Finite Group Databasebeta

← #10692425

Subgroups — #10692425

Order 768.

390 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2985C211yesmaximal, characteristic, complemented
2#83361C211yesmaximal, characteristic
3#71831C211yesmaximal, characteristic, complemented
4#70811C211yesmaximal, characteristic
5#70758C211yesmaximal, characteristic
6#67755C211yesmaximal, characteristic, complemented
7#67747C211yesmaximal, characteristic, complemented
8#9372—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#446—22no—
12C3×#446—21no—
13C3×C8×Q8—21no—
14C3×#433—22no—
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17#5087—21no—
18#5084—21no—
19#5067C412yes—
20#5064C412yes—
21#5003—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4547—21no—
26#4547—21no—
27#4464—22no—
28#4459—21no—
29#4459—21no—
30#3235—31no—
31#2985D311yescharacteristic, complemented
32#1552—31no—
33#1536—31no—
34#1480—31no—
35#1290—31no—
36#1287—31no—
37C2×#235—21no—
38C3×C4⋊Q8D411yescharacteristic
39C3×C4⋊Q8D411yescharacteristic
40C3×#125C2×C412yes—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×#117D412yes—
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930—21no—
49C22×C3⋊C8—21no—
50#875—21no—
51#874C2×C411yescharacteristic
52C4×D12—21no—
53C4×D12—21no—
54#869C2×C411yescharacteristic
55C3×C4⋊C8—22no—
56C3×C4⋊C8—22no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97—22no—
62C3×#97D412yes—
63C3×C4×C8—22no—
64C3×C4×C8D411yescharacteristic
65C3×C4×C8D411yescharacteristic
66#830—21no—
67#830—21no—
68#809—22no—
69#809—22no—
70#808—22no—
71#807—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77#803C2×C412yes—
78#533C2×D311yescharacteristic
79#532—31no—
80#501—32no—
81#490—32no—
82#482—62no—
83#476—61no—
84#474—61no—
85#446—22no—
86#446—21no—
87C8×Q8—21no—
88#433—22no—
89#432—31no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431C2×D311yescharacteristic
93C2×C4⋊C8—31no—
94#332—61no—
95#332—61no—
96#329—62no—
97D4⋊C8—61no—
98D4⋊C8—61no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—41no—
105#235—81no—
106#235—81no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12—21no—
110C2×C4×D3—41no—
111C2×Dic6—21no—
112C2×C3×C8—21no—
113C2×C3×C8—22no—
114C2×C3×C8—21no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4C22⋊C412yes—
120C3×C4⋊C4C22⋊C412yes—
121C3×C22⋊C4—22no—
122C3×C42—22no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—81no—
135C2×#41—61no—
136C22×C8—61no—
137C4⋊Q8—31no—
138C4⋊Q8D1211yescharacteristic
139C4⋊Q8D1211yescharacteristic
140#127—31no—
141#125C4×D312yes—
142#124—61no—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
145#121—61no—
146#121—61no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×D4—61no—
150C4×D4—61no—
151#117#6712yes—
152C2×C42C22×D311yescharacteristic
153#107—62no—
154#106—62no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#102—62no—
164#102—62no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—22no—
170#97—32no—
171#97—61no—
172#97#6712yes—
173C4×C8—22no—
174C4×C8#6711yescharacteristic
175C4×C8#6711yescharacteristic
176C23×C3#12311yescharacteristic
177C22×D3—41no—
178C3×Q8—41no—
179C3×Q8—41no—
180C3×Q8—81no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4C2×C22⋊C411yescharacteristic
188C2×C3×C4C4×D411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12111yescharacteristic
193#67—82no—
194C2×Dic3—41no—
195D12—42no—
196D12—42no—
197D12—81no—
198C4×D3—81no—
199C4×D3—81no—
200Dic6—81no—
201C3×C8—41no—
202C3×C8—42no—
203C3×C8—41no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C3⋊C8—81no—
207#41—241no—
208#41—241no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—121no—
216C22×C4—121no—
217C22×C4#23511yescharacteristic
218C22×C4#23511yescharacteristic
219C22×C4C2×#6711yescharacteristic
220C2×C8—21no—
221C2×C8—22no—
222C2×C8—21no—
223C2×C8—62no—
224C2×C8—121no—
225C2×C8—121no—
226C2×C8—241no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4(C2×D3)⋊C412yes—
234C4⋊C4(C2×D3)⋊C412yes—
235C22⋊C4—22no—
236C22⋊C4—121no—
237C22⋊C4—121no—
238C42—22no—
239C42C2×C4×D311yescharacteristic
240C42C2×D1211yescharacteristic
241C42C2×#6711yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#38611yescharacteristic
247C2×D3—81no—
248C2×D3—81no—
249C2×D3—81no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—42no—
256C3×C4—42no—
257C3×C4#47111yescharacteristic
258C3×C4#47211yescharacteristic
259Dic3—81no—
260C23—121no—
261C23#92911yescharacteristic
262Q8—41no—
263Q8—41no—
264Q8—81no—
265Q8—241no—
266D4—122no—
267D4—122no—
268D4—241no—
269D4—242no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—121no—
277C2×C4—241no—
278C2×C4—241no—
279C2×C4C2×(C2×D3)⋊C411yescharacteristic
280C2×C4C4×#6711yescharacteristic
281C2×C4C4×#6711yescharacteristic
282C2×C4#93011yescharacteristic
283C2×C4#93011yescharacteristic
284C2×C4#95311yescharacteristic
285C8—41no—
286C8—42no—
287C8—41no—
288C8—122no—
289C8—241no—
290C8—241no—
291C2×C3—41no—
292C2×C3#188111yescharacteristic
293C2×C3#190711yescharacteristic
294C2×C3#191211yescharacteristic
295D3—161no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—241no—
300C22—241no—
301C22—241no—
302C22#521311yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—241no—
311C4#514111yescharacteristic
312C4#514211yescharacteristic
313C3#937211yescharacteristic, complemented, Sylow
314C2—41no—
315C2—481no—
316C2#7166311yescharacteristic
317C2#7201811yescharacteristic
318C2#7202011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2985C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2985D3
hypocenterC3#9372
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2985

Complements

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

NH ≅ G/Ntype
1. C3×#2985C2semidirect
3. #71831C2semidirect
6. #67755C2semidirect
7. #67747C2semidirect
31. #2985D3semidirect
313. C3#9372semidirect

Maximal subgroups

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

subgroupindex
C3×#29852
#833612
#718312
#708112
#707582
#677552
#677472
#93723