Finite Group Databasebeta

← #10848600

Subgroups — #10848600

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
2C3×#3039C211yesmaximal, characteristic
3#83253C211yesmaximal, characteristic, complemented
4#83198C211yesmaximal, characteristic, complemented
5#72108C214yesmaximal
6C4×#837C211yesmaximal, characteristic
7#70522C211yesmaximal, characteristic, complemented
8#70514C211yesmaximal, characteristic, complemented
9#70345C214yesmaximal, complemented
10#25589—31nomaximal, Sylow
11C2×C3×#137C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C3×C4×#41C2211yescharacteristic
14#5825C2212yes—
15C2×#944C2214yes—
16#5567—22no—
17#5542—22no—
18#5541—24no—
19#5518—22no—
20#5516C2212yes—
21#5470—22no—
22#5470C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C3×#462—22no—
25C3×#461—22no—
26C3×#423C2211yescharacteristic
27C3×#419C2212yes—
28C2×#837C2212yes—
29#5225C2214yes—
30C2×#824C2214yes—
31#5082—22no—
32#5079—22no—
33#4975—24no—
34C42×Dic3C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4555C2214yes—
37#4532C2212yes—
38#3509—31no—
39#3139—31no—
40#3039D311yescharacteristic
41#2868—31no—
42#2005—34no—
43#1994—31no—
44#1966—31no—
45#1886—34no—
46C4×#104—31no—
47C2×C3×#41C2311yescharacteristic
48C22×C3×Q8C2312yes—
49C22×Dic6C2312yes—
50C3×#137—44no—
51C2×C3×Q16—22no—
52C2×C3×SD16—22no—
53C2×C3×M16C2312yes—
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—21no—
57C3×C4×Q8D412yes—
58C3×C4×D4—22no—
59C3×C4×D4D412yes—
60C3×#117—21no—
61C3×#117C2311yescharacteristic
62C2×C3×C4⋊C4—22no—
63C2×C3×C42—21no—
64C2×C3×C42C2311yescharacteristic
65#944—22no—
66#944—28no—
67#944—44no—
68#936—22no—
69#935—24no—
70#926—24no—
71#926C2312yes—
72C2×C4⋊Dic3D414yes—
73#924—24no—
74#924—22no—
75#924—24no—
76C2×Dic3⋊C4—28no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3C2312yes—
79C2×#208C2312yes—
80#889—28no—
81#887—24no—
82#879—28no—
83#877—24no—
84#870—24no—
85#870D412yes—
86#869—21no—
87#869—21no—
88#869—22no—
89#869D412yes—
90C4×Dic6—24no—
91C4×Dic6—22no—
92C4×Dic6—24no—
93C3×C4⋊C8—22no—
94C3×#103—24no—
95#837—24no—
96#835—28no—
97#824—44no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×#137C2×D312yes—
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⋊Q8C2×D311yescharacteristic
111#519—32no—
112C4×#41C2×D311yescharacteristic
113#491—62no—
114#487—62no—
115#462—22no—
116#461—22no—
117#423C2×D311yescharacteristic
118C2×#104—32no—
119#419—34no—
120#419C2×D312yes—
121C2×#101—34no—
122C4×M16—31no—
123#384—64no—
124C43—31no—
125#339—34no—
126#339—32no—
127C3×#41—22no—
128C3×#41—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—24no—
132C2×C3×Q8—24no—
133C2×C3×Q8C2×D411yescharacteristic
134C2×C3×D4—21no—
135C2×C3×D4C2×D411yescharacteristic
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—22no—
142C22×Dic3C2×D414yes—
143C2×Dic6—22no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—24no—
147C2×Dic6—24no—
148C3×Q16—44no—
149C3×SD16—44no—
150C3×M16—24no—
151C2×C3×C8—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—42no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4C2×D412yes—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—22no—
158C3×C42—22no—
159C3×C42—21no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D412yes—
163#217—28no—
164#217—22no—
165C4⋊Dic3—24no—
166C4⋊Dic3—22no—
167C4⋊Dic3—24no—
168C4⋊Dic3C2×D414yes—
169Dic3⋊C4—28no—
170Dic3⋊C4—28no—
171Dic3⋊C4—24no—
172Dic3⋊C4—416no—
173C4×Dic3—28no—
174C4×Dic3—28no—
175C4×Dic3—22no—
176C4×Dic3—22no—
177C4×Dic3#4114yes—
178#208—22no—
179#208—22no—
180C2×C3⋊C8—22no—
181C2×#41C22×D311yescharacteristic
182C22×Q8—32no—
183C22×Q8C22×D312yes—
184#137—44no—
185C2×Q16—22no—
186C2×SD16—22no—
187C2×M16—32no—
188C2×M16C22×D312yes—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—22no—
192C4⋊Q8—32no—
193C4⋊Q8—68no—
194C4⋊Q8—62no—
