Finite Group Databasebeta

← #10692456

Subgroups — #10692456

Order 768.

430 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 93 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#83214C211yesmaximal, characteristic
2#79016C211yesmaximal, characteristic
3C3×#1987C211yesmaximal, characteristic, complemented
4#71776C211yesmaximal, characteristic, complemented
5#69470C211yesmaximal, characteristic
6#67708C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9376—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5479C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5228—21no—
14#5227—21no—
15#5111—21no—
16C8×#67—21no—
17#5090—21no—
18#5086C412yescomplemented
19#5083—21no—
20#5082C412yescomplemented
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4840—21no—
25#4839—21no—
26#4836—21no—
27#4828—21no—
28C3×#340—21no—
29C3×#340—21no—
30#4547—21no—
31#4547—21no—
32#4452—21no—
33#4451—21no—
34#3218—31no—
35#2992—31no—
36#1987D311yescharacteristic, complemented
37#1556—31no—
38#1509—31no—
39#1290—31no—
40#1285—31no—
41C22×C3×Q8—21no—
42C22×C3×C8D411yescharacteristic
43C22×C3×C8D411yescharacteristic
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8C2×C412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—22no—
54C2×#215—22no—
55#930D411yescharacteristic
56#929—21no—
57C4×#67—21no—
58#924D411yescharacteristic
59C2×ΓL(1,𝔽25)—21no—
60C2×D3×C8—21no—
61#876—21no—
62#875C2×C411yescharacteristic
63C4×D12D411yescharacteristic
64#872—21no—
65#870C2×C411yescharacteristic
66C4×Dic6D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×#95—21no—
70#835—22no—
71#835—22no—
72#830—21no—
73#830—21no—
74#830—21no—
75(C2×D3)⋊C8—21no—
76(C2×D3)⋊C8—21no—
77#815—21no—
78Dic3⋊C8—21no—
79Dic3⋊C8—21no—
80C8×Dic3—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D411yescharacteristic
84C4⋊(C3⋊C8)D411yescharacteristic
85#803—21no—
86#803—21no—
87#803C2×C412yes—
88C2×C4⋊Q8C2×D311yescharacteristic
89#529—31no—
90#497—32no—
91#491—32no—
92#462—61no—
93#461—61no—
94#451—61no—
95#448—61no—
96#446—61no—
97#435—61no—
98#435—61no—
99C8×D4—61no—
100#433—61no—
101#432—31no—
102#432—31no—
103#432—31no—
104#424—61no—
105C2×C4⋊C8C2×D311yescharacteristic
106C4×C4⋊C4C2×D311yescharacteristic
107#340—21no—
108#340—21no—
109#332—61no—
110#332—61no—
111#328—61no—
112#327—61no—
113C2×C3×Q8—22no—
114C2×C3×Q8—41no—
115C2×C3×Q8—82no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—21no—
122C2×D12C22⋊C411yescharacteristic
123C2×C4×D3—21no—
124C2×Dic6C22⋊C411yescharacteristic
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42—81no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—21no—
141#216—82no—
142#215—82no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147C4⋊Dic3C22⋊C411yescharacteristic
148C4⋊Dic3C22⋊C411yescharacteristic
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153C4×Dic3—21no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—41no—
158ΓL(1,𝔽25)—81no—
159D3×C8—81no—
160C22×Q8—21no—
161C2×Q16—62no—
162C2×SD16—62no—
163C2×M16—61no—
164C22×C8—61no—
165C22×C8#6711yescharacteristic
166C22×C8#6711yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C4×D312yes—
171#126—61no—
172#125—31no—
173#124—31no—
174#123—61no—
175C22⋊Q8—31no—
176#121—31no—
177C4×Q8—31no—
178C4×D4—31no—
179C4×D4—61no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C42—21no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—31no—
189C4⋊C8—31no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194#103—62no—
195#103—62no—
196C22⋊C8—61no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#97—32no—
202#97—61no—
203#97—61no—
204#97—61no—
205C4×C8—61no—
206#95—21no—
207C23×C3#12111yescharacteristic
208C22×D3—21no—
209C3×Q8—42no—
210C3×Q8—42no—
211C3×Q8—82no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4—82no—
221C2×C3×C4—81no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12311yescharacteristic
228#67—41no—
229#67—41no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233D12—41no—
234C4×D3—41no—
235C4×D3—41no—
236Dic6—41no—
237C3×C8—41no—
238C3×C8—41no—
239C3⋊C8—42no—
240C3⋊C8—41no—
241C3⋊C8—41no—
242C3⋊C8—81no—
243C2×Q8—22no—
244C2×Q8—31no—
245C2×Q8—41no—
246C2×Q8—82no—
247C2×D4—31no—
248C2×D4—61no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—61no—
252C22×C4D3×D411yescharacteristic
253C22×C4#23711yescharacteristic
254C22×C4C2×#6711yescharacteristic
255Q16—242no—
256SD16—242no—
257M16—241no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—121no—
266C2×C8—241no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C42—61no—
285C42—81no—
286C42C2×C4×D311yescharacteristic
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#38611yescharacteristic
293C22×C3#47111yescharacteristic
294C22×C3#47311yescharacteristic
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4—82no—
305C3×C4#45511yescharacteristic
306C3×C4#45611yescharacteristic
307Dic3—41no—
308Dic3—41no—
309Dic3—41no—
310C23—61no—
311C23#93811yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—82no—
315Q8—121no—
316D4—121no—
317D4—121no—
318D4—121no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4D3×C22⋊C411yescharacteristic
335C2×C4#67⋊C411yescharacteristic
336C2×C4#88311yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4C4×#6711yescharacteristic
339C2×C4#93711yescharacteristic
340C8—41no—
341C8—41no—
342C8—122no—
343C8—121no—
344C8—121no—
345C8—241no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#188211yescharacteristic
349C2×C3#190811yescharacteristic
350C2×C3#191011yescharacteristic
351D3—81no—
352C22—21no—
353C22—21no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22#495611yescharacteristic
358C22#517711yescharacteristic
359C22#519211yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—82no—
367C4—121no—
368C4—121no—
369C4—121no—
370C4#506111yescharacteristic
371C4#509511yescharacteristic
372C3#937611yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—241no—
376C2#6969311yescharacteristic
377C2#7028611yescharacteristic
378C2#7033011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1987C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1987D3
hypocenterC3#9376
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. C3×#1987C2semidirect
4. #71776C2semidirect
6. #67708C2semidirect
7. #67621C2semidirect
18. #5086C4semidirect
20. #5082C4semidirect
36. #1987D3semidirect
372. C3#9376semidirect

Maximal subgroups

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

subgroupindex
#832142
#790162
C3×#19872
#717762
#694702
#677082
#676212
#93763