Finite Group Databasebeta

← #10955180

Subgroups — #10955180

Order 768.

2220 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic
2C3×Q8×C4⋊C4C211yesmaximal, characteristic
3#82461C212yesmaximal
4C4×Q8×Dic3C211yesmaximal, characteristic
5C4×Q8×Dic3C212yesmaximal
6#75989C216yesmaximal
7#75966C216yesmaximal
8#74325C213yesmaximal
9#74320C213yesmaximal
10#73650C213yesmaximal
11C4×#887C213yesmaximal
12C4×#557—31nomaximal, Sylow
13C2×C3×C4×Q8C2212yes—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×Q8×Dic3C2211yescharacteristic
16C2×Q8×Dic3C2212yes—
17C2×Q8×Dic3C2211yescharacteristic
18#5570C4116yescomplemented
19C2×#890C2216yes—
20C2×#887C2213yes—
21C2×#887C2213yes—
22C2×#869C2213yes—
23C2×C4×Dic6C2213yes—
24C3×#389C2216yes—
25C3×#384C2213yes—
26C3×C4×C4⋊C4C2213yes—
27#4978C22112yes—
28#4977C2216yes—
29Dic3×C4⋊C4C2216yes—
30Dic3×C4⋊C4C2216yes—
31Dic3×C4⋊C4C2213yes—
32Dic3×C4⋊C4C2213yes—
33Dic3×C4⋊C4C2216yes—
34Dic3×C4⋊C4C2213yes—
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2216yes—
37#4972C2216yes—
38#4972C2216yes—
39C4×C4⋊Dic3C2213yes—
40C4×C4⋊Dic3C2216yes—
41C4×Dic3⋊C4C2216yes—
42C42×Dic3C2213yes—
43C42×Dic3C2216yes—
44#4928C2216yes—
45#4653C22112yes—
46#4649C2216yes—
47#4649C2216yes—
48C2×#557—31no—
49#2370—36no—
50#2365—33no—
51#2355—33no—
52#2351—32no—
53Q8×C4⋊C4D311yescharacteristic
54C4×C4⋊Q8—33no—
55C4×#125—36no—
56C42×Q8—33no—
57C42×Q8—32no—
58C42×Q8—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×Dic6C2313yes—
61C3×C4×Q8—28no—
62C3×C4×Q8C2×C418yes—
63C2×C3×C4⋊C4C2316yes—
64C2×C3×C4⋊C4C2313yes—
65C2×C3×C4⋊C4C2316yes—
66C2×C3×C4⋊C4C2313yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42C2316yes—
69C2×C3×C42C2313yes—
70Q8×Dic3—24no—
71Q8×Dic3C2×C4116yes—
72Q8×Dic3Q818yescomplemented
73C2×C4⋊Dic3C2313yes—
74C2×C4⋊Dic3C2316yes—
75C2×C4⋊Dic3C2316yes—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2313yes—
78C2×Dic3⋊C4C2316yes—
79C2×Dic3⋊C4C2316yes—
80C2×Dic3⋊C4C2316yes—
81C2×C4×Dic3C2312yes—
82C2×C4×Dic3C2313yes—
83C2×C4×Dic3C2316yes—
84C2×C4×Dic3C2316yes—
85C2×C4×Dic3C2313yes—
86C2×C4×Dic3C2313yes—
87C2×C4×Dic3C2313yes—
88C2×C4×Dic3C2316yes—
89C2×C4×Dic3C2316yes—
90C2×C4×Dic3C2313yes—
91C2×C4×Dic3C2311yescharacteristic
92#890C2×C4148yes—
93#887—212no—
94#887—212no—
95#869C2×C4124yes—
96C4×Dic6C2×C4124yes—
97C3×#95C2316yes—
98C3⋊#95C2312yes—
99C3⋊#95C2316yes—
100C3⋊#95C2316yes—
101C3⋊#95C23112yes—
102C3⋊#95C23112yes—
103#557—316no—
104C2×C4⋊Q8—33no—
105C2×#125—36no—
106C2×C4×Q8—32no—
107C2×C4×Q8—31no—
108C2×C4×Q8—31no—
109C2×C4×Q8—33no—
110C2×C4×Q8—33no—
111C2×C4×Q8—33no—
112C2×C4×Q8C2×D312yes—
113C2×C4×Q8C2×D311yescharacteristic
114#391—36no—
115#389—36no—
116#389—36no—
117#389—36no—
118#389C2×D316yes—
119#387—312no—
120#387—312no—
121#384C2×D313yes—
122#382—36no—
123C4×C4⋊C4—36no—
124C4×C4⋊C4—33no—
125C4×C4⋊C4—36no—
126C4×C4⋊C4—33no—
127C4×C4⋊C4—36no—
128C4×C4⋊C4—33no—
129C4×C4⋊C4—31no—
130C4×C4⋊C4—36no—
