Finite Group Databasebeta

← #10804454

Subgroups — #10804454

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5833C211yesmaximal, characteristic
2#74054C212yesmaximal, complemented
3C3×#1782C211yesmaximal, characteristic, complemented
4#72139C211yesmaximal, characteristic, complemented
5#70030C218yesmaximal, complemented
6#69537C212yesmaximal, complemented
7#21796—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×#235C2211yescharacteristic
10#5833—24no—
11#5833—24no—
12C2×#946C412yescomplemented
13C2×#939C2214yes—
14C2×#938C412yescomplemented
15C2×#930C412yescomplemented
16C22×C4⋊Dic3C2211yescharacteristic
17C2×#924C412yescomplemented
18C2×C3×#99C2214yes—
19C3×#381C2212yes—
20C2×#834C2214yes—
21#5211C2212yes—
22C2×#806C2218yes—
23#4942C2214yes—
24Dic3×C22⋊C4C2214yes—
25#4654—24no—
26#4653—24no—
27C2×#537—31no—
28#2313—32no—
29#1964—38no—
30#1782—32no—
31#1782—31no—
32#1782D311yescharacteristic, complemented
33C2×C3×#41—22no—
34C2×C3×#41—22no—
35C2×C3×#41—44no—
36C22×C3×Q8C2×C411yescharacteristic
37C22×C3×D4C2×C411yescharacteristic
38C22×C3×D4C2312yes—
39C23×C3×C4C2×C412yes—
40C23×C3×C4C2311yescharacteristic
41C22×#67C2×C414yes—
42C22×#67C2312yes—
43C23×Dic3C2312yes—
44C2×#235—22no—
45C2×#235—22no—
46C2×#235—44no—
47C22×D12C2×C411yescharacteristic
48C22×C4×D3C2×C412yes—
49C22×Dic6C2×C411yescharacteristic
50C2×C3×C22⋊C4C2314yes—
51C2×#217C2×C414yes—
52C2×#217C2314yes—
53#946—28no—
54#939—28no—
55#939—28no—
56#938—28no—
57#930—24no—
58#930—24no—
59C2×(C2×D3)⋊C4C2×C414yes—
60C2×C4⋊Dic3C2×C412yes—
61C2×C4⋊Dic3C2×C412yes—
62C2×C4⋊Dic3D414yes—
63C2×C4⋊Dic3D414yes—
64#924—24no—
65#924—24no—
66C2×Dic3⋊C4—24no—
67C2×Dic3⋊C4C2×C414yes—
68C2×C4×Dic3—24no—
69C3×#99—44no—
70C3×#95—22no—
71C3×#95—22no—
72#834—44no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77#806—48no—
78C22×#41—31no—
79C22×#41C2×D311yescharacteristic
80#537—64no—
81#537—64no—
82C2×C22⋊Q8—32no—
83C2×C22⋊Q8—32no—
84C2×#121—32no—
85C2×#121—34no—
86C2×#121—32no—
87C22×C4⋊C4—31no—
88C2×#99—34no—
89C2×#99—38no—
90C2×#99C2×D314yes—
91#387—64no—
92#387—64no—
93#381—32no—
94#381C2×D312yes—
95#380—34no—
96C4×C22⋊C4—34no—
97C24×C3C22×C412yes—
98C24×C3C2411yescharacteristic
99C23×D3C22×C412yes—
100C3×#41—48no—
101C3×#41—48no—
102C2×C3×Q8—22no—
103C2×C3×Q8—42no—
104C2×C3×D4—24no—
105C2×C3×D4—24no—
106C2×C3×D4—24no—
107C2×C3×D4—22no—
108C2×C3×D4—42no—
109C22×C3×C4—24no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—24no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4C22×C412yes—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2×D412yes—
120C2×#67—24no—
121C2×#67—28no—
122C2×#67—28no—
123C2×#67—84no—
124C22×Dic3—24no—
125C22×Dic3—82no—
126C22×Dic3C22×C412yes—
127C22×Dic3C22×C414yes—
128C22×Dic3C2×D414yes—
129C22×Dic3C2×D414yes—
130#235—48no—
131#235—48no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—42no—
135C2×C4×D3—24no—
136C2×C4×D3—44no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C3×C22⋊C4—28no—
141#217—28no—
142#217—28no—
143(C2×D3)⋊C4—28no—
144C4⋊Dic3C22⋊C414yes—
145C4⋊Dic3C22⋊C414yes—
146C4⋊Dic3C22⋊C418yes—
147Dic3⋊C4—28no—
148Dic3⋊C4—84no—
149C4×Dic3—84no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—44no—
153C2×#41—62no—
154C2×#41—62no—
155C2×#41—124no—
156C22×Q8—31no—
157C22×Q8C4×D311yescharacteristic
158C22×D4—32no—
159C22×D4—34no—
160C22×D4—31no—
161C22×D4C4×D311yescharacteristic
162C22×D4C22×D312yes—
163C23×C4—32no—
164C23×C4—32no—
165C23×C4C4×D312yes—
166C23×C4C22×D311yescharacteristic
167C22⋊Q8—64no—
168C22⋊Q8—64no—
169C22⋊Q8—68no—
170#121—64no—
171#121—64no—
172#121—68no—
173#121—68no—
174#121—68no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—34no—
178C2×C4⋊C4—34no—
179C2×C4⋊C4—34no—
180C2×C4⋊C4—64no—
181C2×C22⋊C4—34no—
182C2×C22⋊C4—34no—
183C2×C22⋊C4—34no—
184C2×C22⋊C4C22×D314yes—
185C2×C42—64no—
186#99—44no—
187#99—128no—
188#99—124no—
189#95—22no—
190#95—22no—
191#95—64no—
192#95—64no—
