Finite Group Databasebeta

← #10690114

Subgroups — #10690114

Order 768.

662 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 111 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#82957C211yesmaximal, characteristic, complemented
2C2×#5058C211yesmaximal, characteristic
3C2×C3×#329C211yesmaximal, characteristic, complemented
4#69823C212yesmaximal, complemented
5#67659C212yesmaximal, complemented
6#9001—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C22×#215—21no—
9C22×#213—21no—
10C2×#874C2211yescharacteristic
11C2×C3×#97C2212yes—
12#5229—21no—
13#5220—21no—
14C2×#821—22no—
15C2×#821—22no—
16#5058—22no—
17#5058C414yescomplemented
18#4951—22no—
19#4950—22no—
20#4936C2212yes—
21C2×#803C2211yescharacteristic
22C3×#329—22no—
23C2×#496—31no—
24#2379—31no—
25#1961—32no—
26C2×#329D311yescharacteristic, complemented
27#1295—32no—
28C22×C3×Q8D411yescharacteristic
29C22×C3×D4D411yescharacteristic
30C23×Dic3—21no—
31C22×D12—21no—
32C22×D12—22no—
33C22×C3×C8—22no—
34C3×#124—22no—
35C3×#124C2×C412yes—
36C3×#124D412yes—
37C2×C3×C22⋊C4—22no—
38C2×C3×C42C2311yescharacteristic
39C2×#217—22no—
40C2×#217—21no—
41C2×#215—24no—
42C2×#215—42no—
43C2×#215—42no—
44C2×#213—24no—
45C2×#213—42no—
46C2×#213—42no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D412yes—
49C2×C4⋊Dic3D412yes—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52#874—21no—
53#874—22no—
54#874C2×C412yescomplemented
55C3×#97—22no—
56C3×#97—22no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59C3⋊#95—22no—
60#821—44no—
61#821—44no—
62#803—21no—
63#803C2×C412yescomplemented
64C22×SD16—61no—
65C22×D8—61no—
66C2×#127—31no—
67C2×#124C2×D311yescharacteristic
68#496—34no—
69#496—62no—
70C2×#102—62no—
71C2×#102—62no—
72C2×#97—31no—
73C2×#97C2×D312yes—
74#391—61no—
75#388—62no—
76#386—61no—
77#385—62no—
78#382—32no—
79#329—22no—
80C24×C3—21no—
81C23×D3—41no—
82C2×C3×Q8—21no—
83C2×C3×Q8—41no—
84C2×C3×Q8—41no—
85C2×C3×Q8C22⋊C412yes—
86C2×C3×D4—21no—
87C2×C3×D4—41no—
88C2×C3×D4—41no—
89C2×C3×D4C22⋊C412yes—
90C22×C3×C4—21no—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×Dic3—22no—
94C22×Dic3—22no—
95C22×Dic3—82no—
96C22×Dic3—81no—
97C2×D12—22no—
98C2×D12—24no—
99C2×D12—42no—
100C2×D12—42no—
101C2×D12—41no—
102C2×D12—41no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—42no—
106C2×D12—42no—
107C2×D12—42no—
108C2×D12—41no—
109C2×C3×C8—24no—
110C2×C3×C8—42no—
111C2×C3×C8—42no—
112C3×C22⋊C4—24no—
113C3×C22⋊C4—24no—
114C3×C42C22×C411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4112yes—
117#217—81no—
118#217—82no—
119#215—48no—
120#213—48no—
121C4⋊Dic3—42no—
122C4⋊Dic3—42no—
123C4⋊Dic3—81no—
124C4×Dic3—81no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—22no—
128C22×Q8D1211yescharacteristic
129C22×D4—61no—
130C22×D4—62no—
131C22×D4D1211yescharacteristic
132C23×C4—61no—
133C2×SD16—64no—
134C2×SD16—122no—
135C2×SD16—122no—
136C2×D8—64no—
137C2×D8—122no—
138C2×D8—122no—
139C22×C8—22no—
140C22×C8—61no—
141#127—32no—
142#127—62no—
143#127—61no—
144#124—22no—
145#124C4×D312yes—
146#124#6712yes—
147C2×C4⋊C4—32no—
148C2×C4⋊C4—32no—
149C2×C4⋊C4—61no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—61no—
153C2×C42—61no—
154C2×C42C22×D311yescharacteristic
155#102—124no—
156#102—124no—
157#97—22no—
158#97—22no—
159#97—32no—
160#97—61no—
161#95—62no—
162#95—62no—
163#95—62no—
164C23×C3—22no—
165C23×C3—42no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—42no—
169C23×C3#12011yescharacteristic
170C22×D3—42no—
