Finite Group Databasebeta

← #10699225

Subgroups — #10699225

Order 768.

390 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 95 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#79271C211yesmaximal, characteristic
2#70746C211yesmaximal, characteristic
3#70522C211yesmaximal, characteristic
4#70509C211yesmaximal, characteristic
5#69493C211yesmaximal, characteristic
6#69490C211yesmaximal, characteristic
7C3×#1303C211yesmaximal, characteristic
8#10507—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#824—21no—
15C2×#824—21no—
16C2×C4⋊(C3⋊C8)C2211yescharacteristic
17C4×#208C2211yescharacteristic
18#4851—21no—
19#4851—21no—
20#4721—22no—
21#4681—21no—
22#4680—21no—
23#4529—22no—
24#4523—21no—
25#4523—21no—
26#4486—21no—
27#4481—21no—
28#4480—21no—
29#4479—21no—
30#3229—31no—
31#1988—31no—
32#1986—31no—
33#1966—31no—
34#1565—31no—
35#1550—31no—
36#1303D311yescharacteristic
37C22×Dic6—21no—
38C2×C3×M16—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3D412yes—
45#924—21no—
46#924—21no—
47C2×#208D411yescharacteristic
48C2×#208D411yescharacteristic
49C22×C3⋊C8—21no—
50#870—21no—
51#869—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D411yescharacteristic
62C3×C4⋊C8D411yescharacteristic
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65C3×#95—21no—
66#824—42no—
67#824—42no—
68#817—22no—
69#817—22no—
70#816—22no—
71#816—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75C4×C3⋊C8—21no—
76#533—31no—
77C2×C4⋊Q8—31no—
78#498—61no—
79#487—61no—
80#483—62no—
81#475—61no—
82#475—61no—
83#431C2×D311yescharacteristic
84#423C2×D311yescharacteristic
85C2×C4⋊C8—31no—
86C2×#101—61no—
87C2×#101—61no—
88C4×M16—31no—
89C4×C4⋊C4C2×D311yescharacteristic
90#340—62no—
91#339—61no—
92#339—61no—
93#335—61no—
94#334—61no—
95#333—61no—
96#332—61no—
97C22×C3×C4—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—22no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×Dic6—41no—
110C3×M16—41no—
111C3×M16—41no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—41no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—41no—
117C3×C42—41no—
118C3×C42—41no—
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123#217—41no—
124C4⋊Dic3—22no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—22no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C4×Dic3—41no—
135#208—21no—
136#208—21no—
137#208—21no—
138#208SD1612yes—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—42no—
142C2×C3⋊C8—41no—
143C22×Q8—61no—
144C2×M16—21no—
145C2×M16—31no—
146C2×M16—31no—
147C22×C8—61no—
148C4⋊Q8—31no—
149C4⋊Q8—31no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154#125—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C4×Q8—61no—
158C4×Q8—61no—
159#117—61no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—32no—
162C2×C4⋊C4—61no—
163C2×C42—21no—
164C2×C42C22×D311yescharacteristic
165#106—62no—
166#106—62no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8D1211yescharacteristic
172C4⋊C8D1211yescharacteristic
173#103—62no—
174#103—62no—
175#101—122no—
176#101—122no—
177C22⋊C8—21no—
178#97—61no—
179C4×C8—21no—
180C4×C8—61no—
181#95—21no—
182C23×C3#12111yescharacteristic
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12311yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204Dic6—41no—
205Dic6—41no—
206Dic6—81no—
207Dic6—81no—
208Dic6—81no—
209Dic6—81no—
210Dic6—81no—
211C3×C8—41no—
212C3×C8—41no—
213C3×C8—81no—
214C3⋊C8—22no—
215C3⋊C8—41no—
216C3⋊C8—81no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×Q8—121no—
223C2×Q8—121no—
224C2×Q8—121no—
225C22×C4—41no—
226C22×C4—62no—
227C22×C4#23511yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4D3×D411yescharacteristic
230M16—32no—
231M16—41no—
232M16—41no—
233M16—61no—
234M16—61no—
235M16—61no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—61no—
240C2×C8—61no—
241C2×C8—122no—
242C2×C8—121no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—62no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—62no—
250C4⋊C4—61no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—121no—
256C42—41no—
257C42—41no—
258C42—121no—
259C42C2×D1211yescharacteristic
260C42#23511yescharacteristic
261C42D3×D411yescharacteristic
262C42#23911yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#39411yescharacteristic
266C22×C3#46111yescharacteristic
267C22×C3#46311yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—82no—
272C3×C4#45411yescharacteristic
273C3×C4#45611yescharacteristic
274C3×C4#46011yescharacteristic
275C3×C4#46411yescharacteristic
276C3×C4#46511yescharacteristic
277C3×C4#46811yescharacteristic
278Dic3—81no—
279Dic3—82no—
280Dic3—81no—
281C23#88411yescharacteristic
282Q8—121no—
283Q8—121no—
284Q8—241no—
285Q8—241no—
286Q8—241no—
287Q8—241no—
288Q8—241no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4#87511yescharacteristic
301C2×C4#88211yescharacteristic
302C2×C4#88411yescharacteristic
303C2×C4#89411yescharacteristic
304C2×C4#89411yescharacteristic
305C2×C4#89511yescharacteristic
306C2×C4#91111yescharacteristic
307C2×C4D3×SD1611yescharacteristic
308C2×C4#91611yescharacteristic
309C2×C4#91811yescharacteristic
310C8—41no—
311C8—41no—
312C8—62no—
313C8—81no—
314C8—121no—
315C8—241no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#203811yescharacteristic
319C2×C3#204911yescharacteristic
320C2×C3#205511yescharacteristic
321C22—21no—
322C22—21no—
323C22#467211yescharacteristic
324C22#478711yescharacteristic
325C22#481811yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—241no—
331C4—241no—
332C4—242no—
333C4#476211yescharacteristic
334C4#479811yescharacteristic
335C4#482611yescharacteristic
336C4#482711yescharacteristic
337C4#486411yescharacteristic
338C4#488611yescharacteristic
339C3#1050711yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2#6873411yescharacteristic
343C2#6886311yescharacteristic
344C2#6901911yescharacteristic
subgroupG/H
centerC22#4672
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1303D3
hypocenterC3#10507
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
339. C3#10507semidirect

Maximal subgroups

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

subgroupindex
#792712
#707462
#705222
#705092
#694932
#694902
C3×#13032
#105073