Finite Group Databasebeta

← #10692084

Subgroups — #10692084

Order 768.

612 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 115 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×#3110C211yesmaximal, characteristic
2#82317C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic, complemented
4#71836C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#9323—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#484—21no—
13C3×#474C412yescomplemented
14C3×#461C412yescomplemented
15C3×#450—21no—
16C3×#440—22no—
17C3×#432C2211yescharacteristic
18C4×#217—21no—
19C2×C3⋊#95—21no—
20Dic3×C4⋊C4—21no—
21Dic3×C22⋊C4—22no—
22C4×C4⋊Dic3C2211yescharacteristic
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4538—21no—
27#4538—21no—
28D4⋊(C3⋊C8)—22no—
29#4532—22no—
30#3110D311yescharacteristic
31#2275—31no—
32#1986—31no—
33#1506—31no—
34#1490—31no—
35#1300—31no—
36#1285—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×Dic3—22no—
40C2×C3×SD16—22no—
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4C2×C412yes—
51C3×C4×D4D412yes—
52C3×#117—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#217—22no—
57D4×Dic3—28no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3D412yes—
62C2×#208—21no—
63C3×#107—22no—
64C3×C4⋊C8C2×C412yescomplemented
65C3×#103—22no—
66C3×#102—22no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95D411yescharacteristic
73C3⋊#95D411yescharacteristic
74#830—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77#803—21no—
78C4×C3⋊C8—21no—
79#533C2×D311yescharacteristic
80C2×C4×D4—61no—
81C2×C4×D4C2×D311yescharacteristic
82#484—21no—
83#474Dic312yescomplemented
84#461Dic312yescomplemented
85#450—21no—
86#440—22no—
87#432—31no—
88#432C2×D311yescharacteristic
89C4×M16—31no—
90C4×C4⋊C4—31no—
91C4×C4⋊C4—61no—
92C4×C22⋊C4—61no—
93C4×C22⋊C4—62no—
94#376—31no—
95C2×#95—61no—
96#339—62no—
97#328—61no—
98#328—61no—
99D4⋊C8—62no—
100C24×C3—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×D4—24no—
104C2×C3×D4—21no—
105C2×C3×D4C22⋊C412yes—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—82no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—21no—
116C22×Dic3—28no—
117C22×Dic3—82no—
118C22×Dic3—82no—
119C3×SD16—44no—
120C2×C3×C8—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4C22⋊C412yes—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—22no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137#217—28no—
138C4⋊Dic3—24no—
139C4×Dic3—24no—
140C4×Dic3—42no—
141C4×Dic3—82no—
142#208—22no—
143#208—41no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C22×D4#6711yescharacteristic
147C23×C4—21no—
148C23×C4—62no—
149C2×SD16—22no—
150C2×M16—61no—
151C4⋊Q8C2×Dic311yescharacteristic, complemented
152C4⋊Q8C2×Dic311yescharacteristic, complemented
153#125—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—68no—
161C4×D4C2×Dic312yes—
162C4×D4#6712yes—
163#117—21no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4#6711yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—62no—
168C2×C42—32no—
169C2×C42—61no—
170C2×C42—62no—
171C2×C42C22×D311yescharacteristic
172#107—22no—
173C4⋊C8—61no—
174C4⋊C8C2×Dic312yescomplemented
175#103—22no—
176#102—22no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179#97—21no—
180#97—61no—
181#97—61no—
182C4×C8—61no—
183#95—31no—
184#95—31no—
185#95—62no—
186#95—61no—
187#95—61no—
188C23×C3—24no—
189C23×C3—22no—
190C23×C3—42no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3#12011yescharacteristic
194C3×Q8—42no—
195C3×Q8—81no—
196C3×D4—24no—
197C3×D4—22no—
198C3×D4#10414yes—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—82no—
208C2×C3×C4—82no—
209C2×C3×C4C2×C22⋊C411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—24no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—88no—
225C3×C8—42no—
226C3⋊C8—42no—
227C3⋊C8—81no—
228C24—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4#21712yes—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—68no—
241C22×C4—82no—
242C22×C4—242no—
243C22×C4—242no—
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4C2×#6711yescharacteristic
247SD16—44no—
248M16—62no—
249M16—121no—
250C2×C8—22no—
251C2×C8—121no—
252C2×C8—121no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—64no—
261C4⋊C4#21712yes—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—41no—
265C22⋊C4—68no—
266C42—41no—
267C42—64no—
268C42—122no—
269C42—242no—
270C42#23711yescharacteristic
271C42#23711yescharacteristic
272C42C22×Dic311yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—24no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3#38611yescharacteristic
282C22×C3#45511yescharacteristic
283C22×C3#45611yescharacteristic
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—42no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4C2×#10412yes—
293Dic3—42no—
294Dic3—44no—
295Dic3—42no—
296C23—24no—
297C23—22no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23#93711yescharacteristic
302Q8—42no—
303Q8—81no—
304D4—24no—
305D4—22no—
306D4#83714yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—64no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—121no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—122no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—248no—
327C2×C4D4×Dic311yescharacteristic
328C2×C4D4×Dic311yescharacteristic
329C2×C4#93511yescharacteristic
330C2×C4#93911yescharacteristic
331C2×C4#93911yescharacteristic
332C2×C4C2×#21711yescharacteristic
333C8—42no—
334C8—122no—
335C8—241no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—44no—
339C2×C3#188211yescharacteristic
340C2×C3#189611yescharacteristic
341C2×C3#190411yescharacteristic
342C22—21no—
343C22—21no—
344C22—24no—
345C22—44no—
346C22—44no—
347C22—42no—
348C22—42no—
349C22#520211yescharacteristic
350C22#520511yescharacteristic
351C22#522011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—122no—
361C4—124no—
362C4—122no—
363C4C2×#83712yes—
364C3#932311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—44no—
368C2#7171111yescharacteristic
369C2#7208911yescharacteristic
370C2#7210011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3110C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3110D3
hypocenterC3#9323
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
3. #71841C2semidirect
4. #71836C2semidirect
6. #67750C2semidirect
7. #67748C2semidirect
13. C3×#474C4semidirect
14. C3×#461C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
42. C3×C4⋊Q8C2×C4semidirect
64. C3×C4⋊C8C2×C4semidirect
83. #474Dic3semidirect
84. #461Dic3semidirect
151. C4⋊Q8C2×Dic3semidirect
152. C4⋊Q8C2×Dic3semidirect
174. C4⋊C8C2×Dic3semidirect
364. C3#9323semidirect

Maximal subgroups

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

subgroupindex
C3×#31102
#823172
#718412
#718362
#715992
#677502
#677482
#93233