Finite Group Databasebeta

← #10693073

Subgroups — #10693073

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#82502C211yesmaximal, characteristic
2#78518C211yesmaximal, characteristic
3C3×#1491C211yesmaximal, characteristic, complemented
4#71623C211yesmaximal, characteristic, complemented
5#70826C211yesmaximal, characteristic
6#67752C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9493—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#872C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5231C411yescharacteristic
14#5230C411yescharacteristic
15#5126—21no—
16#5118—21no—
17#4992C411yescharacteristic
18#4989—21no—
19#4983C411yescharacteristic
20#4978—21no—
21#4940—21no—
22#4937C2211yescharacteristic
23#4933—21no—
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4709—22no—
27#4690—22no—
28C3×Q8⋊C8—22no—
29#4535—22no—
30#4469—21no—
31#4468—21no—
32#2921—31no—
33#2599—31no—
34#1491—31no—
35#1491D311yescharacteristic, complemented
36#1464—31no—
37#1288—31no—
38#1288—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C2×C3×M16—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4C2×C411yescharacteristic
53C2×(C2×D3)⋊C4D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4C2×C411yescharacteristic
57C2×Dic3⋊C4D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C2×#208—21no—
60#872—21no—
61#872—21no—
62#872—21no—
63#872—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⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95C2×C411yescharacteristic
76C3⋊#95C2×C411yescharacteristic
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×Q8C2×D311yescharacteristic
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#383—31no—
103Q8⋊C8—22no—
104Q8⋊C8—62no—
105#324—61no—
106#323—61no—
107C23×D3C22⋊C411yescharacteristic
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—42no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C3×M16—41no—
126C3×M16—41no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42—41no—
134C3×C42—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—42no—
144(C2×D3)⋊C4—42no—
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149Dic3⋊C4—42no—
150Dic3⋊C4—42no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C22×Q8C4×D311yescharacteristic
158C23×C4—31no—
159C2×M16—21no—
160C2×M16—61no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165#117—61no—
166#117—61no—
167#117—61no—
168#117—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4C4×D311yescharacteristic
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C42—21no—
180C2×C42—31no—
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×C8#6712yes—
194#95—31no—
195#95—31no—
196#95—61no—
197#95—61no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202C23×C3C2×C22⋊C411yescharacteristic
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—41no—
207C22×D3—41no—
208C3×Q8—22no—
209C3×Q8—22no—
210C3×Q8—41no—
211C3×Q8—41no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C4×D3—42no—
242C4×D3—42no—
243C3×C8—42no—
244C3×C8—81no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247C24—31no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—42no—
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—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—122no—
282C4⋊C4—122no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—122no—
288C22⋊C4—122no—
289C42—41no—
290C42—41no—
291C42—61no—
292C42—61no—
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C42C2×#6711yescharacteristic
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—42no—
314C3×C4—81no—
315Dic3—41no—
316Dic3—41no—
317Dic3—41no—
318Dic3—81no—
319Dic3—81no—
320C23—61no—
321C23—61no—
322C23—61no—
323C23—121no—
324C23—121no—
325C23D3×C22⋊C411yescharacteristic
326Q8—22no—
327Q8—22no—
328Q8—41no—
329Q8—41no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4#67⋊C411yescharacteristic
356C2×C4#88311yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4C4×#6711yescharacteristic
359C2×C4#93711yescharacteristic
360C2×C4#93811yescharacteristic
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#482111yescharacteristic
379C22#482111yescharacteristic
380C22#495611yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—42no—
388C4—81no—
389C4—121no—
390C4—121no—
391C4—121no—
392C4—241no—
393C4—241no—
394C3#949311yescharacteristic, complemented, Sylow
395C2—21no—
396C2—21no—
397C2—241no—
398C2#6999211yescharacteristic
399C2#7035011yescharacteristic
400C2#7038311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1491C2
socleC22×C3#386
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1491D3
hypocenterC3#9493
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
3. C3×#1491C2semidirect
4. #71623C2semidirect
6. #67752C2semidirect
7. #67666C2semidirect
35. #1491D3semidirect
394. C3#9493semidirect

Maximal subgroups

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

subgroupindex
#825022
#785182
C3×#14912
#716232
#708262
#677522
#676662
#94933