Finite Group Databasebeta

← #10696596

Subgroups — #10696596

Order 768.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3139C211yesmaximal, characteristic, complemented
2#83357C211yesmaximal, characteristic
3#83263C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71777C211yesmaximal, characteristic, complemented
6#69860C211yesmaximal, characteristic
7#67665C211yesmaximal, characteristic, complemented
8#10098—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#875C2211yescharacteristic
12C3×#491—22no—
13C3×#487—21no—
14C3×#487—21no—
15C3×#419—22no—
16C3×C4×M16C2211yescharacteristic
17#5231—22no—
18#5089—21no—
19#5086—21no—
20#5050—21no—
21#5046—21no—
22#5028—22no—
23#4992—22no—
24#4988—21no—
25#4973—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4546—22no—
30#4538—22no—
31#4464—22no—
32#3215—31no—
33#3139D311yescharacteristic, complemented
34#2640—31no—
35#1558—31no—
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8D412yes—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×C4⋊Q8D411yescharacteristic
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—22no—
59C2×#215—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—24no—
63C2×C4×Dic3—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67#875—21no—
68C3×#103—24no—
69C3×#103—24no—
70C3×#97D412yes—
71C3×C4×C8—21no—
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77#810—22no—
78#809—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86#497—61no—
87#491—22no—
88#489—61no—
89#487—21no—
90#487—21no—
91#464—62no—
92#451—61no—
93#450—61no—
94#432—31no—
95#432—31no—
96#419—22no—
97C4×M16C2×D311yescharacteristic
98#397—62no—
99#393—62no—
100#391—61no—
101#390—61no—
102#376—31no—
103#333—62no—
104#329—62no—
105#328—62no—
106C23×D3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—44no—
112C22×C3×C4—22no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—22no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—42no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—42no—
124C3×M16—42no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#216—82no—
143#215—82no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—84no—
149C4×Dic3—42no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C22×Q8—31no—
153C22×Q8#6712yes—
154C22×D4—31no—
155C2×Q16—62no—
156C2×SD16—62no—
157C2×M16—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8#6711yescharacteristic
163C4⋊Q8#6711yescharacteristic
164#125—21no—
165#124—61no—
166#124—61no—
167#124—61no—
168#124—61no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173#117—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—64no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181C4⋊C8—32no—
182C4⋊C8—61no—
183#103—24no—
184#103—24no—
185#103—62no—
186#102—62no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189#97—61no—
190#97—61no—
191#97#6712yes—
192C4×C8—21no—
193#95—32no—
194#95—64no—
195#95—61no—
196C23×C3#12011yescharacteristic
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—41no—
200C22×D3—42no—
201C3×Q8—44no—
202C3×Q8—42no—
203C3×Q8—81no—
204C3×Q8—82no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—44no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223D12—42no—
224D12—81no—
225Dic6—42no—
226Dic6—81no—
227C3×C8—42no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C24—61no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—122no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—122no—
242C22×C4—22no—
243C22×C4—62no—
244C22×C4—61no—
245C22×C4D3×D411yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4C2×#6711yescharacteristic
248Q16—242no—
249SD16—242no—
250M16—42no—
251C2×C8—22no—
252C2×C8—62no—
253C2×C8—121no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—244no—
270C42—41no—
271C42—122no—
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C42C2×#6711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#39211yescharacteristic
279C22×C3#45512yes—
280C2×D3—84no—
281C2×D3—81no—
282C2×D3—81no—
283C2×D3—81no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—82no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4#45512yes—
293Dic3—82no—
294Dic3—81no—
295C23—122no—
296C23—122no—
297C23—121no—
298C23—122no—
299C23#93711yescharacteristic
300Q8—44no—
301Q8—42no—
302Q8—82no—
303Q8—81no—
304Q8—122no—
305Q8—241no—
306D4—122no—
307D4—241no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4#93711yescharacteristic
321C2×C4#93811yescharacteristic
322C2×C4#93911yescharacteristic
323C2×C4#93911yescharacteristic
324C2×C4#94011yescharacteristic
325C2×C4#95311yescharacteristic
326C8—42no—
327C8—122no—
328C8—241no—
329C2×C3—22no—
330C2×C3#200411yescharacteristic
331C2×C3#200411yescharacteristic
332C2×C3#200911yescharacteristic
333D3—161no—
334C22—21no—
335C22—21no—
336C22—244no—
337C22—241no—
338C22—241no—
339C22—241no—
340C22#506112yes—
341C22#522311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—81no—
349C4—81no—
350C4—242no—
351C4—241no—
352C4#520212yes—
353C3#1009811yescharacteristic, complemented, Sylow
354C2—22no—
355C2—481no—
356C2#7173111yescharacteristic
357C2#7210611yescharacteristic
358C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3139C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3139D3
hypocenterC3#10098
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
1. C3×#3139C2semidirect
4. #71836C2semidirect
5. #71777C2semidirect
7. #67665C2semidirect
33. #3139D3semidirect
353. C3#10098semidirect

Maximal subgroups

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

subgroupindex
C3×#31392
#833572
#832632
#718362
#717772
#698602
#676652
#100983