Finite Group Databasebeta

← #11798775

Subgroups — #11798775

Order 1152.

464 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 68 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×#5672C211yesmaximal, characteristic, complemented
2C3×#5062C211yesmaximal, characteristic
3#10591648C212yesmaximal, complemented
4#10589408C211yesmaximal, characteristic, complemented
5#10589184C212yesmaximal, complemented
6#71959—31nomaximal
7#70989—31nomaximal
8C3×#1007C2211yescharacteristic
9C32×#124C2211yescharacteristic
10C3×#944C2212yes—
11C3×#935—21no—
12C3×#932—21no—
13C3×#879—22no—
14C3×#878—22no—
15C3×#870—21no—
16#63827—22no—
17#63663C2212yes—
18C3×#836C2211yescharacteristic
19C3×#805—21no—
20#63569—22no—
21C3×#563—31no—
22#5672D311yescharacteristic, complemented
23C3×#454—31no—
24#5246—32no—
25#5095—32no—
26#5062D311yescharacteristic
27#4548—32no—
28#4542—31no—
29#4542—31no—
30#4542—61no—
31#4478—32no—
32C2×C32×Q8C2311yescharacteristic
33C2×C32×D4D411yescharacteristic
34C22×C3×Dic3—21no—
35C3×#239—41no—
36C3×D3×Q8—41no—
37C3×#235—21no—
38C3×#235—22no—
39C2×C3×Dic6D412yes—
40C32×C22⋊C4—22no—
41C32×C42D411yescharacteristic
42#3748—22no—
43C3×#217—22no—
44C3×#217—21no—
45C3×#214—41no—
46C3×#213—41no—
47C3×C4⋊Dic3—21no—
48C3×C4⋊Dic3—21no—
49C3×Dic3⋊C4—22no—
50C3×Dic3⋊C4—22no—
51C3×Dic3⋊C4—22no—
52C3×C4×Dic3D412yes—
53C3×#208—21no—
54#3718—42no—
55#3715—42no—
56#2203—91noSylow
57C3×2−1+4—31no—
58#1007C2×D311yescharacteristic
59C3×AGL(1,ℤ/8)—61no—
60C3×C4⋊Q8—32no—
61C3×#126—62no—
62C3×#125—61no—
63C3×#124—21no—
64C3×#124C2×D311yescharacteristic
65C3×#124C2×D311yescharacteristic
66C3×#123—61no—
67C3×C22⋊Q8—62no—
68#951—62no—
69#944C2×D312yes—
70#942—62no—
71#935—21no—
72#932—21no—
73#879—22no—
74#878—22no—
75#870—21no—
76C3×#104—61no—
77C3×#101—31no—
78#837—62no—
79#836—32no—
80#836—32no—
81#836—62no—
82#836C2×D311yescharacteristic
83#805—21no—
84#805—62no—
85C23×C32—21no—
86C32×Q8—41no—
87C32×D4—41no—
88C2×C32×C4—21no—
89C2×C32×C4C2×D412yes—
90C2×C32×C4C2×D411yescharacteristic
91C3×#67—41no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—81no—
95C3×D12—22no—
96C3×D12—21no—
97C3×C4×D3—41no—
98C3×C4×D3—42no—
99C3×Dic6—22no—
100C3×Dic6—21no—
101C3×Dic6—42no—
102C3⋊(C3⋊C8)—42no—
103C3×C3⋊C8—41no—
104#563—31no—
105#456—92no—
106#454—31no—
107#356—92no—
108#355—91no—
109C3×#41—62no—
110C3×#41—61no—
111C3×#41—121no—
112C2×C3×Q8—21no—
113C2×C3×Q8—32no—
114C2×C3×Q8—121no—
115C2×C3×Q8C22×D311yescharacteristic
116C2×C3×Q8C22×D311yescharacteristic
117C2×C3×D4—21no—
118C2×C3×D4#6711yescharacteristic
119C2×C3×D4#6711yescharacteristic
120C22×C3×C4—61no—
121C22×Dic3—21no—
122#239—41no—
123D3×Q8—41no—
124#235—22no—
125#235—21no—
126C2×Dic6#6712yes—
127C3×SD16—121no—
128C3×D8—121no—
129C3×M16—61no—
130C3×C4⋊C4—62no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—62no—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—42no—
138C3×C22⋊C4—62no—
139C3×C22⋊C4—61no—
140C3×C42—21no—
141C3×C42—32no—
142C3×C42#6711yescharacteristic
143C3×C42#6711yescharacteristic
144#217—22no—
145#217—21no—
146#216—122no—
147#216—122no—
148#215—122no—
149#214—41no—
150#214—122no—
151#213—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—22no—
155Dic3⋊C4—22no—
156Dic3⋊C4—22no—
157C4×Dic3#6712yes—
158#208—21no—
159#208—62no—
160#208—62no—
161#208—122no—
162C22×C32—42no—
163C22×C32—41no—
164C22×C32#12011yescharacteristic
165C2×C3×D3—41no—
166C32×C4—21no—
