Finite Group Databasebeta

← #11781806

Subgroups — #11781806

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#10592793C211yesmaximal, characteristic
2C3×#5092C211yesmaximal, characteristic
3#10590307C211yesmaximal, characteristic
4#10590296C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic
6#10589097C211yesmaximal, characteristic
7#10589093C211yesmaximal, characteristic
8#71309—31nomaximal
9#68473—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C4×#634C2211yescharacteristic
12C2×C3×#216—21no—
13C2×C3×#216—21no—
14C3×#869C2211yescharacteristic
15#63637—21no—
16#63635—21no—
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
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—
25C3×#494—31no—
26#5094—31no—
27#5094—31no—
28#5094—31no—
29#5094—61no—
30#5092D311yescharacteristic
31#4851—31no—
32#4831—31no—
33#4679—31no—
34C3×#332—31no—
35#4547—31no—
36#4547D311yescharacteristic
37#4482—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8—21no—
41C2×#634D411yescharacteristic
42C2×C3×Dic6—21no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C42C2311yescharacteristic
46C4⋊(C3⋊Dic3)D411yescharacteristic
47#3750—21no—
48C4×C3⋊Dic3—21no—
49C2×C3⋊(C3⋊C8)—21no—
50C3×#216—42no—
51C3×#216—42no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8D411yescharacteristic
57C2×C3×C3⋊C8D411yescharacteristic
58#1690—91noSylow
59C2×C3×Q16—61no—
60C2×C3×Q16—61no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—31no—
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4⋊Q8C2×D311yescharacteristic
65C2×#216—21no—
66C2×#216—21no—
67#869C2×D311yescharacteristic
68C4×Dic6—31no—
69C4×Dic6—31no—
70C4×Dic6—61no—
71C3×C4⋊C8—31no—
72C3×C4⋊C8—31no—
73C3×C4×C8—31no—
74#817—61no—
75#816—61no—
76#815—61no—
77#815—61no—
78Dic3⋊C8—61no—
79Dic3⋊C8—61no—
80#810—61no—
81#810—61no—
82#810—61no—
83#810—62no—
84#808—61no—
85#808—61no—
86#808—61no—
87#808—62no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4×C3⋊C8C2×D311yescharacteristic
94C32×Q8—41no—
95C32×Q8—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102#634—21no—
103#634—21no—
104C2×C3×Dic3—41no—
105C3×Dic6—41no—
106C3×Dic6—41no—
107C3⋊(C3⋊C8)—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8SD1612yes—
112#494—31no—
113#475—91no—
114#475—91no—
115#446—91no—
116#334—91no—
117#332—31no—
118#332—91no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—41no—
122C2×C3×Q8—61no—
123C2×Dic6—21no—
124C2×Dic6—31no—
125C2×Dic6—31no—
126C2×Dic6—61no—
127C3×Q16—122no—
128C3×Q16—122no—
129C2×C3×C8—31no—
130C2×C3×C8—31no—
131C2×C3×C8—61no—
132C2×C3×C8—61no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C42—21no—
142C3×C42C22×D311yescharacteristic
143C3×C42C22×D311yescharacteristic
144#216—42no—
145#216—42no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—61no—
151Dic3⋊C4—61no—
152Dic3⋊C4—61no—
153Dic3⋊C4—121no—
154C4×Dic3—61no—
155C4×Dic3—61no—
156C4×Dic3—121no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—62no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8#6711yescharacteristic
163C2×C3⋊C8#6711yescharacteristic
164C22×C32#12111yescharacteristic
165C32×C4—21no—
166C32×C4—81no—
167C32×C4C2×SD1611yescharacteristic
168C32×C4AGL(1,ℤ/8)11yescharacteristic
169C32×C4#13711yescharacteristic
170C32×C4#13711yescharacteristic
171C3⋊Dic3—41no—
172C3⋊Dic3—41no—
