Finite Group Databasebeta

← #11780165

Subgroups — #11780165

Order 1152.

423 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 100 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×#5570C211yesmaximal, characteristic
2C3×#5020C211yesmaximal, characteristic
3#10590384C211yesmaximal, characteristic
4#10590312C211yesmaximal, characteristic
5#10589401C211yesmaximal, characteristic
6#10589092C211yesmaximal, characteristic
7#10589083C211yesmaximal, characteristic
8#71937—31nomaximal
9#71130—31nomaximal
10C32×C4×Q8C2211yescharacteristic
11C3×Q8×Dic3—21no—
12C3×#890—21no—
13C3×#890—22no—
14C3×#869—21no—
15C3×#869C2211yescharacteristic
16C3×#869C2211yescharacteristic
17C3×C4×Dic6—21no—
18C3×C4×Dic6C2211yescharacteristic
19C2×#3718—21no—
20C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C4×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#835—21no—
23C3×#808—21no—
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25#63576—22no—
26#63565—21no—
27C3×#557—31no—
28#5570D311yescharacteristic
29C3×#475—31no—
30#5090—31no—
31#5078—31no—
32#5078—31no—
33#5031—31no—
34#5020D311yescharacteristic
35#4547—31no—
36#4535—31no—
37#4535—31no—
38#4535—31no—
39#4535—61no—
40#4486—31no—
41#4484—31no—
42C2×C32×Q8D411yescharacteristic
43C2×C3×Dic6—21no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6D411yescharacteristic
46C32×C4⋊C4—21no—
47C32×C4⋊C4D411yescharacteristic
48C32×C42—21no—
49C32×C42C2311yescharacteristic
50C2×C3⋊(C3⋊C8)—21no—
51C2×C3⋊(C3⋊C8)—21no—
52C3×C4⋊Dic3—22no—
53C3×C4⋊Dic3—22no—
54C3×C4⋊Dic3—21no—
55C3×C4⋊Dic3—21no—
56C3×C4⋊Dic3—21no—
57C3×C4⋊Dic3D412yes—
58C3×C4⋊Dic3D411yescharacteristic
59C3×Dic3⋊C4—22no—
60C3×Dic3⋊C4—21no—
61C3×C4×Dic3—22no—
62C3×C4×Dic3—21no—
63C2×C3×C3⋊C8—21no—
64#3718—42no—
65#1648—91noSylow
66C3×C4⋊Q8—31no—
67C3×C4⋊Q8—31no—
68C3×C4⋊Q8—61no—
69C3×#125—62no—
70C3×#125—61no—
71C3×C4×Q8—21no—
72C3×C4×Q8—31no—
73C3×C4×Q8—61no—
74C3×C4×Q8—61no—
75C3×C4×Q8C2×D311yescharacteristic
76C3×C4×Q8C2×D311yescharacteristic
77Q8×Dic3—21no—
78C2×#216—61no—
79C2×#216—61no—
80#890—21no—
81#890—22no—
82#869—21no—
83#869C2×D311yescharacteristic
84#869C2×D311yescharacteristic
85C4×Dic6—21no—
86C4×Dic6C2×D311yescharacteristic
87C3×#106—61no—
88C3×C4⋊C8—31no—
89C3×#103—61no—
90#835—21no—
91#835—61no—
92#810—61no—
93#808—21no—
94#807—62no—
95#807—62no—
96C4⋊(C3⋊C8)—31no—
97C4⋊(C3⋊C8)—31no—
98C4⋊(C3⋊C8)—61no—
99C4⋊(C3⋊C8)C2×D311yescharacteristic
100C4×C3⋊C8—31no—
101C4×C3⋊C8—31no—
102C4×C3⋊C8—61no—
103C32×Q8—21no—
104C32×Q8Q1612yescomplemented
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—41no—
114C3×Dic6—21no—
115C3×Dic6—21no—
116C3×Dic6—41no—
117C3×Dic6—41no—
118C3×Dic6—41no—
119C3×Dic6—41no—
120C3⋊(C3⋊C8)—22no—
121C3⋊(C3⋊C8)—41no—
122C3×C3⋊C8—41no—
123#557—31no—
124#500—91no—
125#475—31no—
126#462—91no—
127#332—91no—
128Q8⋊C8—91no—
129Q8⋊C8—91no—
130C2×C3×Q8—21no—
131C2×C3×Q8—31no—
132C2×C3×Q8—61no—
133C2×C3×Q8—61no—
134C2×C3×Q8#6711yescharacteristic
135C2×C3×Q8#6711yescharacteristic
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6#6711yescharacteristic
139C2×C3×C8—61no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—32no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4—62no—
148C3×C4⋊C4—62no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—61no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4#6711yescharacteristic
154C3×C4⋊C4#6711yescharacteristic
155C3×C42—21no—
156C3×C42—21no—
157C3×C42—21no—
158C3×C42—41no—
159C3×C42—62no—
160C3×C42—61no—
161C3×C42C22×D311yescharacteristic
162C3×C42C22×D311yescharacteristic
163#216—122no—
164#216—122no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3#6712yes—
171C4⋊Dic3#6711yescharacteristic
172Dic3⋊C4—22no—
173Dic3⋊C4—21no—
174C4×Dic3—22no—
175C4×Dic3—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—61no—
178C2×C3⋊C8—61no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—61no—
181C2×C3⋊C8—121no—
182C2×C3⋊C8—121no—
183C22×C32#12011yescharacteristic
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4C2×Q1611yescharacteristic
