Finite Group Databasebeta

← #10691034

Subgroups — #10691034

Order 768.

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

normal subgroups only

This group has 436 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
1#83369C211yesmaximal, characteristic
2#79265C211yesmaximal, characteristic
3C3×#1503C211yesmaximal, characteristic
4#71822C211yesmaximal, characteristic
5#69475C211yesmaximal, characteristic
6#67742C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic
8#9166—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5094C411yescharacteristic, complemented
14#5094C411yescharacteristic, complemented
15#5090C411yescharacteristic, complemented
16#5090C411yescharacteristic, complemented
17#5050—21no—
18#5043—21no—
19#5030—21no—
20#5030—21no—
21C2×#810—21no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C2×#803C2211yescharacteristic
25#4828—22no—
26#4719—21no—
27#4718—21no—
28#4679—21no—
29C8×Dic6—21no—
30C3×#328—21no—
31C3×#328—21no—
32#4543—22no—
33#4529—21no—
34#3113—31no—
35#2984—31no—
36#1763—31no—
37#1531—31no—
38#1503D311yescharacteristic
39#1302—31no—
40#1285—31no—
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—21no—
54C2×#216—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C22×C3⋊C8—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6C2×C411yescharacteristic
63C4×Dic6D411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C22⋊C8D411yescharacteristic
67C3×C22⋊C8D411yescharacteristic
68C3×C4×C8—21no—
69C3×#95—21no—
70#835—21no—
71#835—21no—
72#830—21no—
73#830—21no—
74#815—22no—
75Dic3⋊C8—22no—
76Dic3⋊C8—22no—
77C8×Dic3—22no—
78#810—24no—
79#810—21no—
80#810—21no—
81#808—21no—
82#808—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)C2×C411yescharacteristic
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86#803—21no—
87#803—21no—
88#803—21no—
89#803D412yes—
90#533C2×D311yescharacteristic
91C2×C4×Q8—31no—
92#483—61no—
93#475—31no—
94#475—31no—
95#462—31no—
96#462—31no—
97#451—61no—
98#446—61no—
99#446—62no—
100C8×Q8—61no—
101#441—61no—
102#441—61no—
103#432—31no—
104#432—31no—
105#432—61no—
106#431—61no—
107#431C2×D311yescharacteristic
108C2×#103—61no—
109C2×#97—31no—
110C4×C4⋊C4C2×D311yescharacteristic
111#340—61no—
112#330—62no—
113#328—21no—
114#328—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—21no—
123C2×Dic6—22no—
124C2×Dic6—21no—
125C2×Dic6—41no—
126C2×Dic6C22⋊C411yescharacteristic
127C2×Dic6C22⋊C411yescharacteristic
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42—41no—
140C3×C42—81no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#216—42no—
146#216—42no—
147C4⋊Dic3—21no—
148C4⋊Dic3C22⋊C411yescharacteristic
149C4⋊Dic3C22⋊C411yescharacteristic
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C22×Q8—31no—
165C2×Q16—61no—
166C2×Q16—61no—
167C22×C8—61no—
168C4⋊Q8C4×D311yescharacteristic
169C4⋊Q8C4×D311yescharacteristic
170#125—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—31no—
176C4×Q8—31no—
177C4×Q8—31no—
178C4×Q8—31no—
179C4×Q8—61no—
180C4×Q8—61no—
181#117—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
185C2×C42—21no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#106—61no—
189#106—61no—
190C4⋊C8—21no—
191C4⋊C8—31no—
192C4⋊C8—31no—
193C4⋊C8—62no—
194C4⋊C8—62no—
195C4⋊C8—61no—
196#103—61no—
197#103—61no—
198#103—64no—
199#103—61no—
200#103—61no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8#6711yescharacteristic
204C22⋊C8#6711yescharacteristic
205#97—32no—
206#97—62no—
207#97—61no—
208#97—61no—
209#97—61no—
210C4×C8—21no—
211C4×C8—62no—
212#95—21no—
213C23×C3#12111yescharacteristic
214C3×Q8—41no—
215C3×Q8—41no—
216C3×Q8—81no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4C2×C22⋊C411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240Dic6—21no—
241Dic6—21no—
242Dic6—21no—
243Dic6—42no—
244Dic6#10412yes—
245C3×C8—41no—
246C3×C8—41no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249C3⋊C8—41no—
250C3⋊C8—41no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—31no—
254C2×Q8—31no—
255C2×Q8—62no—
256C2×Q8—61no—
257C2×Q8—121no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4D3×D411yescharacteristic
262C22×C4#23711yescharacteristic
263C22×C4C2×#6711yescharacteristic
264Q16—122no—
265Q16—122no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—62no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—62no—
272C2×C8—121no—
273C2×C8—121no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—31no—
280C4⋊C4—31no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C22⋊C4—41no—
290C42—41no—
291C42—61no—
292C42—61no—
293C42—61no—
294C42—61no—
295C42—81no—
296C42C2×C4×D311yescharacteristic
297C42#23511yescharacteristic
298C42D3×D411yescharacteristic
299C42D3×D411yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3#38611yescharacteristic
303C22×C3#47111yescharacteristic
304C22×C3#47311yescharacteristic
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—82no—
311C3×C4—81no—
312C3×C4—81no—
313C3×C4C2×#10411yescharacteristic
314C3×C4#42111yescharacteristic
315Dic3—22no—
316Dic3—41no—
317Dic3—41no—
318Dic3—41no—
319C23#93811yescharacteristic
320Q8—32no—
321Q8—41no—
322Q8—41no—
323Q8—61no—
324Q8—61no—
325Q8—61no—
326Q8—81no—
327Q8—122no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—122no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4D3×C22⋊C411yescharacteristic
346C2×C4#67⋊C411yescharacteristic
347C2×C4#88311yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4C4×#6711yescharacteristic
350C2×C4#93711yescharacteristic
351C8—41no—
352C8—41no—
353C8—122no—
354C8—122no—
355C8—121no—
356C8—121no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3#187111yescharacteristic
360C2×C3#190511yescharacteristic
361C2×C3#190811yescharacteristic
362C22—21no—
363C22—21no—
364C22#495611yescharacteristic
365C22#517711yescharacteristic
366C22#519211yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—62no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4—121no—
377C4—121no—
378C4—121no—
379C4D3×#10411yescharacteristic
380C4#482111yescharacteristic
381C3#916611yescharacteristic, complemented, Sylow
382C2—21no—
383C2—21no—
384C2#7005211yescharacteristic
385C2#7028611yescharacteristic
386C2#7038111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9166
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
13. #5094C4semidirect
14. #5094C4semidirect
15. #5090C4semidirect
16. #5090C4semidirect
381. C3#9166semidirect

Maximal subgroups

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

subgroupindex
#833692
#792652
C3×#15032
#718222
#694752
#677422
#676212
#91663