Finite Group Databasebeta

← #11778251

Subgroups — #11778251

Order 1152.

613 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 80 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#10594981C211yesmaximal, characteristic
2#10592930C211yesmaximal, characteristic
3C3×#5228C212yesmaximal, complemented
4#10589152C212yesmaximal, complemented
5#10589146C211yesmaximal, characteristic
6#70980—32nomaximal
7C22×C32×Q8C2211yescharacteristic
8C2×#3752—21no—
9C2×#3752C2211yescharacteristic
10C2×C3×#216—22no—
11C3×#924C2212yes—
12#63982—21no—
13#63974—21no—
14#63895—22no—
15#63888—22no—
16#63658—21no—
17#63658C2211yescharacteristic
18C3×#835—22no—
19C3×#830C2212yes—
20#5634—32no—
21C3×#451—32no—
22#5231—32no—
23#5231—61no—
24#5228D312yescomplemented
25#4538—32no—
26#4468—32no—
27#4467—32no—
28C23×C3⋊D3D411yescharacteristic
29C2×C32×Q8—22no—
30C2×C32×Q8—42no—
31C22×C32×C4—21no—
32C22×C32×C4C2311yescharacteristic
33C22×C3⋊Dic3—21no—
34C22×C3⋊Dic3D411yescharacteristic
35C2×C3×Dic6D412yes—
36C2×#627—21no—
37C2×#624—21no—
38C2×#624—21no—
39#3752—22no—
40#3752—42no—
41C3×#217—22no—
42C3×#216—44no—
43C3×C4⋊Dic3D412yes—
44C3×Dic3⋊C4—22no—
45C2×C3×C3⋊C8—22no—
46#3724—21no—
47Dic3⋊Dic3—22no—
48#3722—22no—
49#1602—91noSylow
50C22×C3×Q8—21no—
51C22×C3×Q8C2×D312yes—
52C2×C3×Q16—62no—
53C3×C22⋊Q8—32no—
54#946—62no—
55C2×#216—22no—
56C2×(C2×D3)⋊C4—32no—
57C2×(C2×D3)⋊C4—61no—
58C2×(C2×D3)⋊C4—62no—
59C2×(C2×D3)⋊C4—62no—
60#924C2×D312yes—
61#897—62no—
62#896—62no—
63#896—62no—
64D3×C22⋊C4—62no—
65#878—62no—
66C3×#103—62no—
67C3×C22⋊C8—32no—
68#835—22no—
69C3⋊#95—32no—
70C3⋊#95—61no—
71C3⋊#95—62no—
72C3⋊#95—62no—
73#830C2×D312yes—
74C23×C32C2×D411yescharacteristic
75C22×C3⋊D3—21no—
76C22×C3⋊D3—21no—
77C22×C3⋊D3—41no—
78C22×C3⋊D3—41no—
79C22×C3⋊D3Q1612yescomplemented
80C32×Q8—24no—
81C32×Q8—42no—
82C2×C32×C4—22no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4C2×D412yes—
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—41no—
89C2×C3⋊Dic3—41no—
90C2×C3⋊Dic3—41no—
91C2×C3⋊Dic3—41no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C3×Dic6—42no—
95#627—81no—
96#624—81no—
97#624—81no—
98C3×C3⋊C8—42no—
99#543—91no—
100#451—32no—
101#393—91no—
102#328—92no—
103#323—91no—
104C23×D3—32no—
105C23×D3—61no—
106C2×C3×Q8—22no—
107C2×C3×Q8—22no—
108C2×C3×Q8—32no—
109C2×C3×Q8—42no—
110C2×C3×Q8—44no—
111C2×C3×Q8—44no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4C22×D312yes—
116C22×Dic3—32no—
117C22×Dic3—61no—
118C22×Dic3—62no—
119C22×Dic3—62no—
120C2×C4×D3—62no—
121C2×C4×D3—62no—
122C2×Dic6—62no—
123C2×Dic6#6712yes—
124C3×Q16—124no—
125C2×C3×C8—62no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—62no—
128C3×C22⋊C4—62no—
129#217—22no—
130#216—44no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4—62no—
133(C2×D3)⋊C4—62no—
134(C2×D3)⋊C4—62no—
135(C2×D3)⋊C4—122no—
136(C2×D3)⋊C4—124no—
137(C2×D3)⋊C4—124no—
138C4⋊Dic3—62no—
139C4⋊Dic3#6712yes—
140Dic3⋊C4—22no—
141Dic3⋊C4—62no—
142Dic3⋊C4—62no—
143C2×C3⋊C8—22no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32#12011yescharacteristic
147C22×C32C2×Q1611yescharacteristic
148C22×C32#13711yescharacteristic
149C2×C3⋊D3—22no—
150C2×C3⋊D3—42no—
151C2×C3⋊D3—41no—
152C2×C3⋊D3—41no—
153C2×C3⋊D3—81no—
154C32×C4—22no—
155C32×C4—42no—
156C3⋊Dic3—41no—
157C3⋊Dic3—81no—
158C3×Dic3—42no—
159C3×Dic3—42no—
160C22×Q8D3211yescharacteristic
