Finite Group Databasebeta

← #10692423

Subgroups — #10692423

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#83361C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic, complemented
3C3×#1552C211yesmaximal, characteristic, complemented
4#70811C211yesmaximal, characteristic
5#70426C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic, complemented
7#67708C211yesmaximal, characteristic, complemented
8#9372—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13#5087—21no—
14#5084—21no—
15#5067C412yes—
16#5064C412yes—
17#5024—22no—
18#5024—21no—
19Q8×C3⋊C8—21no—
20#5003—22no—
21#5000—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25C3×#332—21no—
26C3×#332—21no—
27#4483—21no—
28#4483—21no—
29#4464—22no—
30#3235—31no—
31#2985—31no—
32#1552D311yescharacteristic, complemented
33#1536—31no—
34#1480—31no—
35#1290—31no—
36#1287—31no—
37C2×#235—21no—
38C22×C3×C8—21no—
39C3×C4⋊Q8D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×#125C2×C412yes—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×#117D412yes—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930—21no—
50#875—21no—
51#874C2×C411yescharacteristic
52C4×D12—21no—
53C4×D12—21no—
54#869C2×C411yescharacteristic
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×#97—21no—
60#830—21no—
61#830—21no—
62#830—21no—
63#809—22no—
64#809—22no—
65#808—22no—
66#807—22no—
67C4⋊(C3⋊C8)—22no—
68C4⋊(C3⋊C8)—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—22no—
73#803C2×C412yes—
74#803D412yes—
75C4×C3⋊C8—22no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533C2×D311yescharacteristic
79#532—31no—
80#501—32no—
81#490—32no—
82#482—62no—
83#476—61no—
84#474—61no—
85#446—61no—
86#446—62no—
87C8×Q8—61no—
88#433—62no—
89#432—31no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431—31no—
93C2×C4⋊C8C2×D311yescharacteristic
94#332—21no—
95#332—21no—
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—41no—
113C2×C3×C8—41no—
114C2×C3×C8—81no—
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—21no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—21no—
135C2×#41—61no—
136C22×C8—21no—
137C4⋊Q8—31no—
138C4⋊Q8#6711yescharacteristic
139C4⋊Q8#6711yescharacteristic
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#117D1212yes—
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—61no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—21no—
170#97—32no—
171#97—32no—
172#97—62no—
173C4×C8—31no—
174C4×C8—31no—
175C4×C8—62no—
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—81no—
202C3×C8—81no—
203C3⋊C8—41no—
204C3⋊C8—42no—
205C3⋊C8—41no—
206C3⋊C8—42no—
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×C4C2×D1211yescharacteristic
218C22×C4#23711yescharacteristic
219C22×C4#23711yescharacteristic
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—62no—
225C2×C8—62no—
226C2×C8—81no—
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
240C42D3×D411yescharacteristic
241C42D3×D411yescharacteristic
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#88611yescharacteristic
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×C4#67⋊C411yescharacteristic
280C2×C4#67⋊C411yescharacteristic
281C2×C4#88211yescharacteristic
282C2×C4C2×(C2×D3)⋊C411yescharacteristic
283C2×C4#93911yescharacteristic
284C2×C4#93911yescharacteristic
285C8—81no—
286C8—81no—
287C8—121no—
288C8—121no—
289C8—122no—
290C8—122no—
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#495811yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—241no—
311C4#519911yescharacteristic
312C4#520311yescharacteristic
313C3#937211yescharacteristic, complemented, Sylow
314C2—41no—
315C2—481no—
316C2#7010511yescharacteristic
317C2#7029311yescharacteristic
318C2#7030311yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1552C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1552D3
hypocenterC3#9372
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
2. #83331C2semidirect
3. C3×#1552C2semidirect
6. #67757C2semidirect
7. #67708C2semidirect
32. #1552D3semidirect
313. C3#9372semidirect

Maximal subgroups

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

subgroupindex
#833612
#833312
C3×#15522
#708112
#704262
#677572
#677082
#93723