Finite Group Databasebeta

← #10848595

Subgroups — #10848595

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
1C3×#3509C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic, complemented
3#83253C211yesmaximal, characteristic, complemented
4#82533C211yesmaximal, characteristic
5#71868C214yesmaximal
6#71839C211yesmaximal, characteristic, complemented
7#71789C211yesmaximal, characteristic, complemented
8#71660C214yesmaximal, complemented
9C4×#837C211yesmaximal, characteristic
10#25589—31nomaximal, Sylow
11C3×#557—22no—
12C3×#556—22no—
13C3×#552—24no—
14C3×#541—22no—
15C3×#536C2212yes—
16C3×#533—22no—
17C3×#533C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C3×#519C2212yes—
20C3×C4×#41C2211yescharacteristic
21C2×#951C2212yes—
22C2×#944C2214yes—
23C2×#869C2211yescharacteristic
24C2×#837C2212yes—
25C2×#836C2214yes—
26#5093—22no—
27#5069—22no—
28#5031—22no—
29#5019—22no—
30#5004C2212yes—
31#5004C2214yes—
32#4975—24no—
33C42×Dic3C2211yescharacteristic
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4532C2214yes—
37#4532C2212yes—
38#3509D311yescharacteristic
39#3139—31no—
40#3039—31no—
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×C4⋊Q8—22no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8D412yes—
55C3×#126—28no—
56C3×#125—24no—
57C3×#125—28no—
58C3×#125D412yes—
59C3×#124—22no—
60C3×#123—24no—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—24no—
64C3×C4×Q8—24no—
65C3×C4×Q8—24no—
66C3×C4×Q8—24no—
67C3×C4×Q8—22no—
68C3×C4×Q8D412yes—
69C3×C4×D4—22no—
70C3×C4×D4D412yes—
71C3×#117—24no—
72C3×#117—24no—
73C3×#117—21no—
74C3×#117C2311yescharacteristic
75C3×#117C2312yes—
76C2×C3×C4⋊C4D414yes—
77C2×C3×C42—21no—
78C2×C3×C42C2311yescharacteristic
79#951—44no—
80#944—28no—
81#944—44no—
82C2×#216—22no—
83C2×#214—22no—
84C2×C4⋊Dic3—22no—
85C2×Dic3⋊C4—28no—
86C2×C4×Dic3—22no—
87C2×C4×Dic3C2312yes—
88C2×#208C2312yes—
89C2×#208C2312yes—
90#869—22no—
91#869—21no—
92#869—21no—
93#837—24no—
94#836—44no—
95#810—24no—
96#810—28no—
97C4⋊(C3⋊C8)—22no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×#137—32no—
101#557—22no—
102#556—22no—
103#552—24no—
104#541—22no—
105#536C2×D312yes—
106#533—22no—
107#533C2×D311yescharacteristic
108C2×C4⋊Q8—34no—
109C2×C4⋊Q8—31no—
110C2×C4⋊Q8C2×D311yescharacteristic
111#519C2×D312yes—
112C4×#41C2×D311yescharacteristic
113#491—62no—
114#487—62no—
115#462—62no—
116#461—62no—
117#423—31no—
118C2×#104—32no—
119#419—34no—
120#419—32no—
121C2×#101—34no—
122C4×M16—31no—
123#384—64no—
124C43—31no—
125#339—32no—
126#339—34no—
127C3×#41—22no—
128C3×#41—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—24no—
133C2×C3×Q8—24no—
134C2×C3×Q8C2×D411yescharacteristic
135C2×C3×D4—21no—
136C2×C3×D4C2×D411yescharacteristic
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4C2×D414yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2411yescharacteristic
142C22×Dic3—22no—
143C22×Dic3—22no—
144C2×Dic6—22no—
145C2×Dic6—22no—
146C2×Dic6—24no—
147C2×Dic6—24no—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4C2×D412yes—
156C3×C4⋊C4C2×D414yes—
157C3×C22⋊C4—28no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C42—28no—
162C3×C42—22no—
163C3×C42—21no—
164C3×C42—22no—
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42C2×D412yes—
168#216—44no—
169#214—44no—
170C4⋊Dic3—42no—
171C4⋊Dic3—42no—
172Dic3⋊C4—416no—
173C4×Dic3—28no—
174C4×Dic3—22no—
175C4×Dic3#4114yes—
176#208—22no—
177#208—22no—
178#208—24no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×#41C22×D311yescharacteristic
182C22×Q8—32no—
183C22×Q8C22×D312yes—
184#137—124no—
185C2×Q16—62no—
186C2×SD16—62no—
187C2×M16—32no—
188C2×M16—32no—
189C4⋊Q8—22no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—22no—
