Finite Group Databasebeta

← #10697309

Subgroups — #10697309

Order 768.

486 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 109 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#79313C211yesmaximal, characteristic
2#79037C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4#70792C211yesmaximal, characteristic
5#70459C211yesmaximal, characteristic
6#69868C211yesmaximal, characteristic
7C3×#1283C211yesmaximal, characteristic
8#10199—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5121—22no—
16#5116—22no—
17C4×C4⋊Dic3C2211yescharacteristic
18#4850—21no—
19#4850—21no—
20#4849—21no—
21#4849—21no—
22#4849—22no—
23#4739—22no—
24#4702—22no—
25#4699—21no—
26#4682—21no—
27#4681—21no—
28#4680—21no—
29#4555—22no—
30#4551—22no—
31#4467—22no—
32#3249—31no—
33#3209—31no—
34#3165—31no—
35#1987—31no—
36#1986—31no—
37#1503—31no—
38#1283D311yescharacteristic
39C22×Dic6—21no—
40C2×C3×M16D412yes—
41C22×C3×C8D412yes—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44#926—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48#924—21no—
49#924—21no—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53C2×Dic12—24no—
54C2×Dic12—22no—
55#870—21no—
56#870—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D411yescharacteristic
64#869D412yes—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76C3⋊#95—21no—
77#818—24no—
78#818—22no—
79#817—24no—
80#816—24no—
81#816—22no—
82#816—22no—
83#533—31no—
84#533—31no—
85C2×C4⋊Q8—31no—
86#501—62no—
87#500—61no—
88#498—61no—
89#494—61no—
90#487—61no—
91#477—61no—
92#477—61no—
93#467—62no—
94#462—62no—
95#462—61no—
96#462—61no—
97#451—62no—
98#431C2×D311yescharacteristic
99#428—62no—
100C2×C4⋊C8C2×D311yescharacteristic
101C4×M16C2×D311yescharacteristic
102C4×C4⋊C4—31no—
103#340—62no—
104#339—62no—
105#328—62no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×Dic3—41no—
110C22×Dic3—41no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—22no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—41no—
117C2×Dic6—82no—
118C3×M16—22no—
119C3×M16Q1614yescomplemented
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—42no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128#217—41no—
129#217—41no—
130C4⋊Dic3—22no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148C4×Dic3—41no—
149C4×Dic3—41no—
150C4×Dic3—42no—
151Dic12—84no—
152Dic12—82no—
153C22×Q8—61no—
154C2×Q16—64no—
155C2×Q16—62no—
156C2×M16D1212yes—
157C22×C8D1212yes—
158C4⋊Q8—32no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166#125—61no—
167#125—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176#117—61no—
177#117—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—64no—
184#107—62no—
185#106—64no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189#103—64no—
190#103—62no—
191#103—62no—
192C22⋊C8D1212yes—
193#97—21no—
194C4×C8—21no—
195C4×C8—21no—
196#95—61no—
197C23×C3#12711yescharacteristic
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4C2×Q1612yes—
208C2×C3×C4#13712yes—
209C2×Dic3—42no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—82no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218Dic6—42no—
219Dic6—44no—
220Dic6—41no—
221Dic6—41no—
222Dic6—41no—
223Dic6—41no—
224Dic6—82no—
225Dic6—81no—
226C3×C8—24no—
227C3×C8—42no—
228C3×C8—42no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—62no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—121no—
235C2×Q8—242no—
236C22×C4—121no—
237C22×C4—121no—
238C22×C4C2×D1211yescharacteristic
239C22×C4C2×D1211yescharacteristic
240C22×C4C2×D1211yescharacteristic
241Q16—244no—
242Q16—242no—
243M16—22no—
244M16Dic1214yescomplemented
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—42no—
249C4⋊C4—62no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—62no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—121no—
270C42—121no—
271C42—122no—
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C42#23911yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#39211yescharacteristic
279C22×C3#49411yescharacteristic
280C22×C3#49711yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4#45112yes—
285C3×C4#45511yescharacteristic
286C3×C4#45611yescharacteristic
287C3×C4#46212yes—
288Dic3—82no—
289Dic3—81no—
290Dic3—81no—
291Dic3—82no—
292Dic3—81no—
293Dic3—81no—
294C23#87411yescharacteristic
295Q8—122no—
296Q8—124no—
297Q8—121no—
298Q8—121no—
299Q8—121no—
300Q8—121no—
301Q8—241no—
302Q8—242no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—122no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—122no—
312C2×C4—242no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4#88211yescharacteristic
316C2×C4#88211yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#89511yescharacteristic
319C2×C4#89511yescharacteristic
320C2×C4#89511yescharacteristic
321C2×C4C2×Dic1212yes—
322C2×C4#90912yes—
323C8—24no—
324C8—42no—
325C8—42no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#200711yescharacteristic
329C2×C3#200811yescharacteristic
330C2×C3#201611yescharacteristic
331C22—21no—
332C22—21no—
333C22#467111yescharacteristic
334C22#469911yescharacteristic
335C22#471511yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—242no—
340C4—241no—
341C4—241no—
342C4—242no—
343C4—241no—
344C4—241no—
345C4#473912yes—
346C4#474611yescharacteristic
347C4#475211yescharacteristic
348C4#484912yes—
349C3#1019911yescharacteristic, complemented, Sylow
350C2—21no—
351C2—21no—
352C2#6868811yescharacteristic
353C2#6900911yescharacteristic
354C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1283D3
hypocenterC3#10199
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
119. C3×M16Q16semidirect
244. M16Dic12semidirect
349. C3#10199semidirect

Maximal subgroups

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

subgroupindex
#793132
#790372
#785062
#707922
#704592
#698682
C3×#12832
#101993