Finite Group Databasebeta

← #10708376

Subgroups — #10708376

Order 768.

362 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 55 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#78012C211yesmaximal, characteristic
2#69713C211yesmaximal, characteristic, complemented
3#68995C211yesmaximal, characteristic
4#68139C211yesmaximal, characteristic, complemented
5C3×#1346C211yesmaximal, characteristic, complemented
6#67494C211yesmaximal, characteristic, complemented
7#67454C211yesmaximal, characteristic
8#12276—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#398C2211yescharacteristic
11#5131—21no—
12#4994—21no—
13#4991—21no—
14#4988C2211yescharacteristic
15#4980—21no—
16#4979C2211yescharacteristic
17#4971—21no—
18#4739—21no—
19#4675—21no—
20#4674—21no—
21#4673—21no—
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4555—21no—
25#4528C2211yescharacteristic
26#4467—21no—
27#4452—21no—
28#2830—31no—
29#2027—31no—
30#1611—31no—
31#1605—31no—
32#1354—31no—
33#1346D311yescharacteristic, complemented
34#1318—31no—
35C22×D12D411yescharacteristic
36C22×Dic6D411yescharacteristic
37C2×C3×M16—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C4⋊C4C2311yescharacteristic
41C2×(C2×D3)⋊C4—21no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47#926—21no—
48C2×C4⋊Dic3—21no—
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×Dic12—41no—
54#890—41no—
55#888—41no—
56C3×C22⋊C8—21no—
57C3×#95—21no—
58C3×#95D411yescharacteristic
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95D411yescharacteristic
64#830—21no—
65#821—41no—
66#821—41no—
67#816—41no—
68#810—41no—
69#808—41no—
70#544—31no—
71#483—61no—
72#451—61no—
73#418—61no—
74#398C2×D311yescharacteristic
75#397—61no—
76#397—61no—
77#396—61no—
78#395—31no—
79#394—61no—
80#393—61no—
81#390—31no—
82#387—61no—
83#343C2×D311yescharacteristic
84#339—61no—
85#328—61no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C23×D3—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—41no—
101C2×D12—41no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—41no—
105C2×Dic6—81no—
106C3×M16—41no—
107C3×M16—41no—
108C2×C3×C8—41no—
109C2×C3×C8—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114#217—41no—
115(C2×D3)⋊C4—81no—
116(C2×D3)⋊C4—81no—
117(C2×D3)⋊C4—81no—
118(C2×D3)⋊C4—81no—
119(C2×D3)⋊C4—81no—
120(C2×D3)⋊C4—81no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—81no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128Dic3⋊C4—81no—
129C4×Dic3—41no—
130C4×Dic3—41no—
131C2×C3⋊C8—41no—
132Dic12—82no—
133C22×Q8—31no—
134C22×D4—31no—
135C2×Q16—121no—
136C2×M16—21no—
137C4⋊Q8—121no—
138#125—121no—
139C22⋊Q8—61no—
140C22⋊Q8—61no—
141#117—61no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4D1211yescharacteristic
146C2×C4⋊C4C22×D311yescharacteristic
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42—31no—
154#106—121no—
155#103—121no—
156#103—121no—
157#102—121no—
158#102—121no—
159C22⋊C8—21no—
160C22⋊C8—61no—
161#95—21no—
162#95—31no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95D1211yescharacteristic
168C23×C3#12011yescharacteristic
169C22×D3—41no—
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4#13511yescharacteristic
189C2×C3×C4#13711yescharacteristic
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201D12—42no—
202D12—81no—
203Dic6—42no—
204Dic6—81no—
205Dic6—81no—
206Dic6—81no—
207C3×C8—81no—
208C3×C8—81no—
209C3⋊C8—81no—
210C24—61no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8—121no—
214C2×Q8—241no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4—121no—
218C2×D4—121no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C2×D1211yescharacteristic
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D411yescharacteristic
227Q16—242no—
228M16—41no—
229M16—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—121no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—241no—
244C4⋊C4—241no—
245C22⋊C4—121no—
246C22⋊C4—241no—
247C22⋊C4—241no—
248C22⋊C4—241no—
249C22⋊C4—241no—
250C22⋊C4—241no—
251C22⋊C4—241no—
252C42—121no—
253C42—121no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3#45211yescharacteristic
257C22×C3#45511yescharacteristic
258C22×C3B(4,𝔽2)11yescharacteristic
259C2×D3—81no—
260C2×D3—81no—
261C2×D3—81no—
262C2×D3—81no—
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—81no—
271C3×C4—81no—
272Dic3—81no—
273Dic3—81no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277C23—121no—
278C23—121no—
279C23—121no—
280C23—121no—
281C23—121no—
282C23—121no—
283C23—121no—
284C23#88211yescharacteristic
285Q8—122no—
286Q8—241no—
287Q8—241no—
288Q8—241no—
289D4—122no—
290D4—241no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4#91211yescharacteristic
315C2×C4#91511yescharacteristic
316C8—81no—
317C8—81no—
318C8—241no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3#160611yescharacteristic
322C2×C3#220011yescharacteristic
323C2×C3#220311yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22—241no—
331C22—241no—
332C22—241no—
333C22—241no—
334C22#474111yescharacteristic
335C22#478311yescharacteristic
336C22#482511yescharacteristic
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—241no—
344C4—241no—
345C4—241no—
346C4—241no—
347C4—241no—
348C3#1227611yescharacteristic, complemented, Sylow
349C2—21no—
350C2—21no—
351C2—481no—
352C2#6828711yescharacteristic
353C2#6913911yescharacteristic
354C2#6916711yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1346C2
socleC22×C3B(4,𝔽2)
prime radicalC22×D12D4
hypercenter#1346D3
hypocenterC3#12276
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
2. #69713C2semidirect
4. #68139C2semidirect
5. C3×#1346C2semidirect
6. #67494C2semidirect
33. #1346D3semidirect
348. C3#12276semidirect

Maximal subgroups

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

subgroupindex
#780122
#697132
#689952
#681392
C3×#13462
#674942
#674542
#122763