161C2×Q16—62no—
162C22⋊Q8—32no—
163C22⋊Q8—181no—
164C22⋊Q8—182no—
165C22⋊Q8—182no—
166C2×C22⋊C4—91no—
167C2×C22⋊C4—181no—
168C2×C22⋊C4—181no—
169#103—62no—
170C22⋊C8—32no—
171#95—91no—
172#95—181no—
173C23×C3—21no—
174C23×C3D3×D412yes—
175C22×D3—34no—
176C22×D3—62no—
177C22×D3—62no—
178C22×D3—62no—
179C22×D3—121no—
180C22×D3—121no—
181C22×D3—122no—
182C22×D3—122no—
183C22×D3—122no—
184C22×D3—122no—
185C3×Q8—24no—
186C3×Q8—24no—
187C3×Q8—44no—
188C3×Q8—42no—
189C3×Q8—42no—
190C3×Q8—44no—
191C3×Q8—122no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4D3×D412yes—
204C2×C3×C4C2×#6712yes—
205C2×Dic3—22no—
206C2×Dic3—22no—
207C2×Dic3—62no—
208C2×Dic3—121no—
209C2×Dic3—122no—
210C2×Dic3—122no—
211C2×Dic3—122no—
212C2×Dic3—122no—
213C2×Dic3—122no—
214C2×Dic3—122no—
215C2×Dic3—122no—
216C2×Dic3—122no—
217C4×D3—242no—
218C4×D3—242no—
219Dic6—42no—
220Dic6—242no—
221C3×C8—122no—
222C3⋊C8—42no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32#45111yescharacteristic
226C2×C32#45511yescharacteristic
227C2×C32B(4,𝔽2)11yescharacteristic
228C3⋊D3—42no—
229C24—91no—
230C2×Q8—22no—
231C2×Q8—32no—
232C2×Q8—42no—
233C2×Q8—181no—
234C22×C4—21no—
235C22×C4—91no—
236C22×C4—181no—
237C22×C4—181no—
238C22×C4—181no—
239C22×C4C2×D3211yescharacteristic
240Q16—124no—
241C2×C8—62no—
242C4⋊C4—32no—
243C4⋊C4—62no—
244C4⋊C4—181no—
245C4⋊C4—182no—
246C22⋊C4—62no—
247C22⋊C4—182no—
248C22⋊C4—182no—
249C22⋊C4—362no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3D3×Q1612yes—
258C22×C3#91812yes—
259C22×C3#93712yes—
260C2×D3—64no—
261C2×D3—122no—
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—124no—
265C2×D3—124no—
266C2×D3—241no—
267C2×D3—241no—
268C2×D3—242no—
269C2×D3—242no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—44no—
274C3×C4—44no—
275C3×C4—122no—
276C3×C4—122no—
277Dic3—42no—
278Dic3—42no—
279Dic3—122no—
280Dic3—241no—
281Dic3—242no—
282Dic3—242no—
283C32#160211yescharacteristic, complemented, Sylow
284C23—92no—
285C23—181no—
286C23—181no—
287C23—361no—
288C23—361no—
289C23#381111yescharacteristic
290Q8—24no—
291Q8—42no—
292Q8—122no—
293Q8—721no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—181no—
301C2×C4—361no—
302C2×C4—361no—
303C2×C4—361no—
304C2×C4—361no—
305C2×C4—721no—
306C2×C4—721no—
307C2×C4D3×#6712yes—
308C8—122no—
309C2×C3—22no—
310C2×C3—22no—
311C2×C3—22no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3#503212yes—
317C2×C3#518612yes—
318C2×C3#520212yes—
319D3—124no—
320D3—242no—
321C22—21no—
322C22—21no—
323C22—182no—
324C22—362no—
325C22—361no—
326C22—361no—
327C22—721no—
328C22#6394411yescharacteristic
329C22#6394511yescharacteristic
330C22#6398311yescharacteristic
331C4—22no—
332C4—42no—
333C4—122no—
334C4—122no—
335C4—361no—
336C4—721no—
337C3—21no—
338C3#7098012yescomplemented
339C2—21no—
340C2—21no—
341C2—362no—
342C2#1059165411yescharacteristic
343C2#1059165811yescharacteristic
344C2#1059172511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radicalC23×C3⋊D3D4
hypercenterC22×Q8D32
hypocenterC32#1602
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#5228C2semidirect
4. #10589152C2semidirect
24. #5228D3semidirect
79. C22×C3⋊D3Q16semidirect
283. C32#1602semidirect
338. C3#70980semidirect

Maximal subgroups

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

subgroupindex
#105949812
#105929302
C3×#52282
#105891522
#105891462
#709803