Finite Group Databasebeta

← #10955321

Subgroups — #10955321

Order 768.

2250 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic
2C3×#2313C211yesmaximal, characteristic
3#82155C211yesmaximal, characteristic
4#74964C218yesmaximal, complemented
5#74963C218yesmaximal, complemented
6#74053C212yesmaximal
7#74051C212yesmaximal
8#73867C212yesmaximal
9#73864C212yesmaximal
10#73437C212yesmaximal
11#73435C212yesmaximal
12#34187—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×Dic3⋊C4C2211yescharacteristic
16C22×Dic3⋊C4C2211yescharacteristic
17C22×C4×Dic3C2211yescharacteristic
18C22×C4×Dic3C2211yescharacteristic
19C22×C4×Dic3C2211yescharacteristic
20C2×#887C4116yescomplemented
21C2×#887C2214yes—
22C2×#887C2214yes—
23C2×#877C2214yes—
24C3×#387C2218yes—
25C3×#381C2212yes—
26C3×#380C2212yes—
27C3×C4×C22⋊C4C2212yes—
28#5212C2212yes—
29#5211C2212yes—
30C4×#217C2212yes—
31#4980C2218yes—
32#4978C2218yes—
33#4975C2214yes—
34Dic3×C4⋊C4C2214yes—
35#4943C2214yes—
36#4942C2214yes—
37Dic3×C22⋊C4C2214yes—
38#4932C2214yes—
39C4×Dic3⋊C4C2214yes—
40C4×Dic3⋊C4C2218yes—
41#4655C2218yes—
42#4651C2214yes—
43#4651C2214yes—
44#4650C2214yes—
45#4650C2214yes—
46#4650C2214yes—
47#4649C2214yes—
48#4649C2214yes—
49#4649C2218yes—
50#4648C2214yes—
51#4646C2218yes—
52#4646C2218yes—
53C22×C4×Q8—31no—
54#2356—38no—
55#2355—38no—
56#2313—32no—
57#2313—31no—
58#2313D311yescharacteristic
59#2310—32no—
60#2285—32no—
61#2283—32no—
62#2283—32no—
63#2274—32no—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2311yescharacteristic
71C22×Dic6—28no—
72C22×Dic6C2×C418yes—
73C22×Dic6C2314yes—
74C2×C3×C4⋊C4C2×C418yes—
75C2×C3×C4⋊C4C2314yes—
76C2×C3×C4⋊C4C2314yes—
77C2×C3×C22⋊C4C2314yes—
78C2×C3×C42C2314yes—
79C2×#217C2314yes—
80C2×C4⋊Dic3C2314yes—
81C2×Dic3⋊C4C2×C418yes—
82C2×Dic3⋊C4C2×C418yes—
83C2×Dic3⋊C4C2318yes—
84C2×Dic3⋊C4C2318yes—
85C2×Dic3⋊C4C2314yes—
86C2×Dic3⋊C4C2314yes—
87C2×C4×Dic3C2×C418yes—
88C2×C4×Dic3C2×C418yes—
89C2×C4×Dic3C2×C418yes—
90C2×C4×Dic3C2318yes—
91C2×C4×Dic3C2318yes—
92C2×C4×Dic3C2314yes—
93C2×C4×Dic3C2312yes—
94C2×C4×Dic3C2312yes—
95C2×C4×Dic3C2312yes—
96C2×C4×Dic3C2312yes—
97C2×C4×Dic3C2312yes—
98C2×C4×Dic3C2312yes—
99#887C2×C4132yes—
100#887C2×C4132yes—
101#877—216no—
102C3×#95C2314yes—
103C3×#95C2314yes—
104C3×#95C2314yes—
105C3×#95C2314yes—
106C3⋊#95C2314yes—
107C3⋊#95C2314yes—
108C3⋊#95C2318yes—
109C3⋊#95C2318yes—
110C3⋊#95C2318yes—
111C3⋊#95C2318yes—
112C3⋊#95C2314yes—
113C3⋊#95C2314yes—
114C23×Q8—31no—
115C2×C4×Q8—316no—
116C2×C4×Q8—34no—
117C2×C4×Q8—34no—
118C2×#117—34no—
119C22×C4⋊C4—31no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C42—31no—
123C22×C42—31no—
124C22×C42—31no—
125#391—38no—
126#391—38no—
127#387—38no—
128#387—38no—
129#387—38no—
130#387C2×D318yes—
131#384—34no—
132#383—34no—
133#383—34no—
134#382—34no—
135#382—34no—
136#382—34no—
137#381—32no—
138#381—34no—
139#381C2×D312yes—
140#380—32no—
