Finite Group Databasebeta

← #10696890

Subgroups — #10696890

Order 768.

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

normal subgroups only

This group has 494 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
1C2×C3×#489C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic, complemented
3#82947C211yesmaximal, characteristic
4#71777C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic, complemented
6#69798C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#10140—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#489—22no—
13C3×#489—22no—
14C2×C3×#103—21no—
15C2×C3×#102—21no—
16C2×C3×#97C2211yescharacteristic
17#5227—21no—
18#5217—21no—
19#5088—21no—
20#5087—22no—
21#5078—21no—
22#5058—21no—
23#5057—22no—
24#5055—21no—
25#4967—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4546—22no—
30#4525—22no—
31#4464—22no—
32#3226—31no—
33#3213—31no—
34C2×#489D311yescharacteristic, complemented
35#1558—31no—
36#1530—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C2×#235—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#215—22no—
58C2×#213—22no—
59#930—21no—
60#930—21no—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#869D411yescharacteristic
66C3×#103—24no—
67C3×#102—24no—
68C3×#97—21no—
69C3×#97D412yes—
70#835—22no—
71#832—22no—
72#830—21no—
73#830—21no—
74#830D412yes—
75#809—24no—
76#807—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124C2×D311yescharacteristic
86#500—61no—
87#496—61no—
88#489—22no—
89#489—22no—
90#489—61no—
91#486—61no—
92#480—62no—
93#474—62no—
94#463—62no—
95#448—61no—
96#447—61no—
97#432—31no—
98#432—31no—
99#431—31no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#97C2×D311yescharacteristic
103#333—62no—
104#329—62no—
105#323—62no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C2×C3×Q8—82no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—42no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—41no—
122#235—82no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×C4×D3—41no—
127C2×Dic6—21no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142#215—82no—
143#213—82no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×#41—61no—
152C22×Q8—21no—
153C22×Q8#6711yescharacteristic
154C22×D4#6711yescharacteristic
155C2×SD16—62no—
156C2×D8—62no—
157C22×C8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—31no—
162C4⋊Q8#6712yes—
163#127—31no—
164#124—21no—
165#124—21no—
166#124—21no—
167#124—21no—
168#124—61no—
169#121—61no—
170#121—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#103—24no—
183#103—62no—
184#102—24no—
185#102—64no—
186#102—62no—
187C22⋊C8—32no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190#97—21no—
191#97—61no—
192#97—61no—
193#97#6712yes—
194C4×C8—61no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—81no—
198C23×C3—81no—
199C23×C3#12111yescharacteristic
200C22×D3—41no—
201C3×Q8—42no—
202C3×Q8—42no—
203C3×Q8—82no—
204C3×Q8—82no—
205C3×D4—42no—
206C3×D4—42no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—82no—
213C2×C3×C4—81no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220#67—82no—
221C2×Dic3—41no—
222D12—42no—
223D12—42no—
224D12—81no—
225C4×D3—82no—
226Dic6—81no—
227C3×C8—42no—
228C3⋊C8—42no—
229C3⋊C8—42no—
230C3⋊C8—81no—
231C24—21no—
232#41—242no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—61no—
239C2×Q8—82no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—42no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—121no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—121no—
250C22×C4D3×D411yescharacteristic
251C22×C4#23711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253SD16—242no—
254D8—242no—
255C2×C8—22no—
256C2×C8—42no—
257C2×C8—62no—
258C2×C8—62no—
259C2×C8—121no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C42C2×#6711yescharacteristic
275C42C2×#6711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3—81no—
281C22×C3—162no—
282C22×C3#39211yescharacteristic
283C22×C3#47212yes—
284C2×D3—81no—
285C2×D3—81no—
286C2×D3—81no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—81no—
293C3×C4—82no—
294C3×C4#45512yes—
295Dic3—81no—
296C23—41no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—121no—
301C23#93911yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—82no—
305Q8—82no—
306Q8—241no—
307D4—42no—
308D4—42no—
309D4—122no—
310D4—122no—
311D4—241no—
312D4—242no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—81no—
320C2×C4—121no—
321C2×C4—242no—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#94011yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—122no—
330C8—122no—
331C8—241no—
332C2×C3—22no—
333C2×C3—161no—
334C2×C3#200411yescharacteristic
335C2×C3#200911yescharacteristic
336C2×C3#201811yescharacteristic
337D3—161no—
338C22—21no—
339C22—21no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—162no—
344C22—241no—
345C22—241no—
346C22—241no—
347C22#519912yes—
348C22#522311yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—81no—
355C4—82no—
356C4—241no—
357C4#506112yes—
358C3#1014011yescharacteristic, complemented, Sylow
359C2—22no—
360C2—161no—
361C2—481no—
362C2#7173111yescharacteristic
363C2#7209511yescharacteristic
364C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#489C2
socleC22×C3#392
prime radical#82947C2
hypercenterC2×#489D3
hypocenterC3#10140
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×C3×#489C2semidirect
2. #83215C2semidirect
4. #71777C2semidirect
5. #71720C2semidirect
6. #69798C2semidirect
7. #67662C2semidirect
34. C2×#489D3semidirect
358. C3#10140semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
#832152
#829472
#717772
#717202
#697982
#676622
#101403