Finite Group Databasebeta

← #11781814

Subgroups — #11781814

Order 1152.

352 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 86 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#10592550C211yesmaximal, characteristic
2#10590386C211yesmaximal, characteristic
3#10590307C211yesmaximal, characteristic
4#10590296C211yesmaximal, characteristic
5C3×#4486C211yesmaximal, characteristic
6C3×#4480C211yesmaximal, characteristic
7#10589094C211yesmaximal, characteristic
8#68551—31nomaximal
9#68473—31nomaximal
10#64078C2211yescharacteristic
11C4×#634C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3719—21no—
14C2×#3719—21no—
15C32×C4⋊C8C2211yescharacteristic
16#63644—21no—
17#63643—21no—
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63573—21no—
22#63572—21no—
23#63567—21no—
24#63555—21no—
25#5094—31no—
26#5092—31no—
27#4851—31no—
28#4851—31no—
29#4851—31no—
30#4851—61no—
31#4831—31no—
32#4699—31no—
33#4679—31no—
34C3×#334—31no—
35C3×#332—31no—
36#4486—31no—
37#4486D311yescharacteristic
38#4482—31no—
39#4480D311yescharacteristic
40C2×#634—21no—
41C2×#634D411yescharacteristic
42C2×C3×Dic6—21no—
43C2×C32×C8—21no—
44C32×C42C2311yescharacteristic
45C4⋊(C3⋊Dic3)—21no—
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)D411yescharacteristic
48#3750—21no—
49C4×C3⋊Dic3—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3719—42no—
57#3719—42no—
58#1690—91noSylow
59C3×C4⋊Q8—31no—
60C2×#216—61no—
61C2×#216—61no—
62C2×Dic12—61no—
63C2×Dic12—61no—
64#869—31no—
65#869—31no—
66#869—61no—
67#869C2×D311yescharacteristic
68C4×Dic6—31no—
69C4×Dic6—31no—
70C4×Dic6—61no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×C4⋊C8C2×D311yescharacteristic
75C3×C4⋊C8C2×D311yescharacteristic
76C3×C4×C8—31no—
77#817—61no—
78#817—61no—
79#817—61no—
80#817—62no—
81#816—61no—
82#816—61no—
83#816—61no—
84#816—62no—
85#815—61no—
86#815—61no—
87Dic3⋊C8—61no—
88Dic3⋊C8—61no—
89#810—61no—
90#808—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4×C3⋊C8C2×D311yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4111yescharacteristic
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—41no—
100#634—21no—
101#634—21no—
102#634—41no—
103#634—41no—
104C2×C3×Dic3—41no—
105C3×Dic6—41no—
106C3×Dic6—41no—
107C32×C8—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8SD1612yes—
112#494—91no—
113#475—91no—
114#475—91no—
115#446—91no—
116#334—31no—
117#332—31no—
118#332—91no—
119C2×C3×Q8—61no—
120C2×Dic6—21no—
121C2×Dic6—31no—
122C2×Dic6—31no—
123C2×Dic6—61no—
124C2×Dic6—61no—
125C2×Dic6—61no—
126C2×Dic6—121no—
127C2×C3×C8—22no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—61no—
133C2×C3×C8—61no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C42—21no—
137C3×C42C22×D311yescharacteristic
138C3×C42C22×D311yescharacteristic
139#216—122no—
140#216—122no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—31no—
144C4⋊Dic3—31no—
145C4⋊Dic3—61no—
146C4⋊Dic3—62no—
147C4⋊Dic3—61no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—61no—
151C4⋊Dic3—121no—
152Dic3⋊C4—61no—
153Dic3⋊C4—61no—
154Dic3⋊C4—121no—
155C4×Dic3—61no—
156C4×Dic3—61no—
157C4×Dic3—121no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8D1211yescharacteristic
161C2×C3⋊C8D1211yescharacteristic
162Dic12—122no—
163Dic12—122no—
164C22×C32#12111yescharacteristic
165C32×C4—21no—
166C32×C4C2×SD1611yescharacteristic
167C32×C4AGL(1,ℤ/8)11yescharacteristic
168C32×C4#13711yescharacteristic
169C32×C4#13711yescharacteristic
170C3⋊Dic3—41no—
171C3⋊Dic3—41no—
172C3⋊Dic3—81no—
173C3×Dic3—81no—
174C2×Q16—181no—