141#380—34no—
142#380C2×D312yes—
143C4×C4⋊C4—34no—
144C4×C4⋊C4—34no—
145C4×C4⋊C4—38no—
146C4×C4⋊C4—34no—
147C4×C4⋊C4—34no—
148C4×C4⋊C4—38no—
149C4×C22⋊C4—32no—
150C4×C22⋊C4—34no—
151C4×C22⋊C4C2×D312yes—
152#376—34no—
153#376—34no—
154C24×C3C2411yescharacteristic
155C22×C3×C4—24no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4C22×C414yes—
159C22×C3×C4C22×C414yes—
160C22×C3×C4C22×C414yes—
161C22×C3×C4C2418yes—
162C22×C3×C4C2412yes—
163C22×C3×C4C2412yes—
164C22×C3×C4C2412yes—
165C22×Dic3—24no—
166C22×Dic3—24no—
167C22×Dic3—42no—
168C22×Dic3C22×C414yes—
169C22×Dic3C22×C414yes—
170C22×Dic3C22×C414yes—
171C22×Dic3C22×C414yes—
172C22×Dic3#4118yes—
173C22×Dic3C2418yes—
174C22×Dic3C2411yescharacteristic
175C22×Dic3C2412yes—
176C22×Dic3C2412yes—
177C22×Dic3C2412yes—
178C22×Dic3C2411yescharacteristic
179C2×Dic6—232no—
180C2×Dic6—28no—
181C2×Dic6—28no—
182C2×Dic6C22×C4116yes—
183C3×C4⋊C4—44no—
184C3×C4⋊C4C22×C4116yes—
185C3×C22⋊C4—28no—
186C3×C42—44no—
187#217—28no—
188C4⋊Dic3—44no—
189Dic3⋊C4—216no—
190Dic3⋊C4—216no—
191Dic3⋊C4C22×C4116yes—
192Dic3⋊C4C22×C4116yes—
193C4×Dic3—216no—
194C4×Dic3—48no—
195C4×Dic3—44no—
196C4×Dic3C22×C418yes—
197C4×Dic3C22×C418yes—
198C4×Dic3C22×C418yes—
199C4×Dic3C22×C418yes—
200C4×Dic3C22×C418yes—
201C4×Dic3C22×C418yes—
202C22×Q8—38no—
203C22×Q8—34no—
204C22×Q8—68no—
205C23×C4—31no—
206C23×C4—31no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4C22×D311yescharacteristic
210C23×C4C22×D311yescharacteristic
211C23×C4C22×D311yescharacteristic
212C4×Q8—332no—
213C4×Q8—332no—
214#117—616no—
215C2×C4⋊C4—38no—
216C2×C4⋊C4—38no—
217C2×C4⋊C4—34no—
218C2×C4⋊C4—34no—
219C2×C4⋊C4—38no—
220C2×C4⋊C4—38no—
221C2×C4⋊C4—34no—
222C2×C4⋊C4C4×D318yes—
223C2×C4⋊C4C22×D314yes—
224C2×C4⋊C4C22×D314yes—
225C2×C22⋊C4—34no—
226C2×C22⋊C4C22×D314yes—
227C2×C42—38no—
228C2×C42—38no—
229C2×C42—38no—
230C2×C42—32no—
231C2×C42—32no—
232C2×C42—32no—
233C2×C42—32no—
234C2×C42—32no—
235C2×C42—32no—
236C2×C42—38no—
237C2×C42—38no—
238C2×C42—34no—
239C2×C42C22×D314yes—
240#95—38no—
241#95—38no—
242#95—34no—
243#95—38no—
244#95—34no—
245#95—34no—
246#95—38no—
247#95—34no—
248#95C22×D314yes—
249#95C22×D314yes—
250#95C22×D314yes—
251#95C22×D314yes—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3C23×C412yes—
255C23×C3C2×#4114yes—
256C23×C3C2511yescharacteristic
257C2×C3×C4—28no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—28no—
262C2×C3×C4—28no—
263C2×C3×C4—28no—
264C2×C3×C4—44no—
265C2×C3×C4—44no—
266C2×C3×C4C23×C414yes—
267C2×C3×C4C23×C414yes—
268C2×C3×C4C23×C414yes—
269C2×Dic3—28no—
270C2×Dic3—24no—
271C2×Dic3—22no—
272C2×Dic3—24no—
273C2×Dic3—28no—
274C2×Dic3—28no—
275C2×Dic3—28no—
276C2×Dic3—28no—
277C2×Dic3—44no—
278C2×Dic3#117116yes—
279C2×Dic3C23×C412yes—
280C2×Dic3C23×C414yes—
281C2×Dic3C23×C414yes—
282C2×Dic3C23×C414yes—
283C2×Dic3C23×C412yes—
284C2×Dic3C2×#4114yes—
285C2×Dic3C2×#4114yes—
