Finite Group Databasebeta

← #10691212

Subgroups — #10691212

Order 768.

478 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 105 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#83261C211yesmaximal, characteristic
2#79039C211yesmaximal, characteristic
3C3×#1546C211yesmaximal, characteristic, complemented
4#71773C211yesmaximal, characteristic, complemented
5#70782C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9196—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—21no—
11C2×#872C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5229—21no—
15#5126—21no—
16Dic3×M16—21no—
17#4992—21no—
18#4992C412yes—
19#4983—21no—
20#4983C412yes—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4840—22no—
25#4840—22no—
26C3×#332—22no—
27#4547—22no—
28#4469—21no—
29#4468—21no—
30#2921—31no—
31#2621—31no—
32#1546—31no—
33#1546D311yescharacteristic, complemented
34#1458—31no—
35#1286—31no—
36#1286—31no—
37C22×C3×Q8—21no—
38C22×C4×D3D411yescharacteristic
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4C2×C412yes—
46C2×C3×C42C2311yescharacteristic
47#946—42no—
48#946—42no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4C2×C411yescharacteristic
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—22no—
54C2×Dic3⋊C4C2×C411yescharacteristic
55C2×Dic3⋊C4D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×#208—21no—
58#872—21no—
59#872—21no—
60#872—21no—
61#872—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8D412yes—
65C3×C22⋊C8—21no—
66C3×C4×C8—21no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95C2×C412yes—
71#830—21no—
72(C2×D3)⋊C8—24no—
73#815—22no—
74C8×Dic3—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C22⋊Q8—61no—
81C2×#117—31no—
82#433—62no—
83#433—62no—
84#431—31no—
85#431C2×D311yescharacteristic
86#423—31no—
87#423C2×D311yescharacteristic
88#407—61no—
89C4×M16—61no—
90#398—32no—
91#398—61no—
92#397—32no—
93#397—61no—
94#391—61no—
95#382—31no—
96#332—22no—
97#332—62no—
98#324—61no—
99#323—61no—
100C23×D3C22⋊C411yescharacteristic
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C22×C3×C4—22no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3C22⋊C411yescharacteristic
113C22×Dic3C22⋊C411yescharacteristic
114C22×Dic3C22⋊C411yescharacteristic
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C3×M16—41no—
122C3×M16—41no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—21no—
136(C2×D3)⋊C4—21no—
137(C2×D3)⋊C4—44no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140Dic3⋊C4—21no—
141Dic3⋊C4—21no—
142Dic3⋊C4—21no—
143Dic3⋊C4—21no—
144Dic3⋊C4—44no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×Q8—21no—
152C23×C4—31no—
153C2×M16—21no—
154C2×M16—61no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C22⋊Q8—122no—
160C22⋊Q8—122no—
161#117—61no—
162#117—61no—
163#117—61no—
164#117—61no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4C4×D312yes—
171C2×C22⋊C4—31no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—62no—
174C2×C42—31no—
175C2×C42C22×D311yescharacteristic
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—32no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8#6712yes—
182C22⋊C8—21no—
183C22⋊C8—64no—
184C22⋊C8—61no—
185#97—62no—
186C4×C8—21no—
187C4×C8—62no—
188C4×C8—61no—
189#95—32no—
190#95—62no—
191#95—62no—
192#95—61no—
193C23×C3C2×C22⋊C411yescharacteristic
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—41no—
198C22×D3—41no—
199C3×Q8—81no—
200C3×Q8—81no—
201C3×Q8—81no—
202C3×Q8—81no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C4×D3—41no—
229C4×D3—41no—
230C4×D3—41no—
231C4×D3—41no—
232C3×C8—42no—
233C3×C8—81no—
234C3⋊C8—42no—
235C3⋊C8—81no—
236C24—31no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C22×C4—22no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—62no—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257M16—41no—
258M16—41no—
259M16—121no—
260M16—121no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—62no—
264C2×C8—121no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—124no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—124no—
281C42—61no—
282C42—61no—
283C42#23511yescharacteristic
284C42D3×D411yescharacteristic
285C42#23711yescharacteristic
286C42C2×#6711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#38611yescharacteristic
290C22×C3#41611yescharacteristic
291C22×C3#41811yescharacteristic
292C2×D3—42no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4#44911yescharacteristic
303C3×C4#44911yescharacteristic
304C3×C4#48211yescharacteristic
305C3×C4#48211yescharacteristic
306Dic3—41no—
307Dic3—41no—
308Dic3—41no—
309Dic3—82no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—121no—
314C23—121no—
315C23D3×C22⋊C411yescharacteristic
316Q8—81no—
317Q8—81no—
318Q8—81no—
319Q8—81no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4#67⋊C411yescharacteristic
340C2×C4#88311yescharacteristic
341C2×C4#88411yescharacteristic
342C2×C4#91111yescharacteristic
343C2×C4#91211yescharacteristic
344C2×C4#91411yescharacteristic
345C2×C4#91611yescharacteristic
346C2×C4C4×#6711yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93811yescharacteristic
349C8—42no—
350C8—81no—
351C8—122no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3#187511yescharacteristic
356C2×C3#190911yescharacteristic
357C2×C3#191511yescharacteristic
358D3—81no—
359C22—21no—
360C22—21no—
361C22—122no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—241no—
366C22#477011yescharacteristic
367C22#477211yescharacteristic
368C22#495611yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4—242no—
378C4#477911yescharacteristic
379C4#478411yescharacteristic
380C4#515811yescharacteristic
381C4#517211yescharacteristic
382C3#919611yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—241no—
386C2#7012411yescharacteristic
387C2#7020111yescharacteristic
388C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9196
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
3. C3×#1546C2semidirect
4. #71773C2semidirect
6. #67738C2semidirect
7. #67713C2semidirect
33. #1546D3semidirect
382. C3#9196semidirect

Maximal subgroups

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

subgroupindex
#832612
#790392
C3×#15462
#717732
#707822
#677382
#677132
#91963