Finite Group Databasebeta

← #11798810

Subgroups — #11798810

Order 1152.

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

normal subgroups only

This group has 502 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×#5734C211yesmaximal, characteristic, complemented
2C3×#5095C211yesmaximal, characteristic, complemented
3#10591645C212yesmaximal, complemented
4#10589414C211yesmaximal, characteristic, complemented
5#10589182C212yesmaximal, complemented
6#71930—31nomaximal
7#70994—31nomaximal
8C3×#1007C2211yescharacteristic
9C32×C4⋊Q8C2211yescharacteristic
10C3×#947—21no—
11C3×#947C2212yes—
12C3×#942—21no—
13C3×#895—22no—
14C3×#892—22no—
15C3×C4×D12—21no—
16#63823—22no—
17#63663C2212yes—
18C3×#836C2211yescharacteristic
19C3×#805—21no—
20#63569—22no—
21C3×#555—31no—
22#5734D311yescharacteristic, complemented
23C3×#456—31no—
24#5245—32no—
25#5095D311yescharacteristic, complemented
26#5062—32no—
27#4548—31no—
28#4548—31no—
29#4548—61no—
30#4542—32no—
31#4477—32no—
32C2×C32×Q8D411yescharacteristic
33C2×C32×Q8C2311yescharacteristic
34C3×#239—41no—
35C3×D3×Q8—41no—
36C3×#235—21no—
37C3×#235—22no—
38C2×C3×D12—21no—
39C2×C3×D12—21no—
40C2×C3×D12D412yes—
41C2×C3×C4×D3—22no—
42C32×C4⋊C4—22no—
43C32×C42D411yescharacteristic
44#3748—22no—
45C3×#216—41no—
46C3×#215—41no—
47C3×(C2×D3)⋊C4—22no—
48C3×(C2×D3)⋊C4—22no—
49C3×(C2×D3)⋊C4—22no—
50C3×C4⋊Dic3—21no—
51C3×C4×Dic3—21no—
52C3×C4×Dic3D412yes—
53C3×#208—21no—
54#3715—42no—
55#3713—42no—
56#2204—91noSylow
57C3×2−1+4—31no—
58#1007C2×D311yescharacteristic
59C3×#137—61no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×#124—32no—
64C3×#124—61no—
65C3×#121—62no—
66C3×C4×D4—61no—
67C3×#117—62no—
68#949—62no—
69#947—21no—
70#947C2×D312yes—
71#942—21no—
72#932—62no—
73#895—22no—
74#892—22no—
75C4×D12—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—
85C22×C3×D3—22no—
86C32×Q8—41no—
87C32×Q8—41no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4C2×D412yes—
91C2×C32×C4C2×D411yescharacteristic
92C3×#67—41no—
93C2×C3×Dic3—22no—
94C3×D12—22no—
95C3×D12—21no—
96C3×D12—42no—
97C3×D12—42no—
98C3×D12—81no—
99C3×C4×D3—44no—
100C3×C4×D3—41no—
101C3×C4×D3—42no—
102C3×Dic6—22no—
103C3×Dic6—21no—
104C3⋊(C3⋊C8)—42no—
105C3×C3⋊C8—41no—
106#555—31no—
107#456—31no—
108#454—92no—
109#356—91no—
110#355—92no—
111C3×#41—62no—
112C3×#41—61no—
113C3×#41—121no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—121no—
117C2×C3×Q8#6711yescharacteristic
118C2×C3×Q8#6711yescharacteristic
119C2×C3×Q8C22×D311yescharacteristic
120C2×C3×Q8C22×D311yescharacteristic
121C2×C3×D4—32no—
122C2×C3×D4—61no—
123C2×C3×D4—61no—
124C22×C3×C4—62no—
125#239—41no—
126D3×Q8—41no—
127#235—22no—
128#235—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12#6712yes—
132C2×C4×D3—22no—
133C3×Q16—121no—
134C3×SD16—121no—
135C3×M16—61no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—61no—
140C3×C22⋊C4—62no—
141C3×C22⋊C4—62no—
142C3×C22⋊C4—62no—
143C3×C42—21no—
144C3×C42—32no—
145C3×C42—61no—
146C3×C42#6711yescharacteristic
147C3×C42#6711yescharacteristic
148#216—41no—
149#215—41no—
150#215—122no—
151#214—122no—
152#213—122no—
153#213—122no—
154(C2×D3)⋊C4—22no—
155(C2×D3)⋊C4—22no—
156(C2×D3)⋊C4—22no—
157C4⋊Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3#6712yes—
160#208—21no—
161#208—62no—
162#208—62no—
163#208—122no—
164C22×C32#12011yescharacteristic
165C2×C3×D3—42no—
166C2×C3×D3—42no—
167C2×C3×D3—41no—
168C2×C3×D3—42no—
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—41no—
