Finite Group Databasebeta

← #10692463

Subgroups — #10692463

Order 768.

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

normal subgroups only

This group has 434 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#83218C211yesmaximal, characteristic
2#79016C211yesmaximal, characteristic
3C3×#1981C211yesmaximal, characteristic, complemented
4#71776C211yesmaximal, characteristic, complemented
5#69470C211yesmaximal, characteristic
6#67708C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9377—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5479C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×#103—21no—
13C2×C3×#103—21no—
14C3×C4×C4⋊C4C2211yescharacteristic
15#5111—21no—
16C8×#67—21no—
17#5094—21no—
18#5089C412yescomplemented
19#5087—21no—
20#5081C412yescomplemented
21#4970—21no—
22#4968—21no—
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26#4840—21no—
27#4839—21no—
28#4836—21no—
29#4828—21no—
30#4547—21no—
31#4547—21no—
32#4452—21no—
33#4451—21no—
34#3239—31no—
35#2992—31no—
36#1981D311yescharacteristic, 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
53#930D411yescharacteristic
54#929—21no—
55C4×#67—21no—
56#924D411yescharacteristic
57C2×ΓL(1,𝔽25)—21no—
58C2×D3×C8—21no—
59#876—21no—
60#875C2×C411yescharacteristic
61C4×D12D411yescharacteristic
62#872—21no—
63#870C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×#103—42no—
68C3×#103—42no—
69C3×#95—21no—
70#830—21no—
71#830—21no—
72#830—21no—
73(C2×D3)⋊C8—21no—
74(C2×D3)⋊C8—21no—
75#815—21no—
76Dic3⋊C8—21no—
77Dic3⋊C8—21no—
78C8×Dic3—21no—
79#810—22no—
80#809—22no—
81#808—22no—
82#807—22no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D411yescharacteristic
86C4⋊(C3⋊C8)D411yescharacteristic
87#803—21no—
88#803—21no—
89#803C2×C412yes—
90C2×C4⋊Q8C2×D311yescharacteristic
91#529—31no—
92#501—32no—
93#489—32no—
94#484—61no—
95#481—61no—
96#475—61no—
97#474—61no—
98#446—61no—
99#435—61no—
100#435—61no—
101C8×D4—61no—
102#433—61no—
103#432—31no—
104#432—31no—
105#432—31no—
106#424—61no—
107C2×C4⋊C8C2×D311yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C4×C4⋊C4C2×D311yescharacteristic
111#332—61no—
112#332—61no—
113#328—61no—
114#327—61no—
115C2×C3×Q8—22no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—21no—
127C2×D12C22⋊C411yescharacteristic
128C2×C4×D3—21no—
129C2×Dic6C22⋊C411yescharacteristic
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C42—81no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#217—21no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—21no—
150C4⋊Dic3C22⋊C411yescharacteristic
151C4⋊Dic3C22⋊C411yescharacteristic
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156C4×Dic3—21no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—41no—
161ΓL(1,𝔽25)—81no—
162D3×C8—81no—
163C22×Q8—21no—
164C2×M16—61no—
165C22×C8—61no—
166C22×C8#6711yescharacteristic
167C22×C8#6711yescharacteristic
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C4×D312yes—
172#126—61no—
173#125—31no—
174#124—31no—
175#123—61no—
176C22⋊Q8—31no—
177#121—31no—
178C4×Q8—31no—
179C4×D4—31no—
180C4×D4—61no—
181#117—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C42—21no—
186C2×C42C22×D311yescharacteristic
187#107—62no—
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—31no—
192C4⋊C8—31no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—42no—
198#103—42no—
199#103—62no—
200#102—62no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#97—32no—
207#97—61no—
208#97—61no—
209#97—61no—
210C4×C8—61no—
211#95—21no—
212C23×C3#12111yescharacteristic
213C22×D3—21no—
214C3×Q8—42no—
215C3×Q8—42no—
216C3×Q8—81no—
217C3×Q8—81no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12311yescharacteristic
235#67—41no—
236#67—41no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240D12—41no—
241C4×D3—41no—
242C4×D3—41no—
243Dic6—41no—
244C3×C8—41no—
245C3×C8—41no—
246C3⋊C8—42no—
247C3⋊C8—41no—
248C3⋊C8—41no—
249C3⋊C8—81no—
250C2×Q8—22no—
251C2×Q8—31no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×D4—31no—
258C2×D4—61no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—61no—
262C22×C4D3×D411yescharacteristic
263C22×C4#23711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265M16—241no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—62no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—121no—
274C2×C8—241no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—31no—
279C4⋊C4—31no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C42—61no—
293C42—81no—
294C42C2×C4×D311yescharacteristic
295C42#23511yescharacteristic
296C42D3×D411yescharacteristic
297C42D3×D411yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3#38611yescharacteristic
301C22×C3#47111yescharacteristic
302C22×C3#47311yescharacteristic
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—82no—
312C3×C4—82no—
313C3×C4#45311yescharacteristic
314C3×C4#45411yescharacteristic
315Dic3—41no—
316Dic3—41no—
317Dic3—41no—
318C23—61no—
319C23#93811yescharacteristic
320Q8—42no—
321Q8—42no—
322Q8—81no—
323Q8—81no—
324Q8—121no—
325D4—121no—
326D4—121no—
327D4—121no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4D3×C22⋊C411yescharacteristic
345C2×C4#67⋊C411yescharacteristic
346C2×C4#88311yescharacteristic
347C2×C4#88411yescharacteristic
348C2×C4C4×#6711yescharacteristic
349C2×C4#93711yescharacteristic
350C8—41no—
351C8—41no—
352C8—122no—
353C8—121no—
354C8—121no—
355C8—241no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3#188311yescharacteristic
359C2×C3#190811yescharacteristic
360C2×C3#190911yescharacteristic
361D3—81no—
362C22—21no—
363C22—21no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22#495611yescharacteristic
368C22#517711yescharacteristic
369C22#519211yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—82no—
377C4—121no—
378C4—121no—
379C4—121no—
380C4#506211yescharacteristic
381C4#507711yescharacteristic
382C3#937711yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—241no—
386C2#6966311yescharacteristic
387C2#7028611yescharacteristic
388C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1981D3
hypocenterC3#9377
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. C3×#1981C2semidirect
4. #71776C2semidirect
6. #67708C2semidirect
7. #67621C2semidirect
18. #5089C4semidirect
20. #5081C4semidirect
36. #1981D3semidirect
382. C3#9377semidirect

Maximal subgroups

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

subgroupindex
#832182
#790162
C3×#19812
#717762
#694702
#677082
#676212
#93773