Finite Group Databasebeta

← #10691214

Subgroups — #10691214

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#79039C211yesmaximal, characteristic
2#74193C211yesmaximal, characteristic
3#70782C211yesmaximal, characteristic
4#70778C211yesmaximal, characteristic, complemented
5#70506C211yesmaximal, characteristic, complemented
6C3×#1286C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9196—31nomaximal, Sylow
9C2×#897—21no—
10C2×#872C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5126—21no—
16Dic3×M16—21no—
17#4973—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4840—22no—
21#4840—22no—
22#4674—21no—
23#4674C412yes—
24#4661—21no—
25#4661C412yes—
26#4486—22no—
27#4482—22no—
28#4469—21no—
29#4468—21no—
30#2921—31no—
31#2621—31no—
32#1546—31no—
33#1546—31no—
34#1458—31no—
35#1286—31no—
36#1286D311yescharacteristic, complemented
37C22×C4×D3D411yescharacteristic
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42C2311yescharacteristic
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4C2×C411yescharacteristic
44C2×(C2×D3)⋊C4D411yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3C2×C412yes—
48C2×Dic3⋊C4C2×C411yescharacteristic
49C2×Dic3⋊C4D411yescharacteristic
50C2×C4×Dic3D411yescharacteristic
51C2×#208—21no—
52#897—42no—
53#897—42no—
54#872—21no—
55#872—21no—
56#872—21no—
57#872—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8D412yes—
65C3×C22⋊C8—21no—
66C3×C4×C8—21no—
67C3×#95—21no—
68C3×#95C2×C412yes—
69C3⋊#95—22no—
70C3⋊#95—22no—
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⋊Q8—31no—
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#382C2×D311yescharacteristic
96#332—62no—
97#332—62no—
98#324—61no—
99#323—61no—
100C23×D3C22⋊C411yescharacteristic
101C22×C3×C4—22no—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—22no—
106C22×Dic3C22⋊C411yescharacteristic
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C2×C4×D3—21no—
114C2×C4×D3—21no—
115C2×Dic6—41no—
116C2×Dic6—41no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C3×M16—41no—
122C3×M16—41no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131(C2×D3)⋊C4—21no—
132(C2×D3)⋊C4—21no—
133(C2×D3)⋊C4—21no—
134(C2×D3)⋊C4—21no—
135(C2×D3)⋊C4—44no—
136C4⋊Dic3—22no—
137C4⋊Dic3—22no—
138C4⋊Dic3—44no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141Dic3⋊C4—21no—
142Dic3⋊C4—21no—
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C22×Q8—61no—
152C23×C4—31no—
153C2×M16—21no—
154C2×M16—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
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—32no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4—61no—
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⋊C8D1212yes—
182C22⋊C8—21no—
183C22⋊C8—64no—
184C22⋊C8—61no—
185#97—62no—
186C4×C8—21no—
187C4×C8—62no—
188C4×C8—61no—
189#95—21no—
190#95—62no—
191#95—62no—
192#95C4×D312yes—
193C23×C3C2×C22⋊C411yescharacteristic
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—41no—
198C22×D3—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×C3×C4#13511yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C4×D3—41no—
225C4×D3—41no—
226C4×D3—41no—
227C4×D3—41no—
228Dic6—81no—
229Dic6—81no—
230Dic6—81no—
231Dic6—81no—
232C3×C8—42no—
233C3×C8—81no—
234C3⋊C8—42no—
235C3⋊C8—81no—
236C24—31no—
237C2×Q8—121no—
238C2×Q8—121no—
239C2×Q8—121no—
240C2×Q8—121no—
241C2×Q8—121no—
242C2×Q8—121no—
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—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—62no—
272C4⋊C4—62no—
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—
283C42C2×D1211yescharacteristic
284C42#23511yescharacteristic
285C42D3×D411yescharacteristic
286C42#23911yescharacteristic
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—241no—
317Q8—241no—
318Q8—241no—
319Q8—241no—
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×C4C4×D1211yescharacteristic
340C2×C4#88211yescharacteristic
341C2×C4#88411yescharacteristic
342C2×C4#89311yescharacteristic
343C2×C4#89411yescharacteristic
344C2×C4#89511yescharacteristic
345C2×C4#91111yescharacteristic
346C2×C4#91411yescharacteristic
347C2×C4#91611yescharacteristic
348C2×C4#91911yescharacteristic
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#466511yescharacteristic
367C22#477011yescharacteristic
368C22#480411yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4—242no—
378C4#478111yescharacteristic
379C4#481011yescharacteristic
380C4#486611yescharacteristic
381C4#488511yescharacteristic
382C3#919611yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—241no—
386C2#6871211yescharacteristic
387C2#6888111yescharacteristic
388C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9196
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
4. #70778C2semidirect
5. #70506C2semidirect
6. C3×#1286C2semidirect
7. #67631C2semidirect
36. #1286D3semidirect
382. C3#9196semidirect

Maximal subgroups

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

subgroupindex
#790392
#741932
#707822
#707782
#705062
C3×#12862
#676312
#91963