173C3×Dic3—42no—
174C3×Dic3—41no—
1752−1+4—31no—
176#137—61no—
177AGL(1,ℤ/8)—182no—
178C4⋊Q8D3211yescharacteristic
179#124—32no—
180#124—61no—
181#121—62no—
182C4×D4—61no—
183#117—62no—
184#104—61no—
185#104—182no—
186#101—31no—
187#101—92no—
188C23×C3—62no—
189C22×D3—22no—
190C3×Q8—41no—
191C3×Q8—41no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×Q8—62no—
195C3×Q8—61no—
196C3×Q8—81no—
197C3×Q8—81no—
198C3×D4—62no—
199C3×D4—61no—
200C3×D4—122no—
201C3×D4—122no—
202C3×D4—121no—
203C3×D4—241no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—62no—
213C2×C3×C4—124no—
214C2×C3×C4—122no—
215C2×C3×C4—121no—
216C2×C3×C4D3×D412yes—
217C2×C3×C4C2×#6711yescharacteristic
218C2×C3×C4C2×#6711yescharacteristic
219C2×C3×C4C2×#6712yes—
220#67—41no—
221C2×Dic3—22no—
222D12—22no—
223D12—21no—
224D12—42no—
225D12—42no—
226D12—81no—
227C4×D3—44no—
228C4×D3—41no—
229C4×D3—42no—
230Dic6—22no—
231Dic6—21no—
232C3×C8—121no—
233C3⋊C8—41no—
234C3⋊C8—122no—
235C3⋊C8—122no—
236C3⋊C8—242no—
237C2×C32—21no—
238C2×C32B(4,𝔽2)11yescharacteristic
239C3×D3—81no—
240C3×D3—82no—
241#41—61no—
242#41—62no—
243#41—121no—
244C2×Q8—121no—
245C2×Q8#62511yescharacteristic
246C2×Q8C2×D3211yescharacteristic
247C2×D4—32no—
248C2×D4—61no—
249C2×D4—61no—
250C22×C4—62no—
251Q16—121no—
252SD16—121no—
253SD16—362no—
254D8—362no—
255M16—61no—
256M16—182no—
257C4⋊C4—22no—
258C4⋊C4—61no—
259C22⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C42—32no—
263C42—61no—
264C42#62511yescharacteristic
265C22×C3—21no—
266C22×C3—122no—
267C22×C3—122no—
268C22×C3—121no—
269C22×C3—122no—
270C22×C3#93711yescharacteristic
271C22×C3#95311yescharacteristic
272C2×D3—42no—
273C2×D3—42no—
274C2×D3—42no—
275C2×D3—41no—
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—42no—
283C3×C4—41no—
284C3×C4—42no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—81no—
288C3×C4—122no—
289C3×C4—121no—
290Dic3—42no—
291Dic3—41no—
292C32#220411yescharacteristic, complemented, Sylow
293C23—62no—
294Q8—41no—
295Q8—41no—
296Q8—62no—
297Q8—61no—
298D4—62no—
299D4—61no—
300D4—122no—
301D4—122no—
302D4—121no—
303D4—241no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—62no—
307C2×C4—124no—
308C2×C4—121no—
309C2×C4—122no—
310C2×C4C2×#62511yescharacteristic
311C2×C4D3×#6712yes—
312C8—121no—
313C8—362no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—41no—
318C2×C3—242no—
319C2×C3—241no—
320C2×C3#503211yescharacteristic
321C2×C3#524411yescharacteristic
322D3—81no—
323D3—82no—
324C22—122no—
325C22—122no—
326C22—121no—
327C22—122no—
328C22#6397611yescharacteristic
329C4—22no—
330C4—21no—
331C4—22no—
332C4—41no—
333C4—122no—
334C4—121no—
335C3—21no—
336C3#7099411yescharacteristic, complemented
337C3#7193011yescharacteristic, complemented
338C2—21no—
339C2—242no—
340C2—241no—
341C2#1059153011yescharacteristic
subgroupG/H
centerC2#10591530
commutator, cosocleC2×C32×Q8C23
FrattiniC2×Q8C2×D32
FittingC32×C4⋊Q8C22
socleC2×C32B(4,𝔽2)
hypercenterC4⋊Q8D32
hypocenterC32#2204
LCSG ▷ C2×C32×Q8 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5734C2semidirect
2. C3×#5095C2semidirect
3. #10591645C2semidirect
4. #10589414C2semidirect
5. #10589182C2semidirect
22. #5734D3semidirect
25. #5095D3semidirect
292. C32#2204semidirect
336. C3#70994semidirect
337. C3#71930semidirect

Maximal subgroups

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

subgroupindex
C3×#57342
C3×#50952
#105916452
#105894142
#105891822
#719303
#709943