Finite Group Databasebeta

← #10691929

Subgroups — #10691929

Order 768.

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

normal subgroups only

This group has 492 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
1#79312C211yesmaximal, characteristic
2#74154C211yesmaximal, characteristic, complemented
3#70513C211yesmaximal, characteristic
4#70507C211yesmaximal, characteristic, complemented
5#69545C211yesmaximal, characteristic
6C3×#1286C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9301—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#382C2211yescharacteristic
14C2×#825—21no—
15C2×#825—21no—
16Dic3×C4⋊C4—21no—
17#4972—21no—
18#4932C2211yescharacteristic
19#4925C2211yescharacteristic
20#4841—22no—
21#4738—22no—
22#4693C412yescomplemented
23#4681C412yescomplemented
24#4653—22no—
25#4646—22no—
26#4481—22no—
27#4451—22no—
28#3206—31no—
29#2331—31no—
30#1963—31no—
31#1548—31no—
32#1517—31no—
33#1286D311yescharacteristic, complemented
34#1274—31no—
35C2×#235—21no—
36C22×Dic6D412yes—
37C2×C3×M16—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C42C2311yescharacteristic
40#930—21no—
41#930—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×Dic3⋊C4—22no—
45C2×C4×Dic3—22no—
46C2×C4×Dic3D411yescharacteristic
47C2×C4×Dic3D411yescharacteristic
48C2×#204—24no—
49#875—21no—
50#874C2×C411yescharacteristic
51C4×D12—21no—
52C4×D12—21no—
53#869—21no—
54#869—21no—
55#869C2×C411yescharacteristic
56#869C2×C412yes—
57#869D412yes—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C4×C8C2×C412yes—
63C3×#95—21no—
64C3×#95D412yes—
65C3⋊#95—22no—
66C3⋊#95—22no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69#830—21no—
70#825—42no—
71#825—42no—
72#816—24no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75#532—31no—
76C2×C4⋊Q8—31no—
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#382C2×D311yescharacteristic
90C4×C4⋊C4—61no—
91#376—31no—
92#333—62no—
93#327—62no—
94C22×C3×C4—22no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C2×#67—41no—
99C22×Dic3—21no—
100C22×Dic3—22no—
101C22×Dic3—21no—
102#235—41no—
103#235—41no—
104#235—41no—
105#235—41no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×C4×D3—41no—
110C2×Dic6—21no—
111C2×Dic6—22no—
112C2×Dic6—82no—
113C2×Dic6C22⋊C414yes—
114C3×M16—41no—
115C3×M16—41no—
116C2×C3×C8—22no—
117C2×C3×C8—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C42C22×C411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132Dic3⋊C4—82no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C4×Dic3—21no—
136C4×Dic3—82no—
137C4×Dic3SD1612yescomplemented
138C2×C3⋊C8—41no—
139#204—84no—
140C2×#41—61no—
141C22×Q8—32no—
142C2×SD16—64no—
143C2×M16—21no—
144C4⋊Q8—32no—
145C4⋊Q8—31no—
146C4⋊Q8—32no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
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—61no—
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—61no—
173C4×C8C4×D312yes—
174#95—21no—
175#95—62no—
176#95—62no—
177#95—61no—
178#95—61no—
179#95D1212yes—
180C23×C3#12011yescharacteristic
181C22×D3—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4C2×C22⋊C411yescharacteristic
188C2×C3×C4C4×D411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4C2×SD1611yescharacteristic
194C2×C3×C4#13511yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C2×C3×C4#13711yescharacteristic
197#67—81no—
198#67—81no—
199C2×Dic3—24no—
200C2×Dic3—22no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209D12—42no—
210D12—41no—
211D12—41no—
212D12—81no—
213C4×D3—81no—
214C4×D3—81no—
215Dic6—44no—
216Dic6—41no—
217Dic6—41no—
218Dic6—82no—
219C3×C8—42no—
220C3×C8—81no—
221C3⋊C8—81no—
222#41—121no—
223#41—121no—
224#41—121no—
225#41—121no—
226C2×Q8—34no—
227C2×Q8—62no—
228C2×Q8—61no—
229C2×Q8—242no—
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—22no—
246C2×C8—41no—
247C2×C8—121no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—242no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C42—32no—
260C42—61no—
261C42—61no—
262C42—61no—
263C42—242no—
264C42C2×C4×D311yescharacteristic
265C42#23511yescharacteristic
266C42D3×D411yescharacteristic
267C42#23911yescharacteristic
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—82no—
281C3×C4C4×SD1611yescharacteristic
282C3×C4#44011yescharacteristic
283C3×C4#46011yescharacteristic
284C3×C4#46411yescharacteristic
285Dic3—42no—
286Dic3—44no—
287Dic3—81no—
288Dic3—81no—
289C23—121no—
290C23#88211yescharacteristic
291Q8—124no—
292Q8—121no—
293Q8—121no—
294Q8—242no—
295D4—122no—
296D4—121no—
297D4—121no—
298D4—241no—
299D4—241no—
300D4—241no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—64no—
307C2×C4—62no—
308C2×C4—121no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4C4×D1211yescharacteristic
319C2×C4D3×C22⋊C411yescharacteristic
320C2×C4#88411yescharacteristic
321C2×C4#89311yescharacteristic
322C2×C4#89411yescharacteristic
323C2×C4#89511yescharacteristic
324C2×C4#91111yescharacteristic
325C2×C4D3×SD1611yescharacteristic
326C2×C4#91611yescharacteristic
327C2×C4#91811yescharacteristic
328C8—42no—
329C8—81no—
330C8—241no—
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#466511yescharacteristic
343C22#478111yescharacteristic
344C22#480611yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—82no—
350C4—122no—
351C4—124no—
352C4—241no—
353C4—241no—
354C4#476211yescharacteristic
355C4#479811yescharacteristic
356C4#485211yescharacteristic
357C4#488911yescharacteristic
358C3#930111yescharacteristic, complemented, Sylow
359C2—21no—
360C2—21no—
361C2—481no—
362C2#6870511yescharacteristic
363C2#6890911yescharacteristic
364C2#6903011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9301
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
2. #74154C2semidirect
4. #70507C2semidirect
6. C3×#1286C2semidirect
7. #67737C2semidirect
22. #4693C4semidirect
23. #4681C4semidirect
33. #1286D3semidirect
137. 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
#793122
#741542
#705132
#705072
#695452
C3×#12862
#677372
#93013