193#95—62no—
194#95—62no—
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3C23×C411yescharacteristic
203C23×C3C22×D412yes—
204C23×C3H(5,𝔽2)14yes—
205C22×D3—22no—
206C22×D3—24no—
207C22×D3—42no—
208C22×D3—42no—
209C3×Q8—44no—
210C3×D4—44no—
211C3×D4—48no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—44no—
220C2×C3×C4—44no—
221C2×C3×C4—44no—
222C2×C3×C4—44no—
223C2×C3×C4—44no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4C2×C22⋊C412yes—
227#67—48no—
228#67—88no—
229C2×Dic3—22no—
230C2×Dic3—24no—
231C2×Dic3—44no—
232C2×Dic3—44no—
233C2×Dic3—44no—
234C2×Dic3—44no—
235C2×Dic3—44no—
236C2×Dic3—44no—
237C2×Dic3C2×C22⋊C418yes—
238D12—42no—
239D12—42no—
240C4×D3—48no—
241Dic6—42no—
242Dic6—42no—
243C24—32no—
244C24C2×C4×D312yes—
245C24C23×D311yescharacteristic
246#41—48no—
247#41—48no—
248#41—128no—
249#41—128no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—122no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—42no—
260C2×D4—68no—
261C2×D4—64no—
262C2×D4—68no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—122no—
266C2×D4—244no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—34no—
272C22×C4—34no—
273C22×C4—34no—
274C22×C4—32no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—64no—
278C22×C4—64no—
279C22×C4—124no—
280C22×C4—242no—
281C22×C4C2×C4×D312yes—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4C2×C4×D311yescharacteristic
284C22×C4D3×D412yes—
285C22×C4D3×D412yes—
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—38no—
289C4⋊C4—68no—
290C4⋊C4—244no—
291C22⋊C4—28no—
292C22⋊C4—68no—
293C22⋊C4—68no—
294C22⋊C4—68no—
295C42—244no—
296C22×C3—22no—
297C22×C3—21no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—44no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3C22×C22⋊C411yescharacteristic
307C22×C3#51812yes—
308C22×C3#53514yes—
309C2×D3—42no—
310C2×D3—42no—
311C2×D3—42no—
312C2×D3—84no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—22no—
316C3×C4—42no—
317C3×C4—84no—
318Dic3—28no—
319Dic3—42no—
320Dic3—84no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—62no—
329C23—64no—
330C23—122no—
331C23—122no—
332C23C22×C4×D311yescharacteristic
333C23C2×D3×D412yes—
334C23#100414yes—
335Q8—44no—
336Q8—122no—
337Q8—122no—
338D4—44no—
339D4—48no—
340D4—122no—
341D4—122no—
342D4—128no—
343D4—248no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—38no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—62no—
358C2×C4—64no—
359C2×C4—128no—
360C2×C4—124no—
361C2×C4—124no—
362C2×C4—124no—
363C2×C4—124no—
364C2×C4—124no—
365C2×C4—124no—
366C2×C4D3×C22⋊C411yescharacteristic
367C2×C4D3×C22⋊C411yescharacteristic
368C2×C4D3×C22⋊C412yes—
369C2×C3—24no—
370C2×C3—22no—
371C2×C3—82no—
372C2×C3#230911yescharacteristic
373C2×C3#288311yescharacteristic
374C2×C3#288411yescharacteristic
375D3—82no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—122no—
387C22—122no—
388C22—122no—
389C22—244no—
390C22C2×D3×C22⋊C411yescharacteristic
391C22#548612yes—
392C22#559214yes—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—42no—
397C4—68no—
398C4—84no—
399C4—122no—
400C4—244no—
401C3#2179611yescharacteristic, complemented, Sylow
402C2—24no—
403C2—22no—
404C2—82no—
405C2—242no—
406C2#7381911yescharacteristic
407C2#7868911yescharacteristic
408C2#7869411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1782C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5833C2
cosocleC24×C3C24
hypercenter#1782D3
hypocenterC3#21796
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #74054C2semidirect
3. C3×#1782C2semidirect
4. #72139C2semidirect
5. #70030C2semidirect
6. #69537C2semidirect
12. C2×#946C4semidirect
14. C2×#938C4semidirect
15. C2×#930C4semidirect
17. C2×#924C4semidirect
32. #1782D3semidirect
401. C3#21796semidirect

Maximal subgroups

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

subgroupindex
C2×#58332
#740542
C3×#17822
#721392
#700302
#695372
#217963