131C4×C4⋊C4—33no—
132C4×C4⋊C4—36no—
133C4×C4⋊C4—36no—
134C4×C4⋊C4—36no—
135C4×C4⋊C4C2×D313yes—
136C43—36no—
137C43—33no—
138C2×C3×Q8C22×C414yes—
139C2×C3×Q8C2×Q814yes—
140C2×C3×Q8#4114yes—
141C22×C3×C4C2416yes—
142C22×C3×C4C2413yes—
143C22×C3×C4C2412yes—
144C22×C3×C4C2413yes—
145C22×C3×C4C2411yescharacteristic
146C22×Dic3C2413yes—
147C22×Dic3C2412yes—
148C22×Dic3C2416yes—
149C22×Dic3C2413yes—
150C22×Dic3C2411yescharacteristic
151C22×Dic3C2411yescharacteristic
152C2×Dic6—26no—
153C2×Dic6—26no—
154C2×Dic6C22×C4112yes—
155C3×C4⋊C4—26no—
156C3×C4⋊C4—212no—
157C3×C4⋊C4—46no—
158C3×C4⋊C4C22×C4112yes—
159C3×C4⋊C4#4114yes—
160C3×C42—212no—
161C3×C42C22×C4112yes—
162C4⋊Dic3—26no—
163C4⋊Dic3C22×C414yes—
164C4⋊Dic3C22×C4124yes—
165C4⋊Dic3C22×C4124yes—
166C4⋊Dic3C2×Q8112yes—
167Dic3⋊C4—212no—
168Dic3⋊C4—212no—
169Dic3⋊C4C22×C4124yes—
170C4×Dic3—212no—
171C4×Dic3—26no—
172C4×Dic3—26no—
173C4×Dic3—212no—
174C4×Dic3—24no—
175C4×Dic3—26no—
176C4×Dic3—43no—
177C4×Dic3—46no—
178C4×Dic3C22×C4124yes—
179C4×Dic3C2×Q8112yes—
180C4×Dic3#4114yes—
181C22×Q8—33no—
182C22×Q8C22×D311yescharacteristic
183C4⋊Q8—324no—
184#125—348no—
185C4×Q8—28no—
186C4×Q8—316no—
187C4×Q8—38no—
188C4×Q8—324no—
189C4×Q8—612no—
190C4×Q8—64no—
191C4×Q8—612no—
192C4×Q8C4×D318yes—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—36no—
195C2×C4⋊C4—36no—
196C2×C4⋊C4—33no—
197C2×C4⋊C4—36no—
198C2×C4⋊C4—33no—
199C2×C4⋊C4—36no—
200C2×C4⋊C4—36no—
201C2×C4⋊C4C22×D313yes—
202C2×C4⋊C4C22×D316yes—
203C2×C4⋊C4C22×D316yes—
204C2×C4⋊C4C22×D313yes—
205C2×C4⋊C4C22×D311yescharacteristic
206C2×C42—31no—
207C2×C42—36no—
208C2×C42—33no—
209C2×C42—36no—
210C2×C42—36no—
211C2×C42—33no—
212C2×C42—33no—
213C2×C42—32no—
214C2×C42—33no—
215C2×C42—36no—
216C2×C42—33no—
217C2×C42C22×D316yes—
218C2×C42C22×D313yes—
219#95—32no—
220#95—36no—
221#95—36no—
222#95—312no—
223#95—312no—
224#95C22×D316yes—
225C23×C3C2511yescharacteristic
226C3×Q8C4×Q8116yescomplemented
227C2×C3×C4—22no—
228C2×C3×C4—23no—
229C2×C3×C4—22no—
230C2×C3×C4—23no—
231C2×C3×C4—26no—
232C2×C3×C4—26no—
233C2×C3×C4—26no—
234C2×C3×C4C23×C416yes—
235C2×C3×C4C23×C416yes—
236C2×C3×C4C23×C412yes—
237C2×C3×C4C22×Q816yes—
238C2×C3×C4C2×#4116yes—
239C2×C3×C4C2×#4114yes—
240C2×C3×C4C2×#4112yes—
241C2×C3×C4C2×#4112yes—
242C2×Dic3—21no—
243C2×Dic3—23no—
244C2×Dic3—26no—
245C2×Dic3—26no—
246C2×Dic3—22no—
247C2×Dic3—22no—
248C2×Dic3—23no—
249C2×Dic3—21no—
250C2×Dic3—23no—
251C2×Dic3—23no—
252C2×Dic3—23no—
253C2×Dic3C23×C414yes—
254C2×Dic3C23×C4112yes—
255C2×Dic3C22×Q816yes—
256C2×Dic3C22×Q812yes—
257C2×Dic3C2×#4112yes—
258C2×Dic3C2×#4112yes—
259C2×Dic32−1+412yes—
260Dic6—224no—
261C2×Q8—312no—
262C2×Q8—66no—
263C2×Q8—66no—
264C2×Q8C2×C4×D314yes—
265C2×Q8#23714yes—
266C2×Q8D3×Q814yes—
267C22×C4—31no—
268C22×C4—33no—
269C22×C4—32no—
270C22×C4—36no—
271C22×C4—31no—
272C22×C4—33no—
273C22×C4C23×D316yes—
274C22×C4C23×D313yes—
275C22×C4C23×D312yes—