167C32×C4—22no—
168C32×C4—41no—
169C3×Dic3—42no—
170C3×Dic3—41no—
171C3×Dic3—42no—
1722−1+4—31no—
173#137—182no—
174AGL(1,ℤ/8)—61no—
175C4⋊Q8—32no—
176#126—62no—
177#125—61no—
178#124D3211yescharacteristic
179#123—61no—
180C22⋊Q8—62no—
181#104—61no—
182#104—182no—
183#101—31no—
184#101—92no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—41no—
188C3×Q8—41no—
189C3×Q8—41no—
190C3×Q8—62no—
191C3×Q8—61no—
192C3×Q8—81no—
193C3×Q8—122no—
194C3×D4—41no—
195C3×D4—41no—
196C3×D4—62no—
197C3×D4—61no—
198C3×D4—81no—
199C3×D4—121no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—122no—
208C2×C3×C4—121no—
209C2×C3×C4—241no—
210C2×C3×C4D3×D412yes—
211C2×C3×C4C2×#6711yescharacteristic
212C2×C3×C4C2×#6711yescharacteristic
213C2×C3×C4C2×#6712yes—
214#67—41no—
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—81no—
218D12—21no—
219D12—22no—
220C4×D3—41no—
221C4×D3—42no—
222Dic6—21no—
223Dic6—22no—
224Dic6—42no—
225C3×C8—121no—
226C3⋊C8—41no—
227C3⋊C8—122no—
228C3⋊C8—122no—
229C3⋊C8—242no—
230C2×C32—21no—
231C2×C32—81no—
232C2×C32B(4,𝔽2)11yescharacteristic
233C3×D3—81no—
234#41—61no—
235#41—62no—
236#41—121no—
237C2×Q8—32no—
238C2×Q8—121no—
239C2×Q8C2×D3211yescharacteristic
240C2×D4#62511yescharacteristic
241C22×C4—61no—
242Q16—362no—
243SD16—121no—
244SD16—362no—
245D8—121no—
246M16—61no—
247M16—182no—
248C4⋊C4—61no—
249C4⋊C4—62no—
250C4⋊C4—61no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C22⋊C4—22no—
254C22⋊C4—62no—
255C22⋊C4—61no—
256C42—32no—
257C42#62511yescharacteristic
258C22×C3—21no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—81no—
264C22×C3—82no—
265C22×C3—121no—
266C22×C3#93711yescharacteristic
267C22×C3#95311yescharacteristic
268C2×D3—41no—
269C3×C4—22no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—22no—
273C3×C4—42no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—81no—
278C3×C4—122no—
279C3×C4—121no—
280C3×C4—122no—
281Dic3—42no—
282Dic3—42no—
283Dic3—41no—
284C32#220311yescharacteristic, complemented, Sylow
285C23—21no—
286Q8—41no—
287Q8—61no—
288Q8—62no—
289Q8—122no—
290D4—41no—
291D4—61no—
292D4—62no—
293D4—121no—
294C2×C4—21no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—121no—
298C2×C4—122no—
299C2×C4—241no—
300C2×C4C2×#62511yescharacteristic
301C2×C4D3×#6712yes—
302C8—121no—
303C8—362no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—41no—
308C2×C3—81no—
309C2×C3—81no—
310C2×C3—161no—
311C2×C3—241no—
312C2×C3#503211yescharacteristic
313C2×C3#524411yescharacteristic
314D3—81no—
315C22—42no—
316C22—41no—
317C22—121no—
318C22#6397611yescharacteristic
319C4—21no—
320C4—22no—
321C4—41no—
322C4—122no—
323C4—122no—
324C4—121no—
325C3—21no—
326C3#7098911yescharacteristic, complemented
327C3#7195911yescharacteristic, complemented
328C2—21no—
329C2—81no—
330C2—241no—
331C2#1059153011yescharacteristic
subgroupG/H
centerC2#10591530
commutator, cosocleC2×C32×Q8C23
FrattiniC2×Q8C2×D32
FittingC32×#124C22
socleC2×C32B(4,𝔽2)
prime radicalC3×#5062C2
hypercenter#124D32
hypocenterC32#2203
LCSG ▷ C2×C32×Q8 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×#5672C2semidirect
3. #10591648C2semidirect
4. #10589408C2semidirect
5. #10589184C2semidirect
22. #5672D3semidirect
284. C32#2203semidirect
326. C3#70989semidirect
327. C3#71959semidirect

Maximal subgroups

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

subgroupindex
C3×#56722
C3×#50622
#105916482
#105894082
#105891842
#719593
#709893