Finite Group Databasebeta

← #10696601

Subgroups — #10696601

Order 768.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#82978C211yesmaximal, characteristic
2#82949C211yesmaximal, characteristic
3#78537C211yesmaximal, characteristic, complemented
4C3×#1527C211yesmaximal, characteristic
5#70836C211yesmaximal, characteristic, complemented
6#70824C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10098—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5229—21no—
15#5228—21no—
16#5221—21no—
17#5218—21no—
18#5120—22no—
19#5060—21no—
20#5055—21no—
21#5054—22no—
22#4953—22no—
23#4947—22no—
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26#4703—22no—
27#4681—21no—
28#4681—21no—
29C3×#329—22no—
30#4481—22no—
31#4467—22no—
32#3215—31no—
33#3139—31no—
34#2640—31no—
35#1558—31no—
36#1527D311yescharacteristic
37#1506—31no—
38#1300—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6D412yes—
42C2×C3×M16—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#217—24no—
51C2×#216—22no—
52C2×#214—22no—
53#926—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56#924—21no—
57#924—21no—
58C2×C4×Dic3—21no—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3×C4×C8—21no—
73#835—22no—
74#832—22no—
75C3⋊#95—24no—
76C3⋊#95—21no—
77C3⋊#95D412yes—
78#830—21no—
79#816—24no—
80#816—24no—
81C4⋊(C3⋊C8)D412yes—
82#803—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×#124C2×D311yescharacteristic
86#497—61no—
87#491—62no—
88#489—61no—
89#487—61no—
90#487—61no—
91#464—62no—
92#451—61no—
93#450—61no—
94#432—31no—
95#432C2×D311yescharacteristic
96#419—62no—
97C4×M16C2×D311yescharacteristic
98#397—62no—
99#393—62no—
100#391—61no—
101#390—61no—
102#376—31no—
103#333—62no—
104#329—22no—
105#328—62no—
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—22no—
119C2×Dic6—22no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—22no—
123C2×Dic6—44no—
124C3×M16—42no—
125C2×C3×C8—22no—
126C2×C3×C8—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—41no—
136#217—84no—
137#216—82no—
138#214—82no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—42no—
150C4×Dic3—41no—
151C2×C3⋊C8—22no—
152C22×Q8—32no—
153C22×Q8D1211yescharacteristic
154C22×D4D1211yescharacteristic
155C2×Q16—62no—
156C2×SD16—62no—
157C2×M16—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164#125—61no—
165#124—21no—
166#124—21no—
167#124—21no—
168#124—21no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—64no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181C4⋊C8—21no—
182C4⋊C8—32no—
183#103—64no—
184#103—64no—
185#103—62no—
186#102—62no—
187C22⋊C8—21no—
188C22⋊C8—61no—
189#97—21no—
190#97—61no—
191#97#6712yes—
192C4×C8—21no—
193#95—32no—
194#95—64no—
195#95—61no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3#12011yescharacteristic
201C3×Q8—42no—
202C3×Q8—81no—
203C3×D4—42no—
204C3×D4—81no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—41no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—44no—
223Dic6—44no—
224Dic6—42no—
225Dic6—81no—
226Dic6—82no—
227C3×C8—42no—
228C3×C8—81no—
229C3⋊C8—42no—
230C24—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—42no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8—124no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C22×C4—21no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4C2×D1211yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4D3×D411yescharacteristic
248Q16—242no—
249SD16—242no—
250M16—42no—
251C2×C8—22no—
252C2×C8—41no—
253C2×C8—62no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—121no—
269C22⋊C4—244no—
270C42—122no—
271C42—121no—
272C42#23511yescharacteristic
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C42C2×#6711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—84no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—81no—
282C22×C3#39211yescharacteristic
283C22×C3#45512yes—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—81no—
290C3×C4#45512yes—
291Dic3—82no—
292Dic3—82no—
293Dic3—81no—
294Dic3—81no—
295C23—42no—
296C23—42no—
297C23—41no—
298C23—42no—
299C23#88211yescharacteristic
300Q8—42no—
301Q8—81no—
302Q8—124no—
303Q8—122no—
304Q8—242no—
305Q8—241no—
306D4—42no—
307D4—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—124no—
320C2×C4#88211yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#88411yescharacteristic
323C2×C4#93011yescharacteristic
324C2×C4#93711yescharacteristic
325C2×C4#94011yescharacteristic
326C8—42no—
327C8—81no—
328C8—122no—
329C2×C3—22no—
330C2×C3—161no—
331C2×C3#200411yescharacteristic
332C2×C3#200411yescharacteristic
333C2×C3#200911yescharacteristic
334C22—21no—
335C22—21no—
336C22—84no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#482512yes—
341C22#496011yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—242no—
349C4—241no—
350C4—241no—
351C4—242no—
352C4#474612yes—
353C3#1009811yescharacteristic, complemented, Sylow
354C2—22no—
355C2—161no—
356C2#7008411yescharacteristic
357C2#7038611yescharacteristic
358C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1527D3
hypocenterC3#10098
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
3. #78537C2semidirect
5. #70836C2semidirect
6. #70824C2semidirect
7. #67665C2semidirect
353. C3#10098semidirect

Maximal subgroups

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

subgroupindex
#829782
#829492
#785372
C3×#15272
#708362
#708242
#676652
#100983