Finite Group Databasebeta

← #10692428

Subgroups — #10692428

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#83274C211yesmaximal, characteristic
2#79266C211yesmaximal, characteristic, complemented
3C3×#1480C211yesmaximal, characteristic
4#71799C211yesmaximal, characteristic
5#70439C211yesmaximal, characteristic, complemented
6#67747C211yesmaximal, characteristic, complemented
7#67708C211yesmaximal, characteristic, complemented
8#9372—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13#5081C412yes—
14#5071C412yes—
15#5035—22no—
16#5008—21no—
17#5007—21no—
18#4926C2211yescharacteristic
19#4926C2211yescharacteristic
20#4926C2211yescharacteristic
21#4828—22no—
22#4720—22no—
23#4679—21no—
24C8×Dic6—21no—
25C3×D4⋊C8—21no—
26C3×D4⋊C8—21no—
27#4540—22no—
28#4486—21no—
29#4482—21no—
30#3235—31no—
31#2985—31no—
32#1552—31no—
33#1536—31no—
34#1480D311yescharacteristic
35#1290—31no—
36#1287—31no—
37C2×C3×#41—21no—
38C22×C3×C8—21no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#127C2×C411yescharacteristic
41C3×#124—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C42C2311yescharacteristic
47#926D412yes—
48#924—21no—
49#924—21no—
50#870C2×C412yes—
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×C4×C8D411yescharacteristic
60C3×C4×C8D411yescharacteristic
61#832—22no—
62#832—22no—
63#830—21no—
64#830—21no—
65#830—21no—
66#815—22no—
67Dic3⋊C8—22no—
68Dic3⋊C8—22no—
69C8×Dic3—22no—
70#808—22no—
71#807—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76#803C2×C412yes—
77#803D412yes—
78#533—31no—
79#532C2×D311yescharacteristic
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#432—31no—
91#432—31no—
92#431C2×D311yescharacteristic
93C2×C4⋊C8C2×D311yescharacteristic
94#332—61no—
95#332—61no—
96#329—62no—
97D4⋊C8—21no—
98D4⋊C8—21no—
99C3×#41—81no—
100C3×#41—81no—
101C2×C3×Q8—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—41no—
106C22×C3×C4—41no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—41no—
115C2×C3×C8—81no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C42C22×C411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124#217—22no—
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—22no—
134C2×C3⋊C8—41no—
135C2×#41—21no—
136C22×C8—21no—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8C4×D311yescharacteristic
140#127C4×D311yescharacteristic
141#125—32no—
142#124—21no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145#121—21no—
146#121—21no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×D4—21no—
150C4×D4—21no—
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—61no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—32no—
170#97—32no—
171#97—62no—
172#97—61no—
173C4×C8—62no—
174C4×C8#6711yescharacteristic
175C4×C8#6711yescharacteristic
176C23×C3—41no—
177C23×C3#12311yescharacteristic
178C3×Q8—81no—
179C3×D4—42no—
180C3×D4—42no—
181C3×D4—81no—
182C3×D4—82no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4C2×C22⋊C411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—22no—
198Dic6—41no—
199Dic6—41no—
200Dic6—81no—
201C3×C8—41no—
202C3×C8—41no—
203C3×C8—81no—
204C3⋊C8—42no—
205C3⋊C8—42no—
206C3⋊C8—81no—
207#41—81no—
208#41—81no—
209C2×Q8—21no—
210C2×Q8—61no—
211C2×Q8—61no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—41no—
216C22×C4—41no—
217C22×C4#23511yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4#23711yescharacteristic
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—41no—
223C2×C8—62no—
224C2×C8—62no—
225C2×C8—81no—
226C2×C8—121no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—32no—
230C4⋊C4—32no—
231C4⋊C4—62no—
232C4⋊C4—61no—
233C4⋊C4—62no—
234C4⋊C4—61no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—62no—
238C42—62no—
239C42C2×C4×D311yescharacteristic
240C42D3×D411yescharacteristic
241C42D3×D411yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—81no—
247C22×C3—81no—
248C22×C3—81no—
249C22×C3#38611yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—81no—
256C3×C4#47111yescharacteristic
257C3×C4#47211yescharacteristic
258Dic3—42no—
259Dic3—42no—
260C23—41no—
261C23#88311yescharacteristic
262Q8—81no—
263Q8—121no—
264Q8—121no—
265Q8—241no—
266D4—42no—
267D4—42no—
268D4—81no—
269D4—82no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4D3×C22⋊C411yescharacteristic
280C2×C4#67⋊C411yescharacteristic
281C2×C4#88411yescharacteristic
282C2×C4C4×#6711yescharacteristic
283C2×C4#93711yescharacteristic
284C2×C4#93811yescharacteristic
285C8—41no—
286C8—41no—
287C8—81no—
288C8—122no—
289C8—122no—
290C8—241no—
291C2×C3—41no—
292C2×C3—161no—
293C2×C3#188111yescharacteristic
294C2×C3#190711yescharacteristic
295C2×C3#191211yescharacteristic
296C22—21no—
297C22—21no—
298C22—21no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22#495611yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—81no—
309C4—122no—
310C4—122no—
311C4#516011yescharacteristic
312C4#517711yescharacteristic
313C3#937211yescharacteristic, complemented, Sylow
314C2—41no—
315C2—161no—
316C2#7001211yescharacteristic
317C2#7021811yescharacteristic
318C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1480C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1480D3
hypocenterC3#9372
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #79266C2semidirect
5. #70439C2semidirect
6. #67747C2semidirect
7. #67708C2semidirect
313. C3#9372semidirect

Maximal subgroups

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

subgroupindex
#832742
#792662
C3×#14802
#717992
#704392
#677472
#677082
#93723