Finite Group Databasebeta

← #10696500

Subgroups — #10696500

Order 768.

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

normal subgroups only

This group has 526 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#82992C211yesmaximal, characteristic
2C2×#5055C211yesmaximal, characteristic
3C2×#4703C211yesmaximal, characteristic, complemented
4C2×C3×#329C211yesmaximal, characteristic
5#69828C211yesmaximal, characteristic, complemented
6C2×#4465C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10083—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14#5230—21no—
15C2×#835—21no—
16#5222—21no—
17C2×#832—21no—
18C2×#816—21no—
19C2×#816—21no—
20#5055—22no—
21#5055—22no—
22#4953—21no—
23#4949—21no—
24#4948—22no—
25#4936C2211yescharacteristic
26C2×#803C2211yescharacteristic
27#4703—22no—
28#4703—22no—
29C3×#329—22no—
30#4551—22no—
31#4465—22no—
32C2×#491—31no—
33C2×#489—31no—
34#2820—31no—
35#1993—31no—
36C2×#330—31no—
37C2×#329D311yescharacteristic
38#1295—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—22no—
52C2×#217—22no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D412yes—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—22no—
61C22×C3⋊C8—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×#97—21no—
71C3×#97—21no—
72C3×#97—21no—
73C3×#97D412yes—
74#835—24no—
75#832—24no—
76C3⋊#95—22no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#816—24no—
80#816—24no—
81#803—21no—
82#803D412yes—
83C2×C4⋊Q8—31no—
84C2×#125—31no—
85C2×#124C2×D311yescharacteristic
86#491—62no—
87#491—62no—
88#489—62no—
89#489—62no—
90C2×#103—61no—
91C2×#103—61no—
92C2×#103—61no—
93C2×#102—61no—
94C2×#97—31no—
95C2×#97C2×D311yescharacteristic
96C2×#97C2×D311yescharacteristic
97#399—61no—
98#397—61no—
99#396—62no—
100#395—61no—
101#394—61no—
102#382—31no—
103#340—62no—
104#330—62no—
105#329—22no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—42no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C22×C3×C4—21no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—22no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—41no—
121C2×Dic6—22no—
122C2×Dic6—44no—
123C2×Dic6—41no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C2×C3×C8—42no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—82no—
138#217—82no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—42no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—82no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155C22×Q8—61no—
156C22×Q8D1211yescharacteristic
157C22×D4D1211yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—61no—
161C4⋊Q8—32no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168#125—61no—
169#124—21no—
170#124—21no—
171#124—21no—
172#124—21no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—62no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—62no—
185C2×C42C22×D311yescharacteristic
186#103—64no—
187#103—64no—
188#103—64no—
189#102—64no—
190#97—21no—
191#97—21no—
192#97—21no—
193#97—32no—
194#97—61no—
195#97#6712yes—
196#95—62no—
197#95—61no—
198#95—61no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3—41no—
204C23×C3#12011yescharacteristic
205C3×Q8—42no—
206C3×Q8—81no—
207C3×D4—42no—
208C3×D4—81no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—82no—
231Dic6—44no—
232Dic6—82no—
233C3×C8—42no—
234C3×C8—42no—
235C3⋊C8—42no—
236C24—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—42no—
240C2×Q8—62no—
241C2×Q8—124no—
242C2×Q8—121no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—42no—
246C22×C4—21no—
247C22×C4—62no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4C2×D1211yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—122no—
261C4⋊C4—62no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—122no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—242no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—242no—
280C22⋊C4—242no—
281C42#23511yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3#39211yescharacteristic
291C22×C3#45412yes—
292C22×C3#45512yes—
293C22×C3#45511yescharacteristic
294C22×C3#45611yescharacteristic
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—81no—
302Dic3—82no—
303Dic3—82no—
304Dic3—81no—
305Dic3—81no—
306C23—41no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23#88211yescharacteristic
312Q8—42no—
313Q8—81no—
314Q8—124no—
315Q8—242no—
316D4—42no—
317D4—81no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—122no—
333C2×C4—242no—
334C2×C4#88211yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#93011yescharacteristic
338C2×C4#93711yescharacteristic
339C2×C4#94011yescharacteristic
340C8—42no—
341C8—42no—
342C8—122no—
343C2×C3—161no—
344C2×C3#166212yes—
345C2×C3#166412yes—
346C2×C3#200311yescharacteristic
347C2×C3#200411yescharacteristic
348C2×C3#200711yescharacteristic
349C22—82no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22#474612yes—
355C22#482511yescharacteristic
356C22#482611yescharacteristic
357C22#482712yes—
358C22#496011yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—242no—
367C4—241no—
368C4—242no—
369C4—241no—
370C3#1008311yescharacteristic, complemented, Sylow
371C2—161no—
372C2#6835812yes—
373C2#6835912yes—
374C2#7008411yescharacteristic
375C2#7038811yescharacteristic
376C2#7039311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#329C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenterC2×#329D3
hypocenterC3#10083
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
3. C2×#4703C2semidirect
5. #69828C2semidirect
6. C2×#4465C2semidirect
7. #67659C2semidirect
370. C3#10083semidirect

Maximal subgroups

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

subgroupindex
#829922
C2×#50552
C2×#47032
C2×C3×#3292
#698282
C2×#44652
#676592
#100833