276C22×C4C23×D313yes—
277C22×C4C23×D311yescharacteristic
278C4⋊C4—212no—
279C4⋊C4—26no—
280C4⋊C4—34no—
281C4⋊C4—324no—
282C4⋊C4—324no—
283C4⋊C4—324no—
284C4⋊C4—312no—
285C4⋊C4—46no—
286C4⋊C4—612no—
287C4⋊C4—66no—
288C4⋊C4—612no—
289C4⋊C4C2×C4×D3112yes—
290C4⋊C4#23914yes—
291C42—212no—
292C42—324no—
293C42—312no—
294C42—34no—
295C42—612no—
296C42—64no—
297C42—66no—
298C42—66no—
299C42—612no—
300C42—66no—
301C42—123no—
302C42—126no—
303C42C2×C4×D3112yes—
304C22×C3C24×C411yescharacteristic
305C22×C3C23×Q811yescharacteristic
306C22×C3C22×#4111yescharacteristic
307C22×C3C22×#4111yescharacteristic
308C22×C3C22×#4111yescharacteristic
309C22×C3C2×2−1+411yescharacteristic
310C22×C3#58511yescharacteristic
311C3×C4—26no—
312C3×C4—24no—
313C3×C4—46no—
314C3×C4C2×C4×Q8112yes—
315C3×C4C4×#4114yes—
316Dic3—212no—
317Dic3—24no—
318Dic3—26no—
319Dic3—22no—
320Dic3—43no—
321Dic3#55714yescomplemented
322C23C24×D311yescharacteristic
323Q8—624no—
324Q8#887116yescomplemented
325C2×C4—22no—
326C2×C4—23no—
327C2×C4—23no—
328C2×C4—22no—
329C2×C4—26no—
330C2×C4—26no—
331C2×C4—26no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—34no—
336C2×C4—312no—
337C2×C4—36no—
338C2×C4—32no—
339C2×C4—61no—
340C2×C4—63no—
341C2×C4—66no—
342C2×C4—66no—
343C2×C4—62no—
344C2×C4—61no—
345C2×C4—63no—
346C2×C4—62no—
347C2×C4—63no—
348C2×C4—63no—
349C2×C4—63no—
350C2×C4C22×C4×D316yes—
351C2×C4C22×C4×D316yes—
352C2×C4C22×C4×D312yes—
353C2×C4C2×#23716yes—
354C2×C4C2×D3×Q816yes—
355C2×C4C2×#23914yes—
356C2×C4C2×#23912yes—
357C2×C4D3×#4112yes—
358C2×C3C22×C4×Q811yescharacteristic
359C2×C3C2×C4×#4111yescharacteristic
360C2×C3C4×2−1+411yescharacteristic
361C2×C3C2×#55711yescharacteristic
362C2×C3Q8×#4111yescharacteristic
363C2×C3#348011yescharacteristic
364C2×C3#348211yescharacteristic
365C22C23×C4×D311yescharacteristic
366C22C22×#23711yescharacteristic
367C22C22×D3×Q811yescharacteristic
368C22C22×#23911yescharacteristic
369C22C2×D3×#4111yescharacteristic
370C22#596611yescharacteristic
371C22D3×2−1+411yescharacteristic
372C4—26no—
373C4—24no—
374C4—34no—
375C4—46no—
376C4—612no—
377C4—64no—
378C4—62no—
379C4—66no—
380C4—123no—
381C4C2×#887112yes—
382C4C4×#23914yes—
383C3C4×#55711yescharacteristic, complemented, Sylow
384C2C22×#88711yescharacteristic
385C2C2×C4×#23911yescharacteristic
386C2#8588711yescharacteristic
387C2#8674411yescharacteristic
388C2#8676711yescharacteristic
389C2D3×#55711yescharacteristic
390C2Q8×#23911yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×Q8×C4⋊C4C2
socle, prime radical, cosocleC23×C3C25
hypercenterQ8×C4⋊C4D3
hypocenterC3C4×#557
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
18. #5570C4semidirect
72. Q8×Dic3Q8semidirect
226. C3×Q8C4×Q8semidirect
321. Dic3#557semidirect
324. Q8#887semidirect
383. C3C4×#557semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
C3×Q8×C4⋊C42
#824612
C4×Q8×Dic32
#759892
#759662
#743252
#743202
#736502
C4×#8872
C4×#5573