Finite Group Databasebeta

← #10697648

Subgroups — #10697648

Order 768.

500 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 103 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#79068C211yesmaximal, characteristic
3#73722C211yesmaximal, characteristic
4#70766C211yesmaximal, characteristic
5#70507C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7C3×#1303C211yesmaximal, characteristic
8#10252—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5121—21no—
16#5120—21no—
17#4973—22no—
18#4936—21no—
19#4929—21no—
20#4925C2211yescharacteristic
21#4851—21no—
22#4850—21no—
23#4849—22no—
24#4829—22no—
25#4739—22no—
26#4699—21no—
27#4681—21no—
28#4659—22no—
29#4484—22no—
30#4481—22no—
31#4452—22no—
32#3209—31no—
33#3078—31no—
34#2568—31no—
35#1548—31no—
36#1503—31no—
37#1479—31no—
38#1303D311yescharacteristic
39C22×Dic6D411yescharacteristic
40C22×Dic6D412yes—
41C2×C3×M16—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#926—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×Dic12—24no—
54#870—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×#95D412yes—
73C3⋊#95—24no—
74C3⋊#95—24no—
75#830—21no—
76#818—22no—
77#817—22no—
78#816—22no—
79#816—22no—
80#816—24no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×Q8—31no—
86#494—61no—
87#487—61no—
88#479—62no—
89#477—61no—
90#475—61no—
91#462—62no—
92#451—62no—
93#431—31no—
94#431C2×D311yescharacteristic
95#428—61no—
96#423C2×D311yescharacteristic
97#419—61no—
98#400—62no—
99#391—61no—
100#391—62no—
101#382—61no—
102C4×C4⋊C4C2×D311yescharacteristic
103#333—62no—
104#328—62no—
105Q8⋊C8—62no—
106C22×C3×C4—22no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—22no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—22no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—22no—
122C2×Dic6—44no—
123C2×Dic6Q1612yescomplemented
124C3×M16—41no—
125C3×M16—41no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C3×C4⋊C4—42no—
129C3×C42—41no—
130C3×C42—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153C2×C3⋊C8—41no—
154Dic12—84no—
155C22×Q8—31no—
156C22×Q8—32no—
157C2×Q16—64no—
158C2×M16—21no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#125—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174#117—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C42—21no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—61no—
187C4⋊C8D1212yes—
188C4⋊C8D1212yes—
189#103—62no—
190#103—62no—
191#103—64no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C4×C8—21no—
195C4×C8—61no—
196#95—64no—
197#95—64no—
198#95D1212yes—
199C23×C3#12011yescharacteristic
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12711yescharacteristic
211C2×C3×C4C2×Q1611yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4#13711yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×Dic3—22no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—44no—
219C2×Dic3—42no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224Dic6—22no—
225Dic6—22no—
226Dic6—41no—
227Dic6—41no—
228Dic6—42no—
229Dic6—44no—
230Dic6—81no—
231Dic6—82no—
232C3×C8—42no—
233C3×C8—42no—
234C3⋊C8—81no—
235C2×Q8—32no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—62no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—124no—
246C22×C4—22no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—62no—
250C22×C4C2×D1211yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253Q16—244no—
254M16—41no—
255M16—41no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—121no—
259C4⋊C4—42no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C22⋊C4—121no—
275C42—41no—
276C42—41no—
277C42—121no—
278C42—121no—
279C42—121no—
280C42C2×D1211yescharacteristic
281C42C2×D1211yescharacteristic
282C42D3×D411yescharacteristic
283C42#23911yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3#39211yescharacteristic
287C22×C3#45111yescharacteristic
288C22×C3#45211yescharacteristic
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—82no—
293C3×C4#45111yescharacteristic
294C3×C4#45211yescharacteristic
295C3×C4#45512yes—
296C3×C4#46711yescharacteristic
297C3×C4#47011yescharacteristic
298Dic3—42no—
299Dic3—82no—
300Dic3—81no—
301Dic3—81no—
302Dic3—81no—
303C23#88211yescharacteristic
304Q8—62no—
305Q8—62no—
306Q8—121no—
307Q8—121no—
308Q8—122no—
309Q8—124no—
310Q8—242no—
311Q8—241no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—62no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—124no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—122no—
326C2×C4#87411yescharacteristic
327C2×C4#88211yescharacteristic
328C2×C4#88211yescharacteristic
329C2×C4#89511yescharacteristic
330C2×C4#89511yescharacteristic
331C2×C4#89511yescharacteristic
332C2×C4#91211yescharacteristic
333C2×C4#91511yescharacteristic
334C2×C4#91511yescharacteristic
335C2×C4D3×Q1611yescharacteristic
336C8—42no—
337C8—42no—
338C8—241no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#200111yescharacteristic
342C2×C3#200411yescharacteristic
343C2×C3#200911yescharacteristic
344C22—21no—
345C22—21no—
346C22#467111yescharacteristic
347C22#478311yescharacteristic
348C22#480911yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—122no—
354C4—242no—
355C4—241no—
356C4—241no—
357C4—241no—
358C4#478311yescharacteristic
359C4#480911yescharacteristic
360C4#482512yes—
361C4#486711yescharacteristic
362C4#488711yescharacteristic
363C3#1025211yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2#6876211yescharacteristic
367C2#6888811yescharacteristic
368C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1303D3
hypocenterC3#10252
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
123. C2×Dic6Q16semidirect
363. C3#10252semidirect

Maximal subgroups

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

subgroupindex
#793132
#790682
#737222
#707662
#705072
#694712
C3×#13032
#102523