175C2×Q16—181no—
176C4⋊Q8—31no—
177C4⋊Q8—91no—
178C4×Q8—91no—
179#106—181no—
180#106—181no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—181no—
184C4⋊C8D3211yescharacteristic
185#103—181no—
186#103—181no—
187#97—181no—
188C4×C8—31no—
189C3×Q8—121no—
190C3×Q8—121no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—121no—
195C2×C3×C4C2×D1211yescharacteristic
196C2×C3×C4#23511yescharacteristic
197C2×C3×C4D3×D411yescharacteristic
198C2×C3×C4D3×D411yescharacteristic
199C2×C3×C4D3×D411yescharacteristic
200C2×C3×C4#23911yescharacteristic
201C2×Dic3—41no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—121no—
207C2×Dic3—121no—
208C2×Dic3—122no—
209C2×Dic3—121no—
210C2×Dic3—121no—
211Dic6—41no—
212Dic6—41no—
213Dic6—62no—
214Dic6—61no—
215Dic6—61no—
216Dic6—61no—
217Dic6—61no—
218Dic6—121no—
219Dic6—121no—
220Dic6—121no—
221Dic6—121no—
222Dic6—121no—
223Dic6—241no—
224Dic6—241no—
225C3×C8—32no—
226C3×C8—42no—
227C3×C8—41no—
228C3×C8—41no—
229C3×C8—61no—
230C3×C8—121no—
231C3×C8—121no—
232C3⋊C8—21no—
233C3⋊C8—41no—
234C3⋊C8—41no—
235C3⋊C8#20412yes—
236C2×C32#46011yescharacteristic
237C2×C32#47011yescharacteristic
238C2×C32#47311yescharacteristic
239C2×Q8—61no—
240C2×Q8—91no—
241C2×Q8—181no—
242Q16—362no—
243Q16—362no—
244C2×C8—21no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—61no—
248C2×C8—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—91no—
252C4⋊C4—181no—
253C4⋊C4—181no—
254C4⋊C4—181no—
255C42—181no—
256C42C2×D3211yescharacteristic
257C22×C3—21no—
258C22×C3#88411yescharacteristic
259C22×C3#89511yescharacteristic
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—41no—
267C3×C4—241no—
268C3×C4C2×#20411yescharacteristic
269C3×C4#90811yescharacteristic
270C3×C4#91111yescharacteristic
271C3×C4D3×SD1611yescharacteristic
272C3×C4#91511yescharacteristic
273C3×C4#91511yescharacteristic
274C3×C4#91511yescharacteristic
275C3×C4#91811yescharacteristic
276Dic3—81no—
277Dic3—122no—
278Dic3—121no—
279Dic3—121no—
280Dic3—121no—
281Dic3—121no—
282Dic3—241no—
283Dic3—242no—
284Dic3—241no—
285Dic3—241no—
286C32#169011yescharacteristic, complemented, Sylow
287Q8—121no—
288Q8—121no—
289Q8—181no—
290Q8—181no—
291Q8—361no—
292Q8—361no—
293C2×C4—121no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—361no—
297C2×C4#379911yescharacteristic
298C2×C4D3×D1211yescharacteristic
299C2×C4#381111yescharacteristic
300C8—32no—
301C8—41no—
302C8—61no—
303C8—121no—
304C8—121no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3#475511yescharacteristic
309C2×C3#476211yescharacteristic
310C2×C3#481611yescharacteristic
311C2×C3#484811yescharacteristic
312C2×C3#486711yescharacteristic
313C2×C3#489611yescharacteristic
314C22#6391011yescharacteristic
315C4—21no—
316C4—241no—
317C4—361no—
318C4—361no—
319C4—721no—
320C4D3×#20411yescharacteristic
321C4#6379511yescharacteristic
322C4#6393111yescharacteristic
323C4#6395111yescharacteristic
324C3—21no—
325C3#6847311yescharacteristic, complemented
326C3#6855111yescharacteristic, complemented
327C2#1059052411yescharacteristic
328C2#1059087911yescharacteristic
329C2#1059099211yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socle, prime radicalC22×C32#121
hypercenterC4⋊C8D32
hypocenterC32#1690
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
286. C32#1690semidirect
325. C3#68473semidirect
326. C3#68551semidirect

Maximal subgroups

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

subgroupindex
#105925502
#105903862
#105903072
#105902962
C3×#44862
C3×#44802
#105890942
#685513
#684733