Finite Group Databasebeta

← #10713872

Subgroups — #10713872

Order 768.

420 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76543C211yesmaximal, characteristic
2#70097C211yesmaximal, characteristic
3Dic3×#101C211yesmaximal, characteristic
4C3×#1394C211yesmaximal, characteristic
5#67699C211yesmaximal, characteristic
6#67699C211yesmaximal, characteristic
7#67644C211yesmaximal, characteristic
8#13302—31nomaximal, Sylow
9C2×Q8×Dic3C2211yescharacteristic
10C2×#888C411yescharacteristic
11C2×C3×#101C2211yescharacteristic
12C2×C3×#101C2211yescharacteristic
13C3×#391C2211yescharacteristic
14#5229C2211yescharacteristic
15C2×#836C2211yescharacteristic
16C2×#836C2211yescharacteristic
17#5118—21no—
18Dic3×M16—21no—
19#4975C411yescharacteristic
20#4935C411yescharacteristic
21C4×Dic3⋊C4C411yescharacteristic
22#4656—21no—
23#4656—21no—
24#4647—21no—
25#4647—21no—
26#4554—21no—
27#4554—21no—
28#2567—31no—
29#1801—31no—
30C4×#101—31no—
31#1394—31no—
32#1394—31no—
33#1394—31no—
34#1394D311yescharacteristic
35C22×C3×Q8C2311yescharacteristic
36C22×Dic6—21no—
37C2×C3×M16—21no—
38C2×C3×M16—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C42C2×C411yescharacteristic
41Q8×Dic3—41no—
42Q8×Dic3—41no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×#208—21no—
58C2×#208—21no—
59#888—41no—
60#888—41no—
61C3×#101—21no—
62C3×#101—21no—
63C3×#101—21no—
64C3×#101Q812yescomplemented
65C3×#95—21no—
66C3×#95—21no—
67#836—21no—
68#836—21no—
69#836—21no—
70#836Q812yescomplemented
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74#815—41no—
75Dic3⋊C8—41no—
76Dic3⋊C8—41no—
77C8×Dic3—41no—
78C2×C4⋊Q8—31no—
79C2×C4×Q8—31no—
80#423—61no—
81C2×#101—31no—
82C2×#101—31no—
83C2×#101C2×D311yescharacteristic
84C2×#101C2×D311yescharacteristic
85C4×M16—61no—
86#395—61no—
87#395—61no—
88#391—31no—
89#391C2×D311yescharacteristic
90#389—61no—
91#389—61no—
92#384—31no—
93#384—31no—
94C4×C4⋊C4—31no—
95#343—61no—
96#343—61no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8C2×Q811yescharacteristic
100C2×C3×Q8C2×Q811yescharacteristic
101C2×C3×Q8#4111yescharacteristic
102C2×C3×Q8#4111yescharacteristic
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C22×Dic3C22⋊C411yescharacteristic
113C22×Dic3C22⋊C411yescharacteristic
114C2×Dic6—42no—
115C2×Dic6—81no—
116C2×Dic6—81no—
117C3×M16—22no—
118C3×M16—41no—
119C3×M16—41no—
120C3×M16—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C3×C4⋊C4—41no—
124C3×C42—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—41no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129C4⋊Dic3—81no—
130C4⋊Dic3—81no—
131Dic3⋊C4—24no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—81no—
135Dic3⋊C4—81no—
136C4×Dic3—21no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3—21no—
141C4×Dic3—21no—
142C4×Dic3—41no—
143#208—22no—
144#208—41no—
145#208—41no—
146#208—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C22×Q8—61no—
150C22×Q8C22×D311yescharacteristic
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—61no—
154C2×M16—61no—
155C4⋊Q8—121no—
156C4⋊Q8—121no—
157C4×Q8—121no—
158C4×Q8—121no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4C4×D311yescharacteristic
170C2×C42—31no—
171C2×C42—31no—
172C2×C42—31no—
173C2×C42—31no—
174C2×C42C4×D311yescharacteristic
175C4⋊C8—121no—
176C4⋊C8—121no—
177#101—21no—
178#101—21no—
179#101—21no—
180#101—32no—
181#101—61no—
182#101—61no—
183#101—61no—
184#101Dic612yescomplemented
185#97—121no—
186C4×C8—121no—
187#95—21no—
188#95—21no—
189#95—61no—
190#95—61no—
191#95—61no—
192C23×C3C2×C22⋊C411yescharacteristic
193C3×Q8—41no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4C4×Q811yescharacteristic
210C2×C3×C4C4×Q811yescharacteristic
211C2×C3×C4C22⋊Q811yescharacteristic
212C2×C3×C4C22⋊Q811yescharacteristic
213C2×C3×C4#12411yescharacteristic
214C2×C3×C4C4⋊Q811yescharacteristic
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3#9911yescharacteristic
229C2×Dic3#9911yescharacteristic
230Dic6—84no—
231C3×C8—42no—
232C3×C8—81no—
233C3⋊C8—42no—
234C3⋊C8—81no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—122no—
238C2×Q8—241no—
239C2×Q8—241no—
240C2×Q8C2×Dic611yescharacteristic
241C2×Q8#23511yescharacteristic
242C2×Q8#23711yescharacteristic
243C2×Q8D3×Q811yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255M16—22no—
256M16—41no—
257M16—41no—
258M16—41no—
259M16—62no—
260M16—121no—
261M16—121no—
262M16—121no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—121no—
266C2×C8—121no—
267C4⋊C4—41no—
268C4⋊C4—64no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—241no—
276C4⋊C4—241no—
277C4⋊C4—241no—
278C4⋊C4—241no—
279C42—41no—
280C42—61no—
281C42—61no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42—61no—
286C42—121no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#39111yescharacteristic
290C22×C3C2×#9911yescharacteristic
291C22×C3#41011yescharacteristic
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—42no—
299C3×C4—81no—
300Dic3—21no—
301Dic3—42no—
302Dic3—41no—
303Dic3—41no—
304Dic3—41no—
305Dic3—81no—
306Dic3#35612yes—
307C23D3×C22⋊C411yescharacteristic
308Q8—41no—
309Q8—41no—
310Q8—41no—
311Q8—41no—
312Q8—244no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4C4×Dic611yescharacteristic
341C2×C4#87811yescharacteristic
342C2×C4#88511yescharacteristic
343C2×C4#88711yescharacteristic
344C2×C4#88811yescharacteristic
345C2×C4#89611yescharacteristic
346C8—42no—
347C8—81no—
348C8—122no—
349C8—241no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3#199611yescharacteristic
353C2×C3C2×#35611yescharacteristic
354C2×C3#213411yescharacteristic
355C22—21no—
356C22—21no—
357C22#464611yescharacteristic
358C22#474011yescharacteristic
359C22D3×#9911yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—32no—
367C4—42no—
368C4—61no—
369C4—81no—
370C4—122no—
371C4—121no—
372C4—121no—
373C4—121no—
374C4—241no—
375C3#1330211yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2#6895511yescharacteristic
379C2#6916611yescharacteristic
380C2D3×#35611yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC22×Q8C22×D3
FittingC3×#1394C2
socleC22×C3C2×#99
prime radicalC23×C3C2×C22⋊C4
cosocleC22×C3×Q8C23
hypercenter#1394D3
hypocenterC3#13302
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
64. C3×#101Q8semidirect
70. #836Q8semidirect
184. #101Dic6semidirect
375. C3#13302semidirect

Maximal subgroups

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

subgroupindex
#765432
#700972
Dic3×#1012
C3×#13942
#676992
#676442
#133023