Finite Group Databasebeta

← #10697253

Subgroups — #10697253

Order 768.

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

normal subgroups only

This group has 464 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#79272C211yesmaximal, characteristic
2#79068C211yesmaximal, characteristic
3C2×#4851C211yesmaximal, characteristic
4#70786C211yesmaximal, characteristic
5#70507C211yesmaximal, characteristic
6#70429C211yesmaximal, characteristic
7C3×#1301C211yesmaximal, characteristic
8#10191—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5121—21no—
16#5120—21no—
17C2×#817—21no—
18C2×#816—21no—
19#4925C2211yescharacteristic
20#4851—22no—
21#4851—21no—
22#4851—22no—
23#4850—22no—
24#4850—21no—
25#4829—22no—
26#4722—22no—
27#4682—21no—
28#4681—21no—
29#4484—22no—
30#4481—22no—
31#4469—22no—
32#3230—31no—
33#3078—31no—
34C2×#475—31no—
35#1548—31no—
36#1481—31no—
37#1461—31no—
38#1301D311yescharacteristic
39C22×Dic6—21no—
40C22×Dic6D411yescharacteristic
41C2×C3×M16—21no—
42C22×C3×C8—21no—
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48#924—21no—
49#924—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52#870—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56#869D411yescharacteristic
57#869D411yescharacteristic
58#869D412yes—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8D412yes—
71C3×C4×C8—21no—
72#830—21no—
73#818—22no—
74#818—24no—
75#817—22no—
76#817—24no—
77#816—22no—
78#816—22no—
79#816—24no—
80#816—24no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×Q8—31no—
86#500—61no—
87#487—61no—
88#484—62no—
89#479—62no—
90#477—61no—
91#477—62no—
92#475—62no—
93#475—61no—
94#475—62no—
95#431—31no—
96#431C2×D311yescharacteristic
97#428—61no—
98C2×#106—61no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8C2×D311yescharacteristic
101#419—61no—
102C2×#103—61no—
103#333—62no—
104Q8⋊C8—62no—
105#324—62no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—41no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—22no—
120C2×Dic6—41no—
121C2×Dic6—82no—
122C2×Dic6Q1612yes—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134#217—41no—
135C4⋊Dic3—22no—
136C4⋊Dic3—21no—
137C4⋊Dic3—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C2×C3⋊C8—41no—
153C22×Q8—31no—
154C22×Q8—61no—
155C2×M16—21no—
156C22×C8—21no—
157C4⋊Q8—32no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163#125—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171C4×Q8—61no—
172#117—61no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42—61no—
178C2×C42C22×D311yescharacteristic
179#107—62no—
180#107—64no—
181#106—64no—
182#106—62no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—61no—
187C4⋊C8D1212yes—
188C4⋊C8D1212yes—
189#103—64no—
190#103—62no—
191#103—62no—
192#103—64no—
193C22⋊C8—61no—
194C22⋊C8D1212yes—
195C4×C8—21no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C2×C3×C4—22no—
199C2×C3×C4—41no—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12711yescharacteristic
206C2×C3×C4C2×Q1611yescharacteristic
207C2×C3×C4#13511yescharacteristic
208C2×C3×C4#13711yescharacteristic
209C2×C3×C4#13711yescharacteristic
210C2×Dic3—22no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—82no—
218C2×Dic3—81no—
219Dic6—22no—
220Dic6—22no—
221Dic6—41no—
222Dic6—41no—
223Dic6—42no—
224Dic6—42no—
225Dic6—82no—
226Dic6—82no—
227C3×C8—42no—
228C3×C8—42no—
229C3×C8—42no—
230C3⋊C8—81no—
231C2×Q8—32no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—62no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—121no—
241C2×Q8—242no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—121no—
245C22×C4C2×D1211yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4#23911yescharacteristic
248M16—41no—
249M16—41no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—121no—
256C4⋊C4—62no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—121no—
271C42—121no—
272C42—121no—
273C42—121no—
274C42C2×D1211yescharacteristic
275C42C2×D1211yescharacteristic
276C42D3×D411yescharacteristic
277C42D3×D411yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#39211yescharacteristic
281C22×C3#46711yescharacteristic
282C22×C3#47011yescharacteristic
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4#45111yescharacteristic
287C3×C4#45111yescharacteristic
288C3×C4#45211yescharacteristic
289C3×C4#45211yescharacteristic
290C3×C4#47112yes—
291Dic3—42no—
292Dic3—81no—
293Dic3—81no—
294Dic3—81no—
295Dic3—82no—
296C23#89511yescharacteristic
297Q8—62no—
298Q8—62no—
299Q8—122no—
300Q8—121no—
301Q8—121no—
302Q8—122no—
303Q8—242no—
304Q8—242no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—62no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—122no—
314C2×C4—242no—
315C2×C4—241no—
316C2×C4#87411yescharacteristic
317C2×C4#88211yescharacteristic
318C2×C4#88211yescharacteristic
319C2×C4#88211yescharacteristic
320C2×C4#89511yescharacteristic
321C2×C4#89511yescharacteristic
322C2×C4#91211yescharacteristic
323C2×C4#91511yescharacteristic
324C2×C4#91511yescharacteristic
325C2×C4D3×Q1611yescharacteristic
326C8—42no—
327C8—42no—
328C8—42no—
329C8—241no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#200111yescharacteristic
333C2×C3#200911yescharacteristic
334C2×C3#201911yescharacteristic
335C22—21no—
336C22—21no—
337C22#467111yescharacteristic
338C22#486711yescharacteristic
339C22#488711yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—122no—
344C4—242no—
345C4—241no—
346C4—241no—
347C4—241no—
348C4#478311yescharacteristic
349C4#478311yescharacteristic
350C4#480911yescharacteristic
351C4#480911yescharacteristic
352C4#489812yes—
353C3#1019111yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#6888811yescharacteristic
357C2#6894311yescharacteristic
358C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1301D3
hypocenterC3#10191
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
353. C3#10191semidirect

Maximal subgroups

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

subgroupindex
#792722
#790682
C2×#48512
#707862
#705072
#704292
C3×#13012
#101913