Finite Group Databasebeta

← #10693075

Subgroups — #10693075

Order 768.

458 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 89 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#78518C211yesmaximal, characteristic
2#74308C211yesmaximal, characteristic
3#70826C211yesmaximal, characteristic
4#70802C211yesmaximal, characteristic, complemented
5#69857C211yesmaximal, characteristic, complemented
6C3×#1288C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9493—31nomaximal, Sylow
9C2×#872C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#383C2211yescharacteristic
14#5126—21no—
15#5118—21no—
16#4926C2211yescharacteristic
17C4×#208C2211yescharacteristic
18#4709—22no—
19#4690—22no—
20#4674C411yescharacteristic
21#4673C411yescharacteristic
22#4670—21no—
23#4668—21no—
24#4661C411yescharacteristic
25#4656C411yescharacteristic
26#4653—21no—
27#4653—21no—
28#4484—22no—
29#4469—21no—
30#4468—21no—
31#4456—22no—
32#2921—31no—
33#2599—31no—
34#1491—31no—
35#1491—31no—
36#1464—31no—
37#1288—31no—
38#1288D311yescharacteristic, complemented
39C22×C4×D3D411yescharacteristic
40C22×Dic6C2×C411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×C42C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4C2×C411yescharacteristic
46C2×(C2×D3)⋊C4D411yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4C2×C411yescharacteristic
52C2×Dic3⋊C4D411yescharacteristic
53C2×C4×Dic3—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56#872—21no—
57#872—21no—
58#872—21no—
59#872—21no—
60C4×Dic6—21no—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#97—21no—
67C3×#97—21no—
68C3×C4×C8D412yes—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95C2×C411yescharacteristic
72C3×#95C2×C411yescharacteristic
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830—21no—
78(C2×D3)⋊C8—24no—
79Dic3⋊C8—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83C4×C3⋊C8D412yes—
84C2×C4×Q8—31no—
85C2×#117—31no—
86#433—62no—
87#433—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90#423—61no—
91#407—61no—
92C4×M16—31no—
93C4×M16C2×D311yescharacteristic
94#398—31no—
95#397—31no—
96#395—31no—
97#393—31no—
98#388—61no—
99#387—61no—
100#387—61no—
101#385—61no—
102#383C2×D311yescharacteristic
103Q8⋊C8—62no—
104Q8⋊C8—62no—
105#324—61no—
106#323—61no—
107C23×D3C22⋊C411yescharacteristic
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3C22⋊C411yescharacteristic
116C22×Dic3C22⋊C411yescharacteristic
117C22×Dic3C22⋊C411yescharacteristic
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—42no—
125C3×M16—41no—
126C3×M16—41no—
127C2×C3×C8—22no—
128C2×C3×C8—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—42no—
138(C2×D3)⋊C4—42no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—42no—
142C4⋊Dic3—42no—
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C22×Q8—31no—
158C23×C4—31no—
159C2×M16—21no—
160C2×M16—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165#117—61no—
166#117—61no—
167#117—61no—
168#117—61no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C42—31no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182C4⋊C8—21no—
183C4⋊C8—64no—
184C4⋊C8—61no—
185C22⋊C8—21no—
186C22⋊C8—64no—
187C22⋊C8—61no—
188#97—21no—
189#97—21no—
190#97—61no—
191#97—61no—
192C4×C8—32no—
193C4×C8D1212yes—
194#95—21no—
195#95—21no—
196#95—61no—
197#95—61no—
198#95—61no—
199#95—61no—
200#95C4×D311yescharacteristic
201#95C4×D311yescharacteristic
202C23×C3C2×C22⋊C411yescharacteristic
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—41no—
207C22×D3—41no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×Dic3—22no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C4×D3—42no—
238C4×D3—42no—
239Dic6—22no—
240Dic6—22no—
241Dic6—41no—
242Dic6—41no—
243C3×C8—42no—
244C3×C8—81no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247C24—31no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—122no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4C2×C4×D311yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265M16—41no—
266M16—41no—
267M16—121no—
268M16—121no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—62no—
272C2×C8—121no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—122no—
280C4⋊C4—122no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—122no—
288C22⋊C4—122no—
289C42—61no—
290C42—61no—
291C42—121no—
292C42—121no—
293C42C2×D1211yescharacteristic
294C42#23511yescharacteristic
295C42D3×D411yescharacteristic
296C42#23911yescharacteristic
297C22×C3—21no—
298C22×C3—21no—
299C22×C3#38611yescharacteristic
300C22×C3#42111yescharacteristic
301C22×C3#42111yescharacteristic
302C2×D3—42no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—81no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—81no—
314C3×C4—81no—
315Dic3—42no—
316Dic3—41no—
317Dic3—41no—
318Dic3—41no—
319Dic3—81no—
320C23—61no—
321C23—61no—
322C23—61no—
323C23—121no—
324C23—121no—
325C23D3×C22⋊C411yescharacteristic
326Q8—62no—
327Q8—62no—
328Q8—121no—
329Q8—121no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—62no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4C4×D1211yescharacteristic
356C2×C4#88211yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4#89311yescharacteristic
359C2×C4#89411yescharacteristic
360C2×C4#89511yescharacteristic
361C8—42no—
362C8—81no—
363C8—122no—
364C8—241no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3#189111yescharacteristic
368C2×C3#189711yescharacteristic
369C2×C3#190911yescharacteristic
370D3—81no—
371C22—21no—
372C22—21no—
373C22—122no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22—241no—
378C22#466511yescharacteristic
379C22#482111yescharacteristic
380C22#482111yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—81no—
388C4—81no—
389C4—121no—
390C4—121no—
391C4—122no—
392C4—121no—
393C4—241no—
394C3#949311yescharacteristic, complemented, Sylow
395C2—21no—
396C2—21no—
397C2—241no—
398C2#6871211yescharacteristic
399C2#6875711yescharacteristic
400C2#6903611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1288D3
hypocenterC3#9493
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
4. #70802C2semidirect
5. #69857C2semidirect
6. C3×#1288C2semidirect
7. #67633C2semidirect
38. #1288D3semidirect
394. C3#9493semidirect

Maximal subgroups

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

subgroupindex
#785182
#743082
#708262
#708022
#698572
C3×#12882
#676332
#94933