Finite Group Databasebeta

← #10697224

Subgroups — #10697224

Order 768.

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

normal subgroups only

This group has 496 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
1C2×#5031C211yesmaximal, characteristic
2#82431C211yesmaximal, characteristic
3#79271C211yesmaximal, characteristic
4C3×#1461C211yesmaximal, characteristic
5#70778C211yesmaximal, characteristic
6#70768C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic
8#10187—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#216—21no—
11#5470C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#5225—21no—
16#5031—22no—
17#5031—22no—
18#5022—22no—
19#5021—21no—
20#5020—21no—
21C2×#810—21no—
22#4964—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4851—22no—
26#4721—22no—
27#4681—21no—
28#4680—21no—
29C3×#324—22no—
30#4486—22no—
31#4456—22no—
32#3229—31no—
33#3078—31no—
34C2×#462—31no—
35#1546—31no—
36#1481—31no—
37#1461D311yescharacteristic
38#1301—31no—
39C22×C3×Q8D411yescharacteristic
40C22×Dic6—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—24no—
50C2×#216—42no—
51C2×#216—42no—
52#926—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55#924—21no—
56#924—21no—
57C2×#208—21no—
58C22×C3⋊C8—21no—
59#870—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8D412yes—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74#835—22no—
75#835—22no—
76#817—24no—
77#816—24no—
78#810—24no—
79#808—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4⋊(C3⋊C8)D412yes—
85C22×Q16—61no—
86#533—31no—
87C2×C4⋊Q8—31no—
88C2×C4×Q8C2×D311yescharacteristic
89#498—61no—
90#487—61no—
91#483—62no—
92#479—62no—
93#477—61no—
94#475—62no—
95#475—61no—
96#462—62no—
97#462—62no—
98#431C2×D311yescharacteristic
99#431C2×D311yescharacteristic
100#428—61no—
101#423—31no—
102C2×C4⋊C8—31no—
103#419—61no—
104C2×#103—61no—
105#332—62no—
106Q8⋊C8—62no—
107#324—22no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8SD1612yes—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—41no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×C3×C8—22no—
129C2×C3×C8—41no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#217—41no—
141#216—48no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—41no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8—61no—
160C22×Q8D1211yescharacteristic
161C2×Q16—64no—
162C2×Q16—122no—
163C2×Q16—122no—
164C2×M16—61no—
165C22×C8—61no—
166C4⋊Q8—32no—
167C4⋊Q8—31no—
168C4⋊Q8—31no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172#125—61no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—61no—
180C4×Q8—61no—
181#117—61no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4D1211yescharacteristic
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#106—62no—
190#106—64no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—32no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—64no—
198#103—64no—
199#103—62no—
200#103—62no—
201C22⋊C8—21no—
202C22⋊C8#6712yes—
203C4×C8—21no—
204C4×C8—21no—
205C23×C3#12111yescharacteristic
206C3×Q8—22no—
207C3×Q8—22no—
208C3×Q8—41no—
209C3×Q8—41no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×C3×C4C2×SD1611yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4#13711yescharacteristic
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232Dic6—42no—
233Dic6—42no—
234Dic6—42no—
235Dic6—42no—
236Dic6—81no—
237Dic6—81no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—62no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8#20412yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—121no—
259C22×C4C2×D1211yescharacteristic
260C22×C4#23511yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—128no—
263M16—121no—
264M16—121no—
265C2×C8—22no—
266C2×C8—41no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—121no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—62no—
278C4⋊C4—61no—
279C4⋊C4—62no—
280C4⋊C4—61no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—121no—
286C42—41no—
287C42—41no—
288C42—121no—
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C22×C3—21no—
294C22×C3—21no—
295C22×C3#39211yescharacteristic
296C22×C3#46511yescharacteristic
297C22×C3#46811yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—42no—
302C3×C4—81no—
303C3×C4#44811yescharacteristic
304C3×C4#44911yescharacteristic
305C3×C4#45011yescharacteristic
306C3×C4#45211yescharacteristic
307C3×C4#47312yes—
308Dic3—82no—
309Dic3—81no—
310Dic3—81no—
311C23#93011yescharacteristic
312Q8—22no—
313Q8—22no—
314Q8—41no—
315Q8—41no—
316Q8—122no—
317Q8—122no—
318Q8—122no—
319Q8—122no—
320Q8—241no—
321Q8—241no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—122no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4#88211yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4C2×#20411yescharacteristic
339C2×C4#90411yescharacteristic
340C2×C4#90811yescharacteristic
341C2×C4#90911yescharacteristic
342C2×C4#93711yescharacteristic
343C2×C4#94011yescharacteristic
344C8—42no—
345C8—81no—
346C8—122no—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3#200011yescharacteristic
351C2×C3#200911yescharacteristic
352C2×C3#202011yescharacteristic
353C22—21no—
354C22—21no—
355C22#496011yescharacteristic
356C22#511411yescharacteristic
357C22#513411yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—81no—
363C4—242no—
364C4—241no—
365C4—241no—
366C4#473111yescharacteristic
367C4#473311yescharacteristic
368C4#473411yescharacteristic
369C4#473811yescharacteristic
370C4#475512yes—
371C3#1018711yescharacteristic, complemented, Sylow
372C2—21no—
373C2—21no—
374C2#6996211yescharacteristic
375C2#7005311yescharacteristic
376C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1461C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1461D3
hypocenterC3#10187
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
371. C3#10187semidirect

Maximal subgroups

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

subgroupindex
C2×#50312
#824312
#792712
C3×#14612
#707782
#707682
#677452
#101873