Finite Group Databasebeta

← #10691926

Subgroups — #10691926

Order 768.

492 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 113 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#83228C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4#70790C211yesmaximal, characteristic
5#70513C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9301—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5229—22no—
14#5227—22no—
15C2×#825—21no—
16C2×#825—21no—
17#5085C412yescomplemented
18#5083—22no—
19#5079C412yescomplemented
20#4984—21no—
21#4980—22no—
22Dic3×C4⋊C4—21no—
23#4936C2211yescharacteristic
24#4932C2211yescharacteristic
25#4925C2211yescharacteristic
26C3×#333—22no—
27#4470—22no—
28#3206—31no—
29#2331—31no—
30#1963—31no—
31#1548D311yescharacteristic, complemented
32#1517—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×Q8D412yes—
36C2×#235—21no—
37C2×C3×M16—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8C2×C412yes—
41C3×C4⋊Q8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C42C2311yescharacteristic
45C2×#215—24no—
46#930—21no—
47#930—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—22no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3D411yescharacteristic
52C2×C4×Dic3D411yescharacteristic
53#875—21no—
54#874C2×C411yescharacteristic
55C4×D12—21no—
56C4×D12—21no—
57#869C2×C411yescharacteristic
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C4×C8—21no—
63#835—24no—
64C3⋊#95—22no—
65C3⋊#95—22no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95D412yes—
70#830—21no—
71#825—42no—
72#825—42no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8C2×C412yes—
75#532—31no—
76C2×C4⋊Q8C2×D311yescharacteristic
77#492—32no—
78#487—32no—
79#461—62no—
80#448—62no—
81#431—31no—
82#431C2×D311yescharacteristic
83#423C2×D311yescharacteristic
84C2×#104—61no—
85C2×#104—61no—
86#391—62no—
87#389—61no—
88#387—62no—
89#382—31no—
90C4×C4⋊C4—61no—
91#376—31no—
92#333—22no—
93#327—62no—
94C2×C3×Q8—22no—
95C2×C3×Q8—82no—
96C2×C3×Q8C22⋊C414yes—
97C22×C3×C4—22no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C2×#67—41no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—22no—
105#235—41no—
106#235—41no—
107#235—41no—
108#235—41no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×C4×D3—41no—
113C2×Dic6—21no—
114C3×M16—41no—
115C3×M16—41no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C42C22×C411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125C3×C42#4111yescharacteristic
126#215—84no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—82no—
134C4×Dic3—21no—
135C4×Dic3—21no—
136C4×Dic3—21no—
137C4×Dic3—82no—
138C4×Dic3SD1612yescomplemented
139C2×C3⋊C8—22no—
140C2×#41—61no—
141C22×Q8D1212yes—
142C2×SD16—64no—
143C2×M16—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—31no—
147C4⋊Q8C4×D312yes—
148C4⋊Q8#6712yes—
149#127—31no—
150#124—61no—
151#121—61no—
152#121—61no—
153C4×D4—61no—
154C4×D4—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—62no—
159C2×C42—31no—
160C2×C42—31no—
161C2×C42—62no—
162C2×C42C22×D311yescharacteristic
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—61no—
167#104—122no—
168#104—122no—
169#103—64no—
170C22⋊C8—21no—
171C22⋊C8—61no—
172C4×C8—21no—
173C4×C8—32no—
174#95—32no—
175#95—62no—
176#95—62no—
177#95—61no—
178#95—61no—
179#95—61no—
180C23×C3#12011yescharacteristic
181C22×D3—41no—
182C3×Q8—44no—
183C3×Q8—82no—
184C2×C3×C4—24no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×C3×C4#13711yescharacteristic
199#67—81no—
200#67—81no—
201C2×Dic3—22no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—42no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—42no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—42no—
211D12—42no—
212D12—41no—
213D12—41no—
214D12—81no—
215C4×D3—81no—
216C4×D3—81no—
217Dic6—41no—
218Dic6—41no—
219C3×C8—81no—
220C3×C8—81no—
221C3⋊C8—42no—
222#41—121no—
223#41—121no—
224#41—121no—
225#41—121no—
226C2×Q8—22no—
227C2×Q8—61no—
228C2×Q8—82no—
229C2×Q8(C2×D3)⋊C414yes—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—121no—
234C22×C4—22no—
235C22×C4—61no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242SD16—244no—
243M16—41no—
244M16—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—62no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—242no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C42—32no—
260C42—61no—
261C42—61no—
262C42—61no—
263C42—242no—
264C42C2×C4×D311yescharacteristic
265C42#23711yescharacteristic
266C42#23711yescharacteristic
267C42C2×#6711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#38611yescharacteristic
271C22×C3#44811yescharacteristic
272C22×C3#44911yescharacteristic
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—44no—
281C3×C4C4×SD1611yescharacteristic
282C3×C4#44011yescharacteristic
283C3×C4#46011yescharacteristic
284C3×C4#46411yescharacteristic
285Dic3—42no—
286Dic3—82no—
287Dic3—81no—
288Dic3—81no—
289C23—121no—
290C23#88211yescharacteristic
291Q8—44no—
292Q8—82no—
293Q8—121no—
294Q8—121no—
295D4—122no—
296D4—121no—
297D4—121no—
298D4—241no—
299D4—241no—
300D4—241no—
301C2×C4—24no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—62no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—122no—
310C2×C4—121no—
311C2×C4—122no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—122no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4#67⋊C411yescharacteristic
319C2×C4#67⋊C411yescharacteristic
320C2×C4#88611yescharacteristic
321C2×C4D3×SD1611yescharacteristic
322C2×C4#91411yescharacteristic
323C2×C4#91811yescharacteristic
324C2×C4#91911yescharacteristic
325C2×C4C2×(C2×D3)⋊C411yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4#93911yescharacteristic
328C8—81no—
329C8—81no—
330C8—122no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3#188011yescharacteristic
334C2×C3#190511yescharacteristic
335C2×C3#191111yescharacteristic
336D3—161no—
337C22—21no—
338C22—21no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22#480611yescharacteristic
343C22#481011yescharacteristic
344C22#495811yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—44no—
350C4—122no—
351C4—242no—
352C4—241no—
353C4—241no—
354C4#478811yescharacteristic
355C4#481711yescharacteristic
356C4#516311yescharacteristic
357C4#518511yescharacteristic
358C3#930111yescharacteristic, complemented, Sylow
359C2—21no—
360C2—21no—
361C2—481no—
362C2#7007111yescharacteristic
363C2#7024011yescharacteristic
364C2#7033411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9301
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #83228C2semidirect
3. C3×#1548C2semidirect
6. #67737C2semidirect
7. #67713C2semidirect
17. #5085C4semidirect
19. #5079C4semidirect
31. #1548D3semidirect
138. C4×Dic3SD16semidirect
358. C3#9301semidirect

Maximal subgroups

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

subgroupindex
#832282
#832132
C3×#15482
#707902
#705132
#677372
#677132
#93013