Finite Group Databasebeta

← #10692455

Subgroups — #10692455

Order 768.

430 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#3218C211yesmaximal, characteristic
2#83130C211yesmaximal, characteristic
3#71779C211yesmaximal, characteristic, complemented
4#71776C211yesmaximal, characteristic, complemented
5#71755C211yesmaximal, characteristic
6#67755C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9376—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C3×#529C2211yescharacteristic
11C3×#497C412yescomplemented
12C3×#491C412yescomplemented
13C3×#462—21no—
14C3×#461—21no—
15C3×#451—21no—
16C3×#448—21no—
17C3×#432C2211yescharacteristic
18#5024—21no—
19#5011—21no—
20#5011—21no—
21D4×C3⋊C8—21no—
22#5000—21no—
23#4999—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4926C2211yescharacteristic
26#4926C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4547—21no—
29#4547—21no—
30#4538—21no—
31#4537—21no—
32#4529—21no—
33#4529—21no—
34#3218D311yescharacteristic
35#2992—31no—
36#1987—31no—
37#1556—31no—
38#1509—31no—
39#1290—31no—
40#1285—31no—
41C22×C3×Q8—21no—
42C2×C3×Q16—22no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8C2×C412yes—
48C3×#126—21no—
49C3×#125C2×C411yescharacteristic
50C3×#124C2×C411yescharacteristic
51C3×#123—21no—
52C3×C22⋊Q8D411yescharacteristic
53C3×#121D411yescharacteristic
54C3×C4×Q8D411yescharacteristic
55C3×C4×D4—21no—
56C3×C4×D4D411yescharacteristic
57C3×#117—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C42C2311yescharacteristic
61C2×C4⋊Dic3—21no—
62C2×C4×Dic3—21no—
63C2×#208—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D411yescharacteristic
66C22×C3⋊C8D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×#103—22no—
69C3×#103—22no—
70C3×C22⋊C8—21no—
71C3×#97C2×C412yes—
72C3⋊#95—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D411yescharacteristic
84#803—21no—
85#803—21no—
86#803—21no—
87C4×C3⋊C8—21no—
88C2×C4⋊Q8C2×D311yescharacteristic
89#529C2×D311yescharacteristic
90#497Dic312yescomplemented
91#491Dic312yescomplemented
92#462—21no—
93#461—21no—
94#451—21no—
95#448—21no—
96#446—61no—
97#435—61no—
98#435—61no—
99C8×D4—61no—
100#433—61no—
101#432—31no—
102#432—31no—
103#432C2×D311yescharacteristic
104#424—61no—
105C2×C4⋊C8—31no—
106C4×C4⋊C4—31no—
107#340—61no—
108#340—61no—
109#332—61no—
110#332—61no—
111#328—61no—
112#327—61no—
113C2×C3×Q8—22no—
114C2×C3×Q8—41no—
115C2×C3×Q8—82no—
116C2×C3×Q8C22⋊C411yescharacteristic
117C2×C3×D4—21no—
118C2×C3×D4C22⋊C411yescharacteristic
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×Dic3—41no—
125C3×Q16—82no—
126C3×SD16—82no—
127C2×C3×C8—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4C22⋊C411yescharacteristic
137C3×C4⋊C4C22⋊C411yescharacteristic
138C3×C22⋊C4—21no—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C42—21no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148C4⋊Dic3—22no—
149C4⋊Dic3—41no—
150C4×Dic3—81no—
151#208—81no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—81no—
160C22×Q8—21no—
161C2×Q16—22no—
162C2×SD16—22no—
163C2×M16—61no—
164C22×C8—31no—
165C22×C8—31no—
166C22×C8—61no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C2×Dic312yes—
171#126—21no—
172#125C2×Dic311yescharacteristic
173#124C2×Dic311yescharacteristic
174#123—21no—
175C22⋊Q8#6711yescharacteristic
176#121#6711yescharacteristic
177C4×Q8#6711yescharacteristic
178C4×D4—21no—
179C4×D4#6711yescharacteristic
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—61no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—31no—
188C4⋊C8—31no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194#103—22no—
195#103—22no—
196C22⋊C8—21no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#97—61no—
202#97—61no—
203#97—61no—
204#97C2×Dic312yes—
205C4×C8—61no—
206#95—61no—
207C23×C3—21no—
208C23×C3#12111yescharacteristic
209C3×Q8—41no—
210C3×Q8—42no—
211C3×Q8—42no—
212C3×Q8—82no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—41no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—82no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×Dic3—42no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237C3×C8—42no—
238C3⋊C8—41no—
239C3⋊C8—41no—
240C3⋊C8—41no—
241C3⋊C8—41no—
242C3⋊C8—81no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—82no—
246C2×Q8#21711yescharacteristic
247C2×D4—21no—
248C2×D4#21711yescharacteristic
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—121no—
252C22×C4D3×D411yescharacteristic
253C22×C4#23711yescharacteristic
254C22×C4C2×#6711yescharacteristic
255Q16—82no—
256SD16—82no—
257M16—241no—
258C2×C8—22no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—241no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—62no—
276C4⋊C4—121no—
277C4⋊C4#21711yescharacteristic
278C4⋊C4#21711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C42—21no—
285C42—241no—
286C42D3×D411yescharacteristic
287C42#23711yescharacteristic
288C42C22×Dic311yescharacteristic
289C42C2×#6711yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3#38611yescharacteristic
296C22×C3#47111yescharacteristic
297C22×C3#47311yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—82no—
307C3×C4#45511yescharacteristic
308C3×C4#45611yescharacteristic
309Dic3—82no—
310C23—21no—
311C23#93911yescharacteristic
312Q8—41no—
313Q8—42no—
314Q8—42no—
315Q8—82no—
316D4—41no—
317D4—41no—
318D4—41no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—122no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4D4×Dic311yescharacteristic
335C2×C4D4×Dic311yescharacteristic
336C2×C4#93511yescharacteristic
337C2×C4#93711yescharacteristic
338C2×C4#93911yescharacteristic
339C2×C4C2×#21711yescharacteristic
340C8—42no—
341C8—121no—
342C8—121no—
343C8—121no—
344C8—121no—
345C8—241no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—81no—
349C2×C3#188211yescharacteristic
350C2×C3#190811yescharacteristic
351C2×C3#191011yescharacteristic
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22#520311yescharacteristic
358C22#520411yescharacteristic
359C22#522011yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—82no—
369C4—242no—
370C4#520211yescharacteristic
371C4#520511yescharacteristic
372C3#937611yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—81no—
376C2#7185411yescharacteristic
377C2#7210011yescharacteristic
378C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3218C2
socleC22×C3#386
prime radicalC3×#121D4
cosocleC2×C3×C42C23
hypercenter#3218D3
hypocenterC3#9376
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3218

Complements

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

NH ≅ G/Ntype
3. #71779C2semidirect
4. #71776C2semidirect
6. #67755C2semidirect
7. #67749C2semidirect
11. C3×#497C4semidirect
12. C3×#491C4semidirect
90. #497Dic3semidirect
91. #491Dic3semidirect
372. C3#9376semidirect

Maximal subgroups

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

subgroupindex
C3×#32182
#831302
#717792
#717762
#717552
#677552
#677492
#93763