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197C4⋊Q8—62no—
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×Q8#6712yes—
213C4×D4—22no—
214C4×D4#6712yes—
215#117—21no—
216#117—32no—
217#117—64no—
218#117—64no—
219#117C22×D311yescharacteristic
220C2×C4⋊C4—22no—
221C2×C4⋊C4—34no—
222C2×C4⋊C4—68no—
223C2×C42—21no—
224C2×C42—32no—
225C2×C42—62no—
226C2×C42C22×D311yescharacteristic
227C4⋊C8—22no—
228#104—64no—
229#103—24no—
230#103—68no—
231#101—124no—
232#97—61no—
233C4×C8—61no—
234C23×C3—21no—
235C23×C3C22×D411yescharacteristic
236C3×Q8—21no—
237C3×Q8—21no—
238C3×Q8—42no—
239C3×Q8—42no—
240C3×Q8—44no—
241C3×D4—21no—
242C3×D4—21no—
243C3×D4—42no—
244C2×C3×C4—24no—
245C2×C3×C4—22no—
246C2×C3×C4—21no—
247C2×C3×C4—42no—
248C2×C3×C4—44no—
249C2×C3×C4—42no—
250C2×C3×C4—44no—
251C2×C3×C4#12012yes—
252C2×C3×C4#12012yes—
253C2×C3×C4C22×D411yescharacteristic
254C2×C3×C4C22×D411yescharacteristic
255C2×C3×C4C2×#4112yes—
256C2×C3×C4H(5,𝔽2)12yes—
257C2×Dic3—28no—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—24no—
261C2×Dic3—28no—
262Dic6—42no—
263Dic6—42no—
264Dic6—44no—
265Dic6—44no—
266C3×C8—42no—
267C3⋊C8—42no—
268#41—22no—
269#41—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—62no—
275C2×Q8—62no—
276C2×Q8—64no—
277C2×Q8—64no—
278C2×Q8—62no—
279C2×Q8C2×#6711yescharacteristic
280C2×D4—21no—
281C2×D4C2×#6711yescharacteristic
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—34no—
286C22×C4—62no—
287C22×C4D3×D412yes—
288C22×C4C23×D311yescharacteristic
289Q16—44no—
290SD16—44no—
291M16—24no—
292M16—62no—
293M16—62no—
294C2×C8—22no—
295C2×C8—62no—
296C4⋊C4—22no—
297C4⋊C4—34no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—68no—
301C4⋊C4—68no—
302C4⋊C4—64no—
303C4⋊C4—64no—
304C4⋊C4—62no—
305C4⋊C4—64no—
306C4⋊C4—1216no—
307C4⋊C4C2×#6712yes—
308C22⋊C4—21no—
309C22⋊C4—22no—
310C22⋊C4—68no—
311C22⋊C4—62no—
312C42—22no—
313C42—21no—
314C42—34no—
315C42—68no—
316C42—68no—
317C42—62no—
318C42—62no—
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C42C2×#6712yes—
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—42no—
325C22×C3C2×#12011yescharacteristic
326C22×C3#54612yes—
327C3×C4—22no—
328C3×C4—41no—
329C3×C4—42no—
330C3×C4—82no—
331C3×C4#45614yes—
332C3×C4#53211yescharacteristic
333C3×C4#53511yescharacteristic
334C3×C4#55012yes—
335Dic3—28no—
336Dic3—44no—
337Dic3—44no—
338C23—21no—
339C23C2×D3×D411yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—42no—
343Q8—42no—
344Q8—44no—
345Q8—124no—
346Q8—122no—
347Q8—122no—
348Q8—124no—
349D4—21no—
350D4—21no—
351D4—42no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—68no—
360C2×C4—64no—
361C2×C4—64no—
362C2×C4—64no—
363C2×C4—68no—
364C2×C4#93712yes—
365C2×C4#93712yes—
366C2×C4C2×D3×D411yescharacteristic
367C2×C4#100412yes—
368C2×C4D3×#4112yes—
369C2×C4C22×#6711yescharacteristic
370C8—42no—
371C8—122no—
372C2×C3—22no—
373C2×C3—81no—
374C2×C3#243111yescharacteristic
375C2×C3C2×#45612yes—
376C22—22no—
377C22—42no—
378C22—41no—
379C22#561212yes—
380C22C2×#93711yescharacteristic
381C4—22no—
382C4—42no—
383C4—41no—
384C4—68no—
385C4—82no—
386C4—124no—
387C4—124no—
388C4#520514yes—
389C4#559912yes—
390C4#580511yescharacteristic
391C4#583211yescharacteristic
392C3#2558911yescharacteristic, complemented, Sylow
393C2—22no—
394C2—81no—
395C2#8265211yescharacteristic
396C2C2×#520512yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3039C2
socleC22×C3C2×#120
prime radicalC2×C3×#41C23
hypercenter#3039D3
hypocenterC3#25589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
3. #83253C2semidirect
4. #83198C2semidirect
7. #70522C2semidirect
8. #70514C2semidirect
9. #70345C2semidirect
392. C3#25589semidirect

Maximal subgroups

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

subgroupindex
#858302
C3×#30392
#832532
#831982
#721082
C4×#8372
#705222
#705142
#703452
#255893