Finite Group Databasebeta

← #10692424

Subgroups — #10692424

Order 768.

390 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 89 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#79319C211yesmaximal, characteristic
2#79266C211yesmaximal, characteristic, complemented
3#70820C211yesmaximal, characteristic, complemented
4#70811C211yesmaximal, characteristic
5#70758C211yesmaximal, characteristic
6C3×#1287C211yesmaximal, characteristic, complemented
7#67755C211yesmaximal, characteristic, complemented
8#9372—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#4926C2211yescharacteristic
15C2×C4⋊(C3⋊C8)C2211yescharacteristic
16#4846—21no—
17#4842—21no—
18#4828—22no—
19#4737—22no—
20#4720—22no—
21#4713C412yes—
22#4702C412yes—
23#4679—21no—
24C8×Dic6—21no—
25#4486—21no—
26#4482—21no—
27#4464—22no—
28#4459—21no—
29#4459—21no—
30#3235—31no—
31#2985—31no—
32#1552—31no—
33#1536—31no—
34#1480—31no—
35#1290—31no—
36#1287D311yescharacteristic, complemented
37C2×#235—21no—
38C2×C3×C42C2311yescharacteristic
39#930—21no—
40#930—21no—
41#926D412yes—
42#924—21no—
43#924—21no—
44C22×C3⋊C8—21no—
45#875—21no—
46#874C2×C411yescharacteristic
47C4×D12—21no—
48C4×D12—21no—
49#870C2×C412yes—
50#869C2×C411yescharacteristic
51#869D411yescharacteristic
52#869D411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97—21no—
62C3×#97C2×C412yes—
63C3×C4×C8D411yescharacteristic
64C3×C4×C8D411yescharacteristic
65#830—21no—
66#821—22no—
67#821—22no—
68#818—22no—
69#817—22no—
70#815—22no—
71Dic3⋊C8—22no—
72Dic3⋊C8—22no—
73C8×Dic3—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77#803D412yes—
78#533—31no—
79#532—31no—
80#501—32no—
81#490—32no—
82#482—62no—
83#476—61no—
84#474—61no—
85#446—62no—
86#446—61no—
87C8×Q8—61no—
88#433—62no—
89#432—31no—
90#432C2×D311yescharacteristic
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—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×C3×C4#4111yescharacteristic
102C2×#67—41no—
103#235—81no—
104#235—81no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×C4×D3—41no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×C3×C8—22no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—41no—
116C3×C42C22×C411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120#217—22no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3C22⋊C412yes—
128C4⋊Dic3C22⋊C412yes—
129Dic3⋊C4—22no—
130Dic3⋊C4—22no—
131C4×Dic3—22no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—81no—
135C2×#41—61no—
136C22×C8—61no—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140#127—31no—
141#125—32no—
142#124—61no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145#121—61no—
146#121—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×D4—61no—
150C4×D4—61no—
151#117—32no—
152C2×C42C22×D311yescharacteristic
153#107—62no—
154#106—62no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—62no—
159C4⋊C8—62no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#102—62no—
164#102—62no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167C22⋊C8—21no—
168C22⋊C8—61no—
169#97—21no—
170#97—32no—
171#97—62no—
172#97C4×D312yes—
173C4×C8—62no—
174C4×C8D1211yescharacteristic
175C4×C8D1211yescharacteristic
176C23×C3#12311yescharacteristic
177C22×D3—41no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4C2×C22⋊C411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12111yescharacteristic
188#67—82no—
189C2×Dic3—22no—
190C2×Dic3—22no—
191C2×Dic3—41no—
192D12—42no—
193D12—42no—
194D12—81no—
195C4×D3—81no—
196C4×D3—81no—
197Dic6—41no—
198Dic6—41no—
199Dic6—81no—
200Dic6—81no—
201C3×C8—41no—
202C3×C8—41no—
203C3×C8—42no—
204C3×C8—81no—
205C3⋊C8—42no—
206C3⋊C8—81no—
207#41—241no—
208#41—241no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—121no—
216C22×C4—121no—
217C22×C4#23511yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4#23911yescharacteristic
220C2×C8—22no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—41no—
224C2×C8—62no—
225C2×C8—121no—
226C2×C8—241no—
227C4⋊C4—32no—
228C4⋊C4—32no—
229C4⋊C4—62no—
230C4⋊C4—61no—
231C4⋊C4—62no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C22⋊C4—62no—
236C22⋊C4—121no—
237C22⋊C4—121no—
238C42—62no—
239C42C2×C4×D311yescharacteristic
240C42C2×D1211yescharacteristic
241C42D3×D411yescharacteristic
242C42D3×D411yescharacteristic
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#47111yescharacteristic
256C3×C4#47211yescharacteristic
257Dic3—42no—
258Dic3—42no—
259Dic3—81no—
260C23—121no—
261C23#89411yescharacteristic
262Q8—121no—
263Q8—121no—
264Q8—241no—
265Q8—241no—
266D4—122no—
267D4—122no—
268D4—241no—
269D4—242no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—121no—
277C2×C4—241no—
278C2×C4—241no—
279C2×C4C4×D1211yescharacteristic
280C2×C4D3×C22⋊C411yescharacteristic
281C2×C4#88211yescharacteristic
282C2×C4#88411yescharacteristic
283C2×C4#89311yescharacteristic
284C2×C4#89511yescharacteristic
285C8—41no—
286C8—41no—
287C8—42no—
288C8—81no—
289C8—122no—
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#466511yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—122no—
309C4—122no—
310C4—241no—
311C4#489711yescharacteristic
312C4#489811yescharacteristic
313C3#937211yescharacteristic, complemented, Sylow
314C2—41no—
315C2—481no—
316C2#6870811yescharacteristic
317C2#6894811yescharacteristic
318C2#6903511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1287C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1287D3
hypocenterC3#9372
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

Complements

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

NH ≅ G/Ntype
2. #79266C2semidirect
3. #70820C2semidirect
6. C3×#1287C2semidirect
7. #67755C2semidirect
36. #1287D3semidirect
313. C3#9372semidirect

Maximal subgroups

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

subgroupindex
#793192
#792662
#708202
#708112
#707582
C3×#12872
#677552
#93723