Finite Group Databasebeta

← #10691540

Subgroups — #10691540

Order 768.

406 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#5063C211yesmaximal, characteristic
2#79315C211yesmaximal, characteristic
3C3×#1528C211yesmaximal, characteristic
4#71737C211yesmaximal, characteristic
5#69857C211yesmaximal, characteristic
6#67755C211yesmaximal, characteristic
7#67668C211yesmaximal, characteristic
8#9246—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5063—21no—
14#5063—21no—
15#5063—21no—
16#5063—21no—
17C2×#810C411yescharacteristic
18C2×#810C411yescharacteristic
19C2×#808C411yescharacteristic
20C2×#807C411yescharacteristic
21#4926C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4831—21no—
25#4830—21no—
26#4719—21no—
27#4718—21no—
28#4701—22no—
29C3×#330—22no—
30#4528—21no—
31#4528—21no—
32#4456—21no—
33#4456—21no—
34C2×#479—31no—
35#2984—31no—
36#1528D311yescharacteristic
37#1491—31no—
38#1470—31no—
39#1290—31no—
40#1283—31no—
41C22×Dic6C2×C411yescharacteristic
42C2×C3×M16—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C22×C3⋊C8C2×C411yescharacteristic
56C22×C3⋊C8C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6D411yescharacteristic
60C4×Dic6D411yescharacteristic
61C4×Dic6D411yescharacteristic
62C4×Dic6D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97D412yes—
67C3×C4×C8—21no—
68#830—21no—
69#830D411yescharacteristic
70#830D411yescharacteristic
71#815—22no—
72Dic3⋊C8—22no—
73Dic3⋊C8—22no—
74C8×Dic3—22no—
75#810—21no—
76#810—21no—
77#810—21no—
78#810—21no—
79#808—21no—
80#808—21no—
81#807—21no—
82#807—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C2×#125C2×D311yescharacteristic
90C2×C4×Q8—31no—
91#479—61no—
92#479—61no—
93#479—61no—
94#479—61no—
95#446—61no—
96#446—62no—
97C8×Q8—61no—
98#432—31no—
99#432—61no—
100#432C2×D311yescharacteristic
101#431—31no—
102#431—61no—
103C2×#107—31no—
104C2×#106—31no—
105C2×C4⋊C8—31no—
106C2×#103—31no—
107C2×#103—31no—
108C4×M16C2×D311yescharacteristic
109#330—22no—
110Q8⋊C8—61no—
111Q8⋊C8—61no—
112#324—61no—
113#324—61no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×Dic6—22no—
122C2×Dic6—21no—
123C2×Dic6—41no—
124C2×Dic6C22⋊C411yescharacteristic
125C2×Dic6C22⋊C411yescharacteristic
126C3×M16—22no—
127C3×M16—41no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142C4⋊Dic3—21no—
143C4⋊Dic3C22⋊C411yescharacteristic
144C4⋊Dic3C22⋊C411yescharacteristic
145Dic3⋊C4—21no—
146Dic3⋊C4—21no—
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149C4×Dic3—21no—
150C4×Dic3—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—31no—
161C2×M16—21no—
162C22×C8—31no—
163C22×C8—31no—
164#125—21no—
165#125—21no—
166#125—21no—
167#125—21no—
168C4×Q8—31no—
169C4×Q8—31no—
170C4×Q8—31no—
171C4×Q8—31no—
172C4×Q8—61no—
173C4×Q8—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4C4×D311yescharacteristic
179C2×C4⋊C4C4×D311yescharacteristic
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—61no—
183#107—61no—
184#106—61no—
185#106—61no—
186C4⋊C8—21no—
187C4⋊C8—62no—
188C4⋊C8—62no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—61no—
194#103—61no—
195#103—61no—
196#103—61no—
197C22⋊C8—21no—
198C22⋊C8—31no—
199C22⋊C8—31no—
200C22⋊C8—61no—
201#97—21no—
202#97—62no—
203#97—61no—
204#97#6712yes—
205C4×C8—21no—
206C4×C8—62no—
207C4×C8—61no—
208C23×C3C4×D411yescharacteristic
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—81no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232Dic6—21no—
233Dic6—21no—
234Dic6—21no—
235Dic6—42no—
236Dic6#10112yes—
237C3×C8—42no—
238C3×C8—42no—
239C3⋊C8—41no—
240C3⋊C8—41no—
241C3⋊C8—41no—
242C3⋊C8—41no—
243C3⋊C8—81no—
244C2×Q8—31no—
245C2×Q8—31no—
246C2×Q8—62no—
247C2×Q8—61no—
248C2×Q8—121no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4C2×C4×D311yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4#23711yescharacteristic
256M16—22no—
257M16—41no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—121no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—31no—
272C4⋊C4—31no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42—61no—
286C42#23511yescharacteristic
287C42D3×D411yescharacteristic
288C42D3×D411yescharacteristic
289C42C2×#6711yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#38611yescharacteristic
293C22×C3#44311yescharacteristic
294C22×C3#44411yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—81no—
301C3×C4—81no—
302C3×C4C2×#10111yescharacteristic
303C3×C4#41311yescharacteristic
304Dic3—22no—
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308C23#67⋊C411yescharacteristic
309Q8—32no—
310Q8—61no—
311Q8—61no—
312Q8—61no—
313Q8—122no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—81no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4D3×C22⋊C411yescharacteristic
332C2×C4#88311yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4C4×#6711yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93811yescharacteristic
337C8—42no—
338C8—42no—
339C8—121no—
340C8—121no—
341C8—121no—
342C8—121no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#187611yescharacteristic
347C2×C3#190111yescharacteristic
348C2×C3#190511yescharacteristic
349C22—21no—
350C22—21no—
351C22#482311yescharacteristic
352C22#482411yescharacteristic
353C22#495611yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—62no—
360C4—81no—
361C4—81no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4D3×#10111yescharacteristic
366C4#475111yescharacteristic
367C3#924611yescharacteristic, complemented, Sylow
368C2—21no—
369C2—21no—
370C2#7011511yescharacteristic
371C2#7039011yescharacteristic
372C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1528C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1528D3
hypocenterC3#9246
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
367. C3#9246semidirect

Maximal subgroups

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

subgroupindex
C2×#50632
#793152
C3×#15282
#717372
#698572
#677552
#676682
#92463