋊Q8—34no—
109C2×C4⋊Q8—31no—
110C2×C4⋊Q8—31no—
111#519—32no—
112C4×#41C2×D311yescharacteristic
113#491—62no—
114#487—62no—
115#462—62no—
116#461—62no—
117#423—31no—
118C2×#104C2×D312yes—
119#419—34no—
120#419—32no—
121C2×#101—34no—
122C4×M16C2×D311yescharacteristic
123#384—64no—
124C43C2×D311yescharacteristic
125#339—34no—
126#339C2×D312yes—
127C3×#41—22no—
128C3×#41—22no—
129C2×C3×Q8C2×D411yescharacteristic
130C2×C3×D4—21no—
131C2×C3×D4C2×D411yescharacteristic
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—22no—
138C22×Dic3C2×D414yes—
139C2×Dic6—22no—
140C2×Dic6—22no—
141C2×Dic6—22no—
142C2×Dic6—24no—
143C2×Dic6—24no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—24no—
147C2×Dic6—24no—
148C3×M16—22no—
149C3×M16—22no—
150C2×C3×C8—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4C2×D412yes—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—22no—
155C3×C42—28no—
156C3×C42—22no—
157C3×C42—21no—
158C3×C42—22no—
159C3×C42C2×D412yes—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4114yes—
163#217—28no—
164#217—22no—
165#216—44no—
166#214—44no—
167C4⋊Dic3—24no—
168C4⋊Dic3—22no—
169C4⋊Dic3—24no—
170C4⋊Dic3—416no—
171C4⋊Dic3—42no—
172C4⋊Dic3—42no—
173C4⋊Dic3C2×D414yes—
174Dic3⋊C4—28no—
175Dic3⋊C4—28no—
176Dic3⋊C4—24no—
177C4×Dic3—28no—
178C4×Dic3—22no—
179#208—24no—
180C2×C3⋊C8—22no—
181C2×#41C22×D311yescharacteristic
182C22×Q8—32no—
183C22×Q8—32no—
184#137—124no—
185C2×Q16—62no—
186C2×SD16—62no—
187C2×M16—32no—
188C2×M16C22×D312yes—
189C4⋊Q8—32no—
190C4⋊Q8—62no—
191C4⋊Q8—62no—
192C4⋊Q8—62no—
193C4⋊Q8—68no—
194C4⋊Q8—61no—
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197C4⋊Q8—61no—
198C4⋊Q8—124no—
199#126—68no—
200#125—32no—
201#125—64no—
202#125—68no—
203#124—62no—
204#123—64no—
205C22⋊Q8—64no—
206C22⋊Q8—64no—
207C22⋊Q8—62no—
208C4×Q8—64no—
209C4×Q8—64no—
210C4×Q8—64no—
211C4×Q8—62no—
212C4×Q8D1212yes—
213C4×D4—22no—
214C4×D4D1212yes—
215#117—21no—
216#117—32no—
217#117—64no—
218#117—64no—
219#117C22×D311yescharacteristic
220C2×C4⋊C4—34no—
221C2×C4⋊C4—68no—
222C2×C4⋊C4—62no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42C22×D312yes—
226C2×C42C22×D311yescharacteristic
227C4⋊C8—62no—
228#104—24no—
229#103—68no—
230#103—64no—
231#101—124no—
232#97—21no—
233C4×C8—21no—
234C23×C3—21no—
235C23×C3C22×D411yescharacteristic
236C3×Q8—21no—
237C3×Q8—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—42no—
241C2×C3×C4—28no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—42no—
247C2×C3×C4—44no—
248C2×C3×C4#12012yes—
249C2×C3×C4#12012yes—
250C2×C3×C4C22×D411yescharacteristic
251C2×C3×C4C22×D411yescharacteristic
252C2×C3×C4C2×#4112yes—
253C2×C3×C4H(5,𝔽2)12yes—
254C2×Dic3—24no—
255C2×Dic3—24no—
256C2×Dic3—28no—
257C2×Dic3—42no—
258C2×Dic3—44no—
259Dic6—42no—
260Dic6—42no—
261Dic6—44no—
262Dic6—44no—
263Dic6—42no—
264Dic6—42no—
265Dic6—44no—
266C3×C8—42no—
267C3⋊C8—42no—
268#41—22no—
269#41—22no—
270C2×Q8—62no—
271C2×Q8—62no—
272C2×Q8—64no—
273C2×Q8—62no—
274C2×Q8—64no—
275C2×Q8—62no—
276C2×Q8—64no—
277C2×Q8—62no—
278C2×Q8—64no—
279C2×Q8C2×D1211yescharacteristic
280C2×D4—21no—
281C2×D4C2×D1211yescharacteristic
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—34no—
286C22×C4—62no—
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—34no—
298C4⋊C4—68no—
299C4⋊C4—68no—
300C4⋊C4—64no—
301C4⋊C4—64no—
302C4⋊C4—62no—
303C4⋊C4—64no—
304C4⋊C4—1216no—
305C4⋊C4—122no—
306C4⋊C4—122no—
307C4⋊C4C2×D1212yes—
308C22⋊C4—21no—
309C22⋊C4—22no—
310C22⋊C4—68no—
311C22⋊C4—62no—
312C42—28no—
313C42—22no—
314C42—21no—
315C42—22no—
316C42—68no—
317C42—62no—
318C42C2×D1212yes—
319C42#23514yes—
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—42no—
325C22×C3C2×#12011yescharacteristic
326C22×C3#54612yes—
327C3×C4—28no—
328C3×C4—22no—
329C3×C4—41no—
330C3×C4—42no—
331C3×C4#45614yes—
332C3×C4#53211yescharacteristic
333C3×C4#53511yescharacteristic
334C3×C4#55012yes—
335Dic3—44no—
336Dic3—44no—
337Dic3—82no—
338C23—21no—
339C23C2×D3×D411yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—122no—
343Q8—122no—
344Q8—124no—
345Q8—124no—
346Q8—122no—
347Q8—122no—
348Q8—124no—
349D4—21no—
350D4—21no—
351D4—42no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—64no—
360C2×C4—64no—
361C2×C4—68no—
362C2×C4—122no—
363C2×C4—124no—
364C2×C4#88212yes—
365C2×C4#88212yes—
366C2×C4C22×D1211yescharacteristic
367C2×C4C2×#23512yes—
368C2×C4C2×D3×D411yescharacteristic
369C2×C4#100912yes—
370C8—42no—
371C8—122no—
372C2×C3—22no—
373C2×C3—81no—
374C2×C3#243111yescharacteristic
375C2×C3C2×#45612yes—
376C22—22no—
377C22—41no—
378C22—42no—
379C22C2×#88211yescharacteristic
380C22#555012yes—
381C4—28no—
382C4—22no—
383C4—41no—
384C4—42no—
385C4—124no—
386C4—124no—
387C4—242no—
388C4#482614yes—
389C4#547711yescharacteristic
390C4#552511yescharacteristic
391C4#555312yes—
392C3#2558911yescharacteristic, complemented, Sylow
393C2—22no—
394C2—81no—
395C2#7358611yescharacteristic
396C2C2×#482612yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#104C2
socleC22×C3C2×#120
prime radicalC2×C3×#41C23
hypercenterC4×#104D3
hypocenterC3#25589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
3. #78537C2semidirect
4. #73311C2semidirect
7. #69877C2semidirect
8. #69396C2semidirect
9. #68765C2semidirect
392. C3#25589semidirect

Maximal subgroups

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

subgroupindex
#858302
#825332
#785372
#733112
C3×C4×#1042
#703892
#698772
#693962
#687652
#255893