Finite Group Databasebeta

← #10708127

Subgroups — #10708127

Order 768.

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

normal subgroups only

This group has 368 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#74353C211yesmaximal, characteristic
2#69573C211yesmaximal, characteristic, complemented
3#68141C211yesmaximal, characteristic, complemented
4#68138C211yesmaximal, characteristic
5C3×#1325C211yesmaximal, characteristic, complemented
6#67459C211yesmaximal, characteristic, complemented
7#67456C211yesmaximal, characteristic
8#12238—31nomaximal, Sylow
9#5500C2211yescharacteristic
10C3×#384C2211yescharacteristic
11#4994—21no—
12#4991—21no—
13#4991C2211yescharacteristic
14#4990—21no—
15#4990—21no—
16#4986C2211yescharacteristic
17#4978—21no—
18#4964—21no—
19#4940—21no—
20#4731—21no—
21#4722—21no—
22C3×#324C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4530C2211yescharacteristic
25#4523—21no—
26#4467—21no—
27#4467—21no—
28#2795—31no—
29#2060—31no—
30#1617—31no—
31#1599—31no—
32#1353—31no—
33#1345—31no—
34#1325D311yescharacteristic, complemented
35C22×D12D411yescharacteristic
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40C2×C3×C42D411yescharacteristic
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⋊Dic3D411yescharacteristic
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×#208—21no—
54C2×#204—41no—
55#890—41no—
56#888—41no—
57C3×C22⋊C8—21no—
58C3×C22⋊C8—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95D411yescharacteristic
64C3⋊#95D411yescharacteristic
65#821—41no—
66#818—41no—
67#816—41no—
68#808—41no—
69#807—41no—
70#544—31no—
71#484—61no—
72#450—61no—
73#428—61no—
74#400—31no—
75#396—61no—
76#394—31no—
77#394—61no—
78#388—61no—
79#388—61no—
80#388—61no—
81#387—61no—
82#384C2×D311yescharacteristic
83#343—31no—
84#339—61no—
85#328—61no—
86#328—61no—
87#324C2×D311yescharacteristic
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—41no—
99C2×D12—81no—
100C2×Dic6—21no—
101C2×Dic6—21no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—81no—
107C3×C4⋊C4—81no—
108C3×C42—21no—
109C3×C42—21no—
110#217—41no—
111(C2×D3)⋊C4—42no—
112(C2×D3)⋊C4—42no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—81no—
116(C2×D3)⋊C4—81no—
117(C2×D3)⋊C4—81no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—41no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125Dic3⋊C4—42no—
126C4×Dic3—41no—
127C4×Dic3—81no—
128#208—22no—
129#208—41no—
130C2×C3⋊C8—41no—
131#204—82no—
132C22×D4—31no—
133C2×SD16—121no—
134C2×M16—61no—
135C4⋊Q8—121no—
136#125—121no—
137C22⋊Q8—61no—
138C22⋊Q8—61no—
139#117—61no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4D1211yescharacteristic
145C2×C4⋊C4C22×D311yescharacteristic
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C42—61no—
153C2×C42D1211yescharacteristic
154#107—121no—
155#107—121no—
156#106—121no—
157#103—121no—
158#102—121no—
159C22⋊C8—21no—
160C22⋊C8—21no—
161#95—31no—
162#95—31no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167C23×C3#12011yescharacteristic
168C22×D3—41no—
169C22×D3—41no—
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—22no—
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#13511yescharacteristic
188C2×C3×C4#13711yescharacteristic
189C2×Dic3—41no—
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—
200D12—42no—
201D12—81no—
202Dic6—81no—
203Dic6—81no—
204C3×C8—81no—
205C3×C8—81no—
206C3⋊C8—42no—
207C24—61no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×D4—121no—
211C2×D4—241no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4C2×D1211yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4D3×D411yescharacteristic
220SD16—242no—
221M16—62no—
222M16—121no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—121no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—121no—
235C4⋊C4—122no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C22⋊C4—122no—
239C22⋊C4—122no—
240C22⋊C4—121no—
241C22⋊C4—121no—
242C22⋊C4—121no—
243C22⋊C4—241no—
244C22⋊C4—241no—
245C22⋊C4—241no—
246C42—21no—
247C42—21no—
248C42—121no—
249C42—241no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#45211yescharacteristic
253C22×C3#45611yescharacteristic
254C22×C3B(4,𝔽2)11yescharacteristic
255C2×D3—81no—
256C2×D3—81no—
257C2×D3—81no—
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—81no—
261C2×D3—81no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—42no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—81no—
268Dic3—81no—
269Dic3—81no—
270Dic3—81no—
271Dic3—81no—
272Dic3—81no—
273C23—121no—
274C23—121no—
275C23—121no—
276C23—121no—
277C23—121no—
278C23—121no—
279C23—121no—
280C23#88211yescharacteristic
281Q8—241no—
282Q8—241no—
283D4—122no—
284D4—241no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#90411yescharacteristic
309C2×C4#90911yescharacteristic
310C8—81no—
311C8—81no—
312C8—122no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3#160311yescharacteristic
316C2×C3#220011yescharacteristic
317C2×C3#220411yescharacteristic
318D3—161no—
319C22—21no—
320C22—21no—
321C22—241no—
322C22—241no—
323C22—241no—
324C22—241no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22#473311yescharacteristic
329C22#474111yescharacteristic
330C22#475211yescharacteristic
331C4—21no—
332C4—21no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—241no—
338C4—241no—
339C4—241no—
340C4—241no—
341C4—241no—
342C3#1223811yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—481no—
346C2#6813211yescharacteristic
347C2#6914711yescharacteristic
348C2#6916811yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1325C2
socleC22×C3B(4,𝔽2)
prime radicalC22×D12D4
hypercenter#1325D3
hypocenterC3#12238
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #69573C2semidirect
3. #68141C2semidirect
5. C3×#1325C2semidirect
6. #67459C2semidirect
34. #1325D3semidirect
342. C3#12238semidirect

Maximal subgroups

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

subgroupindex
#743532
#695732
#681412
#681382
C3×#13252
#674592
#674562
#122383