Finite Group Databasebeta

← #10691032

Subgroups — #10691032

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#79356C211yesmaximal, characteristic
2#79315C211yesmaximal, characteristic
3#70813C211yesmaximal, characteristic
4#69475C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic
7#67751C211yesmaximal, characteristic
8#9166—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#810—21no—
15#4925C2211yescharacteristic
16C2×#803C2211yescharacteristic
17#4851C411yescharacteristic, complemented
18#4851C411yescharacteristic, complemented
19#4849C411yescharacteristic, complemented
20#4849C411yescharacteristic, complemented
21#4831—21no—
22#4830—21no—
23#4739—21no—
24#4721—21no—
25#4719—21no—
26#4718—21no—
27#4716—21no—
28#4716—21no—
29#4701—22no—
30#4529—21no—
31#4465—22no—
32#4452—21no—
33#4452—21no—
34#3113—31no—
35#2984—31no—
36#1763—31no—
37#1531—31no—
38#1503—31no—
39#1302—31no—
40#1285D311yescharacteristic
41C22×Dic6D411yescharacteristic
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#926—21no—
46C2×C4⋊Dic3D411yescharacteristic
47#924—21no—
48#924—21no—
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52C2×Dic12—21no—
53C2×Dic12—21no—
54#870—21no—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6C2×C411yescharacteristic
63C4×Dic6D411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8C2×C411yescharacteristic
67C3×C4⋊C8C2×C411yescharacteristic
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3×#95—21no—
73#830D411yescharacteristic
74#830D411yescharacteristic
75#817—21no—
76#817—21no—
77#816—21no—
78#816—21no—
79#816—21no—
80#816—21no—
81#815—22no—
82Dic3⋊C8—22no—
83Dic3⋊C8—22no—
84C8×Dic3—22no—
85#810—24no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8—21no—
90#533—31no—
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—61no—
104#432C2×D311yescharacteristic
105#432C2×D311yescharacteristic
106#431—31no—
107#431—61no—
108C2×#103—61no—
109C2×#97—31no—
110C4×C4⋊C4C2×D311yescharacteristic
111#340—61no—
112#330—62no—
113#328—61no—
114#328—61no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×Dic6—22no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—41no—
126C2×Dic6C22⋊C411yescharacteristic
127C2×Dic6C22⋊C411yescharacteristic
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C42—81no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3C22⋊C411yescharacteristic
145C4⋊Dic3C22⋊C411yescharacteristic
146Dic3⋊C4—21no—
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156C4×Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162Dic12—42no—
163Dic12—42no—
164C22×Q8—31no—
165C2×Q16—61no—
166C2×Q16—61no—
167C22×C8—61no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170#125—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173C4×Q8—31no—
174C4×Q8—31no—
175C4×Q8—31no—
176C4×Q8—31no—
177C4×Q8—61no—
178C4×Q8—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181#117—61no—
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—62no—
192C4⋊C8—62no—
193C4⋊C8—61no—
194C4⋊C8C4×D311yescharacteristic
195C4⋊C8C4×D311yescharacteristic
196#103—64no—
197#103—61no—
198#103—61no—
199#103—61no—
200#103—61no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—31no—
204C22⋊C8—31no—
205#97—21no—
206#97—62no—
207#97—61no—
208#97—61no—
209#97D1212yes—
210C4×C8—62no—
211C4×C8—61no—
212#95—21no—
213C23×C3#12111yescharacteristic
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—81no—
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
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237Dic6—21no—
238Dic6—21no—
239Dic6—21no—
240Dic6—42no—
241Dic6—41no—
242Dic6—41no—
243Dic6—81no—
244Dic6#10412yes—
245C3×C8—42no—
246C3×C8—41no—
247C3×C8—41no—
248C3⋊C8—41no—
249C3⋊C8—41no—
250C3⋊C8—42no—
251C2×Q8—31no—
252C2×Q8—31no—
253C2×Q8—62no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—121no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4#23511yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4D3×D411yescharacteristic
264Q16—122no—
265Q16—122no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—62no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—121no—
273C2×C8—121no—
274C4⋊C4—22no—
275C4⋊C4—31no—
276C4⋊C4—31no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C22⋊C4—121no—
290C42—61no—
291C42—61no—
292C42—61no—
293C42—61no—
294C42—81no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42C2×D1211yescharacteristic
298C42D3×D411yescharacteristic
299C42#23911yescharacteristic
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×C4C2×#10411yescharacteristic
312C3×C4#42111yescharacteristic
313Dic3—22no—
314Dic3—41no—
315Dic3—41no—
316Dic3—41no—
317Dic3—81no—
318Dic3—81no—
319C23#88411yescharacteristic
320Q8—32no—
321Q8—61no—
322Q8—61no—
323Q8—61no—
324Q8—122no—
325Q8—121no—
326Q8—121no—
327Q8—241no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—122no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4C4×D1211yescharacteristic
346C2×C4D3×C22⋊C411yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#89311yescharacteristic
349C2×C4#89411yescharacteristic
350C2×C4#89511yescharacteristic
351C8—42no—
352C8—41no—
353C8—41no—
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#466511yescharacteristic
365C22#474811yescharacteristic
366C22#475511yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—62no—
373C4—82no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4—241no—
378C4—241no—
379C4D3×#10411yescharacteristic
380C4#482111yescharacteristic
381C3#916611yescharacteristic, complemented, Sylow
382C2—21no—
383C2—21no—
384C2#6870711yescharacteristic
385C2#6875511yescharacteristic
386C2#6900711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9166
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
17. #4851C4semidirect
18. #4851C4semidirect
19. #4849C4semidirect
20. #4849C4semidirect
381. C3#9166semidirect

Maximal subgroups

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

subgroupindex
#793562
#793152
#708132
#694752
#694712
C3×#12852
#677512
#91663