188C32×C4#13511yescharacteristic
189C32×C4AGL(1,ℤ/8)11yescharacteristic
190C32×C4#13711yescharacteristic
191C3×Dic3—41no—
192C3×Dic3—42no—
193C3×Dic3—41no—
194C3×Dic3—81no—
195C2×Q16—181no—
196C4⋊Q8—31no—
197C4⋊Q8—31no—
198C4⋊Q8—61no—
199#125—62no—
200#125—61no—
201C4×Q8—31no—
202C4×Q8—61no—
203C4×Q8—61no—
204C4×Q8D3211yescharacteristic
205#107—182no—
206#106—61no—
207C4⋊C8—31no—
208C4⋊C8—91no—
209#103—61no—
210#103—181no—
211C4×C8—91no—
212C3×Q8—21no—
213C3×Q8—21no—
214C3×Q8—22no—
215C3×Q8—41no—
216C3×Q8—61no—
217C3×Q8—61no—
218C3×Q8—121no—
219C3×Q8—121no—
220C3×Q8—121no—
221C3×Q8—121no—
222C3×Q8#21612yescomplemented
223C3×Q8#21612yescomplemented
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—61no—
234C2×C3×C4—62no—
235C2×C3×C4—61no—
236C2×C3×C4—121no—
237C2×C3×C4D3×D411yescharacteristic
238C2×C3×C4D3×D411yescharacteristic
239C2×C3×C4C2×#6711yescharacteristic
240C2×C3×C4C2×#6711yescharacteristic
241C2×C3×C4C2×#6711yescharacteristic
242C2×C3×C4C2×#6711yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—22no—
246C2×Dic3—41no—
247Dic6—21no—
248Dic6—21no—
249Dic6—41no—
250Dic6—41no—
251Dic6—41no—
252Dic6—41no—
253C3×C8—121no—
254C3⋊C8—41no—
255C3⋊C8—62no—
256C3⋊C8—62no—
257C3⋊C8—122no—
258C3⋊C8—121no—
259C3⋊C8—121no—
260C3⋊C8—241no—
261C2×C32#44911yescharacteristic
262C2×C32#45111yescharacteristic
263C2×C32#45611yescharacteristic
264C2×Q8—31no—
265C2×Q8—61no—
266C2×Q8—61no—
267C2×Q8#62511yescharacteristic
268Q16—362no—
269C2×C8—61no—
270C2×C8—181no—
271C2×C8—181no—
272C4⋊C4—21no—
273C4⋊C4—31no—
274C4⋊C4—32no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—62no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4#62511yescharacteristic
283C42—21no—
284C42—62no—
285C42—61no—
286C42C2×D3211yescharacteristic
287C22×C3—21no—
288C22×C3#93711yescharacteristic
289C22×C3#95311yescharacteristic
290C3×C4—21no—
291C3×C4—22no—
292C3×C4—21no—
293C3×C4—22no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—41no—
299C3×C4—42no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4—81no—
303C3×C4—121no—
304C3×C4—122no—
305C3×C4—121no—
306C3×C4—241no—
307C3×C4#91411yescharacteristic
308C3×C4#91611yescharacteristic
309C3×C4#94211yescharacteristic
310C3×C4C2×#21611yescharacteristic
311C3×C4C2×#21611yescharacteristic
312C3×C4#94911yescharacteristic
313C3×C4#95011yescharacteristic
314C3×C4#95111yescharacteristic
315Dic3—41no—
316Dic3—42no—
317Dic3—41no—
318Dic3—81no—
319C32#164811yescharacteristic, complemented, Sylow
320Q8—21no—
321Q8—61no—
322Q8—61no—
323Q8—121no—
324Q8—121no—
325Q8—121no—
326Q8—121no—
327Q8#371812yescomplemented
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—61no—
331C2×C4—62no—
332C2×C4—61no—
333C2×C4—121no—
334C2×C4C2×#62511yescharacteristic
335C2×C4D3×#6711yescharacteristic
336C2×C4D3×#6711yescharacteristic
337C8—121no—
338C8—182no—
339C8—361no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3#505011yescharacteristic
344C2×C3#517211yescharacteristic
345C2×C3#520511yescharacteristic
346C2×C3#522811yescharacteristic
347C2×C3#523811yescharacteristic
348C2×C3#524611yescharacteristic
349C22#6397611yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—121no—
354C4—122no—
355C4—121no—
356C4—241no—
357C4#6382711yescharacteristic
358C4C2×#371811yescharacteristic
359C4#6393911yescharacteristic
360C4#6394711yescharacteristic
361C3—21no—
362C3#7113011yescharacteristic, complemented
363C3#7193711yescharacteristic, complemented
364C2#1059118511yescharacteristic
365C2#1059158611yescharacteristic
366C2#1059168711yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socle, prime radicalC22×C32#120
hypercenterC4×Q8D32
hypocenterC32#1648
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
104. C32×Q8Q16semidirect
222. C3×Q8#216semidirect
223. C3×Q8#216semidirect
319. C32#1648semidirect
327. Q8#3718semidirect
362. C3#71130semidirect
363. C3#71937semidirect

Maximal subgroups

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

subgroupindex
C3×#55702
C3×#50202
#105903842
#105903122
#105894012
#105890922
#105890832
#719373
#711303