171C22×D3—81no—
172C22×D3—81no—
173C22×D3—82no—
174C22×D3—82no—
175C3×Q8—42no—
176C3×Q8—81no—
177C3×D4—42no—
178C3×D4—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4C2×C22⋊C411yescharacteristic
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12112yes—
189C2×C3×C4#12311yescharacteristic
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—84no—
197D12—42no—
198D12—42no—
199D12—44no—
200D12—44no—
201D12—82no—
202D12—82no—
203D12—81no—
204D12—81no—
205C3×C8—44no—
206C3⋊C8—24no—
207C24—21no—
208C24—121no—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8(C2×D3)⋊C412yes—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—64no—
217C2×D4—62no—
218C2×D4—122no—
219C2×D4—122no—
220C2×D4—121no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C2×D4—122no—
225C2×D4—122no—
226C2×D4—121no—
227C2×D4—122no—
228C2×D4(C2×D3)⋊C412yes—
229C22×C4—21no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—241no—
233C22×C4—242no—
234C22×C4C2×D1211yescharacteristic
235C22×C4D3×D412yes—
236SD16—128no—
237D8—128no—
238C2×C8—24no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—62no—
244C4⋊C4—122no—
245C4⋊C4—122no—
246C4⋊C4—241no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—241no—
250C22⋊C4—242no—
251C42—241no—
252C42C2×C4×D311yescharacteristic
253C42#23712yes—
254C42C2×#6711yescharacteristic
255C22×C3—44no—
256C22×C3—42no—
257C22×C3—81no—
258C22×C3—82no—
259C22×C3—81no—
260C22×C3#38611yescharacteristic
261C22×C3#45311yescharacteristic
262C22×C3#45311yescharacteristic
263C22×C3#45312yes—
264C22×C3#45412yes—
265C2×D3—84no—
266C2×D3—82no—
267C2×D3—161no—
268C2×D3—162no—
269C2×D3—161no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—42no—
275Dic3—82no—
276Dic3—82no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23—122no—
283C23—241no—
284C23—241no—
285C23—242no—
286C23—242no—
287C23#88211yescharacteristic
288Q8—42no—
289Q8—81no—
290D4—42no—
291D4—81no—
292D4—122no—
293D4—122no—
294D4—124no—
295D4—124no—
296D4—241no—
297D4—241no—
298D4—242no—
299D4—242no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—244no—
314C2×C4#67⋊C412yes—
315C2×C4#88611yescharacteristic
316C2×C4C2×(C2×D3)⋊C411yescharacteristic
317C2×C4#93912yes—
318C8—44no—
319C8—64no—
320C2×C3—82no—
321C2×C3#165511yescharacteristic
322C2×C3#165612yes—
323C2×C3#165711yescharacteristic
324C2×C3#188312yes—
325C2×C3#188711yescharacteristic
326D3—162no—
327C22—42no—
328C22—44no—
329C22—81no—
330C22—81no—
331C22—82no—
332C22—242no—
333C22—244no—
334C22—481no—
335C22—481no—
336C22—482no—
337C22#474711yescharacteristic
338C22#474711yescharacteristic
339C22#482212yes—
340C22#482712yes—
341C22#495811yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—242no—
348C4—242no—
349C3#900111yescharacteristic, complemented, Sylow
350C2—82no—
351C2—482no—
352C2#6835011yescharacteristic
353C2#6835112yes—
354C2#6835411yescharacteristic
355C2#7007011yescharacteristic
356C2#7036612yes—
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#329C2
socleC23×C3#120
prime radicalC2×#5058C2
cosocleC2×C3×C42C23
hypercenterC2×#329D3
hypocenterC3#9001
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
1. #82957C2semidirect
3. C2×C3×#329C2semidirect
4. #69823C2semidirect
5. #67659C2semidirect
17. #5058C4semidirect
26. C2×#329D3semidirect
54. #874C2×C4semidirect
63. #803C2×C4semidirect
349. C3#9001semidirect

Maximal subgroups

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

subgroupindex
#829572
C2×#50582
C2×C3×#3292
#698232
#676592
#90013