173C3×Dic3—81no—
174C2×Q16—61no—
175C2×Q16—61no—
176C4⋊Q8—31no—
177C4⋊Q8D3211yescharacteristic
178C4×Q8—91no—
179#106—181no—
180#106—181no—
181C4⋊C8—31no—
182C4⋊C8—31no—
183C4⋊C8—91no—
184C4⋊C8—181no—
185#103—181no—
186#103—181no—
187#97—181no—
188C4×C8—31no—
189C3×Q8—41no—
190C3×Q8—41no—
191C3×Q8—41no—
192C3×Q8—41no—
193C3×Q8—81no—
194C3×Q8—81no—
195C3×Q8—121no—
196C3×Q8—121no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—121no—
204C2×C3×C4#23511yescharacteristic
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4#23711yescharacteristic
209C2×C3×C4C2×#6711yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—121no—
216C2×Dic3—121no—
217Dic6—41no—
218Dic6—41no—
219Dic6—62no—
220Dic6—61no—
221Dic6—61no—
222Dic6—61no—
223Dic6—61no—
224Dic6—121no—
225C3×C8—32no—
226C3×C8—61no—
227C3×C8—121no—
228C3×C8—121no—
229C3⋊C8—21no—
230C3⋊C8—41no—
231C3⋊C8—41no—
232C3⋊C8—122no—
233C3⋊C8—121no—
234C3⋊C8—121no—
235C3⋊C8#21412yes—
236C2×C32#46011yescharacteristic
237C2×C32#47011yescharacteristic
238C2×C32#47311yescharacteristic
239C2×Q8—21no—
240C2×Q8—61no—
241C2×Q8—91no—
242Q16—122no—
243Q16—122no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8—181no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C42—181no—
256C42C2×D3211yescharacteristic
257C22×C3—21no—
258C22×C3#88411yescharacteristic
259C22×C3#93811yescharacteristic
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—41no—
267C3×C4—82no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4—241no—
271C3×C4#91111yescharacteristic
272C3×C4D3×SD1611yescharacteristic
273C3×C4#91511yescharacteristic
274C3×C4#91811yescharacteristic
275C3×C4#91811yescharacteristic
276C3×C4#91811yescharacteristic
277C3×C4#93211yescharacteristic
278C3×C4C2×#21411yescharacteristic
279Dic3—81no—
280Dic3—122no—
281Dic3—121no—
282Dic3—121no—
283Dic3—121no—
284Dic3—121no—
285Dic3—241no—
286C32#169011yescharacteristic, complemented, Sylow
287Q8—41no—
288Q8—41no—
289Q8—121no—
290Q8—121no—
291Q8—181no—
292Q8—181no—
293C2×C4—41no—
294C2×C4—121no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4#380411yescharacteristic
298C2×C4D3×#6711yescharacteristic
299C2×C4#381111yescharacteristic
300C8—32no—
301C8—61no—
302C8—121no—
303C8—121no—
304C8—361no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3#475511yescharacteristic
309C2×C3#476211yescharacteristic
310C2×C3#481611yescharacteristic
311C2×C3#507511yescharacteristic
312C2×C3#518911yescharacteristic
313C2×C3#519211yescharacteristic
314C22#6395811yescharacteristic
315C4—21no—
316C4—81no—
317C4—241no—
318C4—361no—
319C4—361no—
320C4D3×#21411yescharacteristic
321C4#6393211yescharacteristic
322C4#6394511yescharacteristic
323C4#6395111yescharacteristic
324C3—21no—
325C3#6847311yescharacteristic, complemented
326C3#7130911yescharacteristic, complemented
327C2#1059143911yescharacteristic
328C2#1059154311yescharacteristic
329C2#1059160011yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socle, prime radicalC22×C32#121
hypercenterC4⋊Q8D32
hypocenterC32#1690
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
286. C32#1690semidirect
325. C3#68473semidirect
326. C3#71309semidirect

Maximal subgroups

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

subgroupindex
#105927932
C3×#50922
#105903072
#105902962
C3×#45472
#105890972
#105890932
#713093
#684733