193C4⋊Q8—68no—
194C4⋊Q8—62no—
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197C4⋊Q8—124no—
198C4⋊Q8#6712yes—
199#126—28no—
200#125—24no—
201#125—28no—
202#125#6712yes—
203#124—22no—
204#123—24no—
205C22⋊Q8—24no—
206C22⋊Q8—22no—
207C22⋊Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—22no—
212C4×Q8#6712yes—
213C4×D4—22no—
214C4×D4#6712yes—
215#117—24no—
216#117—24no—
217#117—21no—
218#117C22×D311yescharacteristic
219#117C22×D312yes—
220C2×C4⋊C4—68no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4#6714yes—
223C2×C42—21no—
224C2×C42—32no—
225C2×C42—62no—
226C2×C42C22×D311yescharacteristic
227C4⋊C8—62no—
228#104—64no—
229#103—64no—
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×Q8—44no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—42no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—28no—
251C2×C3×C4—42no—
252C2×C3×C4—44no—
253C2×C3×C4#12012yes—
254C2×C3×C4#12012yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C4H(5,𝔽2)12yes—
259C2×Dic3—28no—
260C2×Dic3—24no—
261C2×Dic3—42no—
262C2×Dic3—44no—
263Dic6—42no—
264Dic6—42no—
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—22no—
275C2×Q8—64no—
276C2×Q8—64no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8C2×#6711yescharacteristic
280C2×D4—21no—
281C2×D4C2×#6711yescharacteristic
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4C2×#6714yes—
287C22×C4C2×#6712yes—
288C22×C4C23×D311yescharacteristic
289Q16—124no—
290SD16—124no—
291M16—62no—
292M16—62no—
293M16—64no—
294C2×C8—62no—
295C2×C8—62no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—1216no—
304C4⋊C4—122no—
305C4⋊C4—122no—
306C4⋊C4C2×#6712yes—
307C4⋊C4C2×#6714yes—
308C22⋊C4—28no—
309C22⋊C4—21no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C42—28no—
313C42—22no—
314C42—21no—
315C42—22no—
316C42—34no—
317C42—68no—
318C42—62no—
319C42C2×#6711yescharacteristic
320C42C2×#6711yescharacteristic
321C42C2×#6712yes—
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—42no—
325C22×C3C2×#12011yescharacteristic
326C22×C3#54612yes—
327C3×C4—22no—
328C3×C4—44no—
329C3×C4—41no—
330C3×C4—42no—
331C3×C4—44no—
332C3×C4#45614yes—
333C3×C4#53211yescharacteristic
334C3×C4#53511yescharacteristic
335C3×C4#55012yes—
336Dic3—28no—
337Dic3—82no—
338C23—21no—
339C23C22×#6711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—44no—
343Q8—42no—
344Q8—42no—
345Q8—44no—
346Q8—122no—
347Q8—122no—
348Q8—124no—
349D4—21no—
350D4—21no—
351D4—42no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—68no—
361C2×C4—64no—
362C2×C4—124no—
363C2×C4—122no—
364C2×C4#95312yes—
365C2×C4#95312yes—
366C2×C4D3×#4112yes—
367C2×C4#100912yes—
368C2×C4C22×#6711yescharacteristic
369C2×C4C22×#6711yescharacteristic
370C8—122no—
371C8—122no—
372C2×C3—22no—
373C2×C3—81no—
374C2×C3#243111yescharacteristic
375C2×C3C2×#45612yes—
376C22—22no—
377C22—41no—
378C22—42no—
379C22#582912yes—
380C22C2×#95311yescharacteristic
381C4—22no—
382C4—44no—
383C4—41no—
384C4—42no—
385C4—44no—
386C4—68no—
387C4—242no—
388C4#524614yes—
389C4#583011yescharacteristic
390C4#583211yescharacteristic
391C4#583412yes—
392C3#2558911yescharacteristic, complemented, Sylow
393C2—22no—
394C2—81no—
395C2#8409211yescharacteristic
396C2C2×#524612yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3509C2
socleC22×C3C2×#120
prime radicalC2×C3×#41C23
hypercenter#3509D3
hypocenterC3#25589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
2. #83337C2semidirect
3. #83253C2semidirect
6. #71839C2semidirect
7. #71789C2semidirect
8. #71660C2semidirect
392. C3#25589semidirect

Maximal subgroups

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

subgroupindex
C3×#35092
#833372
#832532
#825332
#718682
#718392
#717892
#716602
C4×#8372
#255893