Finite Group Databasebeta

← #11778595

Subgroups — #11778595

Order 1152.

459 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C3×#5500C211yesmaximal, characteristic
2C3×#4981C211yesmaximal, characteristic
3#10591101C212yesmaximal
4#10589394C211yesmaximal, characteristic
5#10589153C212yesmaximal
6#71898—31nomaximal
7#69713—31nomaximal
8C2×C32×C4⋊C4C2211yescharacteristic
9C3×#926—21no—
10C2×C3×C4⋊Dic3C2211yescharacteristic
11C3×#924—21no—
12C3×#924C2212yes—
13C2×C3×Dic3⋊C4—21no—
14C3×#890—22no—
15C3×#888—22no—
16#63657C2212yes—
17C3×C3⋊#95—21no—
18C3×C3⋊#95C2211yescharacteristic
19#63574—22no—
20#63565—22no—
21C3×#544—31no—
22#5500D311yescharacteristic
23C3×#398—31no—
24#5043—32no—
25#4981D311yescharacteristic
26#4971—32no—
27#4538—32no—
28#4528—31no—
29#4528—31no—
30#4528—61no—
31#4452—32no—
32C22×C32×C4D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C2×C3×Dic6—21no—
38C2×C3×Dic6D412yes—
39C32×C4⋊C4—22no—
40C2×C3⋊(C3⋊C8)—22no—
41C3×#217—22no—
42C3×C4⋊Dic3—21no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—22no—
46C3×C4⋊Dic3D412yes—
47C3×Dic3⋊C4—22no—
48C3×Dic3⋊C4—22no—
49C3×Dic3⋊C4—42no—
50C3×C4×Dic3—22no—
51#1611—91noSylow
52C3×C4⋊Q8—62no—
53C3×#125—62no—
54C3×C22⋊Q8—32no—
55C3×C22⋊Q8—61no—
56C3×#117—61no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—31no—
59C2×C3×C4⋊C4—61no—
60C2×C3×C4⋊C4C2×D311yescharacteristic
61C2×C3×C4⋊C4C2×D311yescharacteristic
62#926—21no—
63C2×C4⋊Dic3C2×D311yescharacteristic
64#924—21no—
65#924C2×D312yes—
66C2×Dic3⋊C4—21no—
67#890—22no—
68#888—22no—
69C3×#95—31no—
70C3×#95—61no—
71#835—62no—
72C3⋊#95—21no—
73C3⋊#95C2×D311yescharacteristic
74#830—32no—
75#830—32no—
76#830—62no—
77#810—62no—
78#808—62no—
79#808—62no—
80C23×C32C2×D411yescharacteristic
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4SD1612yes—
87C2×C32×C4C2×D412yes—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C3×Dic6—42no—
97C3×Dic6—81no—
98C3⋊(C3⋊C8)—42no—
99#544—31no—
100#483—92no—
101#398—31no—
102#328—92no—
103#324—91no—
104C2×C3×Q8—32no—
105C2×C3×Q8—61no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—61no—
110C22×C3×C4—61no—
111C22×C3×C4#6711yescharacteristic
112C22×C3×C4#6711yescharacteristic
113C22×C3×C4#6711yescharacteristic
114C22×C3×C4#6711yescharacteristic
115C22×C3×C4C22×D311yescharacteristic
116C22×C3×C4C22×D311yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C2×Dic6—21no—
120C2×Dic6#6712yes—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—42no—
125C3×C4⋊C4—62no—
126C3×C4⋊C4—61no—
127C3×C4⋊C4—61no—
128C3×C4⋊C4—61no—
129C3×C4⋊C4—62no—
130C3×C4⋊C4—62no—
131C3×C4⋊C4—122no—
132C3×C22⋊C4—62no—
133C3×C42—62no—
134#217—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—22no—
139C4⋊Dic3#6712yes—
140Dic3⋊C4—22no—
141Dic3⋊C4—22no—
142Dic3⋊C4—42no—
143C4×Dic3—22no—
144C2×C3⋊C8—62no—
145C2×C3⋊C8—62no—
146C2×C3⋊C8—122no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32#12011yescharacteristic
150C22×C32C2×SD1611yescharacteristic
151C22×C32#13711yescharacteristic
152C32×C4—22no—
153C32×C4—22no—
154C32×C4—41no—
155C3×Dic3—42no—
156C3×Dic3—42no—
157C3×Dic3—42no—
158C3×Dic3—81no—
159C4⋊Q8—62no—
160#125—62no—
161C22⋊Q8—32no—
162C22⋊Q8—61no—
163#117—61no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4D3211yescharacteristic
167#106—182no—
168#103—182no—
169C22⋊C8—92no—
170#95—31no—
171#95—61no—
172C23×C3—21no—
173C23×C3C2×#6711yescharacteristic
174C23×C3C2×#6711yescharacteristic
175C3×Q8—122no—
176C3×Q8—241no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
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×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4—121no—
202C2×C3×C4#21412yes—
203C2×C3×C4#21512yes—
204C2×C3×C4D3×D412yes—
205C2×C3×C4C2×#6712yes—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214Dic6—42no—
215Dic6—81no—
216C3⋊C8—122no—
217C3⋊C8—122no—
218C3⋊C8—242no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32#44811yescharacteristic
222C2×C32#45611yescharacteristic
223C2×C32#45811yescharacteristic
224C2×Q8—32no—
225C2×Q8—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4#62511yescharacteristic
229C22×C4#62511yescharacteristic
230C22×C4C2×D3211yescharacteristic
231C2×C8—182no—
232C4⋊C4—22no—
233C4⋊C4—32no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—122no—
241C22⋊C4—62no—
242C42—62no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3C2×#21411yescharacteristic
253C22×C3#93711yescharacteristic
254C22×C3C2×#21511yescharacteristic
255C22×C3#94211yescharacteristic
256C22×C3#95111yescharacteristic
257C22×C3#95311yescharacteristic
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—42no—
263C3×C4—42no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—81no—
267C3×C4—122no—
268C3×C4—122no—
269C3×C4—122no—
270C3×C4—241no—
271Dic3—42no—
272Dic3—42no—
273Dic3—42no—
274Dic3—81no—
275C32#161111yescharacteristic, complemented, Sylow
276C23C2×#62511yescharacteristic
277Q8—122no—
278Q8—241no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4#371512yes—
293C2×C4D3×#6712yes—
294C8—362no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3#504011yescharacteristic
305C2×C3#505311yescharacteristic
306C2×C3#509511yescharacteristic
307C2×C3#522711yescharacteristic
308C2×C3#524311yescharacteristic
309C2×C3#524611yescharacteristic
310C22—21no—
311C22—21no—
312C22C2×#371511yescharacteristic
313C22#6382711yescharacteristic
314C22#6397611yescharacteristic
315C4—22no—
316C4—22no—
317C4—41no—
318C4—122no—
319C4—122no—
320C4—122no—
321C4—241no—
322C3—21no—
323C3#6971311yescharacteristic, complemented
324C3#7189811yescharacteristic, complemented
325C2—21no—
326C2—21no—
327C2#1059117111yescharacteristic
328C2#1059153311yescharacteristic
329C2#1059164811yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC2×C4⋊C4D32
hypocenterC32#1611
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
275. C32#1611semidirect
323. C3#69713semidirect
324. C3#71898semidirect

Maximal subgroups

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

subgroupindex
C3×#55002
C3×#49812
#105911012
#105893942
#105891532
#718983
#697133