286C2×Dic32−1+414yes—
287Dic6—232no—
288C24C23×D311yescharacteristic
289C2×Q8—316no—
290C2×Q8—68no—
291C2×Q8—68no—
292C2×Q8—632no—
293C22×C4—24no—
294C22×C4—34no—
295C22×C4—38no—
296C22×C4—31no—
297C22×C4—32no—
298C22×C4—34no—
299C22×C4—34no—
300C22×C4—31no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—34no—
304C22×C4—38no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—64no—
308C22×C4—64no—
309C22×C4—122no—
310C22×C4C2×C4×D314yes—
311C22×C4C2×C4×D314yes—
312C22×C4C2×C4×D314yes—
313C22×C4C23×D318yes—
314C22×C4C23×D312yes—
315C22×C4C23×D312yes—
316C22×C4C23×D312yes—
317C4⋊C4—316no—
318C4⋊C4—316no—
319C4⋊C4—44no—
320C4⋊C4—616no—
321C4⋊C4—616no—
322C4⋊C4—124no—
323C4⋊C4C2×C4×D3116yes—
324C22⋊C4—28no—
325C22⋊C4—68no—
326C42—38no—
327C42—38no—
328C42—38no—
329C42—38no—
330C42—38no—
331C42—38no—
332C42—44no—
333C42—616no—
334C42—124no—
335C42—128no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—28no—
339C22×C3C2×#11714yes—
340C22×C3C24×C411yescharacteristic
341C22×C3C22×#4112yes—
342C22×C3C2×2−1+412yes—
343C22×C3#58512yes—
344C3×C4—24no—
345C3×C4—24no—
346C3×C4—24no—
347C3×C4—48no—
348Dic3—24no—
349Dic3—22no—
350Dic3—24no—
351Dic3—48no—
352Dic3C2×#11718yes—
353Dic3#51914yes—
354C23—22no—
355C23—22no—
356C23C22×C4×D312yes—
357C23D3×#4114yes—
358C23C24×D311yescharacteristic
359Q8—632no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—32no—
368C2×C4—34no—
369C2×C4—34no—
370C2×C4—34no—
371C2×C4—34no—
372C2×C4—316no—
373C2×C4—32no—
374C2×C4—34no—
375C2×C4—34no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—68no—
379C2×C4—64no—
380C2×C4—64no—
381C2×C4—62no—
382C2×C4—68no—
383C2×C4—68no—
384C2×C4—68no—
385C2×C4—68no—
386C2×C4—124no—
387C2×C4C22×C4×D314yes—
388C2×C4C22×C4×D314yes—
389C2×C4C22×C4×D314yes—
390C2×C3—24no—
391C2×C3C22×#11711yescharacteristic
392C2×C3C2×#51911yescharacteristic
393C2×C3#342811yescharacteristic
394C2×C3#350714yes—
395C22—22no—
396C22—22no—
397C22—28no—
398C22D3×#11714yes—
399C22C23×C4×D311yescharacteristic
400C22C2×D3×#4112yes—
401C22#596612yes—
402C22D3×2−1+412yes—
403C4—24no—
404C4—24no—
405C4—24no—
406C4—34no—
407C4—38no—
408C4—48no—
409C4—64no—
410C4—62no—
411C4—64no—
412C4—128no—
413C3#3418711yescharacteristic, complemented, Sylow
414C2—24no—
415C2C2×D3×#11711yescharacteristic
416C2#8585611yescharacteristic
417C2D3×#51911yescharacteristic
418C2#8643114yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2313C2
socle, cosocleC23×C3C25
prime radicalC24×C3C24
hypercenter#2313D3
hypocenterC3#34187
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2313

Complements

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

NH ≅ G/Ntype
4. #74964C2semidirect
5. #74963C2semidirect
20. C2×#887C4semidirect
413. C3#34187semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C3×#23132
#821552
#749642
#749632
#740532
#740512
#738672
#738642
#734372
#734352
#341873