Finite Group Databasebeta

← #11798314

Subgroups — #11798314

Order 1152.

461 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 84 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×#5748C211yesmaximal, characteristic
2C3×#5137C211yesmaximal, characteristic, complemented
3#10590636C212yesmaximal, complemented
4#10589419C211yesmaximal, characteristic, complemented
5#10589123C212yesmaximal, complemented
6#72113—31nomaximal
7#71091—31nomaximal
8C2×C3×#235—21no—
9C2×C32×M16C2211yescharacteristic
10C3×#924—21no—
11C3×#909—22no—
12C3×#908—22no—
13C2×C3×Dic12C2211yescharacteristic, complemented
14C3×#904—21no—
15C3×#904C2212yes—
16#63639C2212yescomplemented
17C3×#817C2211yescharacteristic, complemented
18C3×#816—21no—
19#63550—22no—
20#63547—22no—
21C3×#575—31no—
22#5748D311yescharacteristic
23C3×#470—31no—
24#5195—32no—
25#5179—32no—
26#5137D311yescharacteristic, complemented
27#4566—32no—
28#4553—31no—
29#4553—31no—
30#4553—61no—
31#4488—32no—
32C22×C32×C4D411yescharacteristic
33C2×C3×#67—41no—
34C3×#235—22no—
35C3×#235—22no—
36C3×#235—41no—
37C3×#235—41no—
38C2×C3×D12—21no—
39C2×C3×C4×D3—41no—
40C2×C3×Dic6—21no—
41C2×C3×Dic6—21no—
42C32×M16—22no—
43C2×C32×C8D411yescharacteristic
44C2×C32×C8C2311yescharacteristic
45C3×#217—41no—
46C3×C4⋊Dic3—21no—
47C3×Dic3⋊C4—41no—
48C3×Dic12—21no—
49C3×Dic12—22no—
50C3×Dic12D412yes—
51C3×D24—21no—
52C3×D24D412yes—
53C3×#204—22no—
54C3×#204—22no—
55C3⋊(C3⋊C16)—22no—
56#2191—91noSylow
57C2×C3×#41—61no—
58C2×#235—21no—
59C3×#137—62no—
60C3×AGL(1,ℤ/8)—62no—
61C3×#135—32no—
62C3×#135—61no—
63C2×C3×Q16—31no—
64C2×C3×M16—21no—
65C2×C3×M16C2×D311yescharacteristic
66C2×C3×M16C2×D311yescharacteristic
67C3×C22⋊Q8—61no—
68#924—21no—
69#909—22no—
70#908—22no—
71C2×Dic12C2×D311yescharacteristic, complemented
72#904—21no—
73#904C2×D312yes—
74C3×#106—31no—
75C3×#103—61no—
76#829—62no—
77#829—62no—
78#828—62no—
79#827—62no—
80#817C2×D311yescharacteristic, complemented
81#816—21no—
82#812—32no—
83#812—32no—
84#812—62no—
85C23×C32D811yescharacteristic
86C22×C3×D3—41no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4D811yescharacteristic
90C2×C32×C4C2×D411yescharacteristic
91C3×#67—42no—
92C3×#67—42no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C3×D12—22no—
97C3×D12—41no—
98C3×C4×D3—42no—
99C3×C4×D3—42no—
100C3×Dic6—22no—
101C3×Dic6—41no—
102C3×Dic6—41no—
103C3×Dic6—41no—
104C32×C8—21no—
105C32×C8C2×D412yes—
106#575—31no—
107#510—92no—
108#470—31no—
109#360—92no—
110#349—91no—
111C3×#41—62no—
112C3×#41—62no—
113C3×#41—121no—
114C3×#41—121no—
115C2×C3×Q8—61no—
116C2×C3×Q8—61no—
117C2×C3×D4—61no—
118C2×C3×D4—121no—
119C22×C3×C4—21no—
120C22×C3×C4—121no—
121C22×C3×C4#6711yescharacteristic
122C22×C3×C4#6711yescharacteristic
123C2×#67—41no—
124#235—22no—
125#235—22no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×C4×D3—41no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C3×Q16—32no—
133C3×Q16—61no—
134C3×Q16—62no—
135C3×SD16—62no—
136C3×SD16—62no—
137C3×D8—32no—
138C3×D8—61no—
139C3×M16—22no—
140C3×M16—22no—
141C3×M16—42no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8#6711yescharacteristic
145C2×C3×C8#6711yescharacteristic
146C2×C3×C8C22×D311yescharacteristic
147C2×C3×C8C22×D311yescharacteristic
148C3×C4⋊C4—61no—
149C3×C4⋊C4—121no—
150C3×C22⋊C4—121no—
151#217—41no—
152C4⋊Dic3—21no—
153Dic3⋊C4—41no—
154Dic12—21no—
155Dic12—22no—
156Dic12#6712yes—
157D24—21no—
158D24#6712yes—
159#204—22no—
160#204—22no—
161C3⋊C16—62no—
162C3⋊C16—62no—
163C3⋊C16—122no—
164C22×C32—21no—
165C22×C32—41no—
166C22×C32C2×D811yescharacteristic
167C2×C3×D3—42no—
168C2×C3×D3—82no—
169C32×C4—21no—
170C32×C4#12011yescharacteristic
171C32×C4AGL(1,ℤ/8)11yescharacteristic
172C3×Dic3—42no—
173C3×Dic3—81no—
174C3×Dic3—81no—
175C2×#41—61no—
176#137—62no—
177AGL(1,ℤ/8)—62no—
178#135—32no—
179#135—61no—
180C2×Q16—31no—
181C2×M16D3211yescharacteristic
182C22⋊Q8—61no—
183Q32—182no—
184SD32—182no—
185M32—92no—
186#106—31no—
187#103—61no—
188C23×C3—21no—
189C23×C3—121no—
190C23×C3#21311yescharacteristic
191C23×C3#21311yescharacteristic
192C22×D3—41no—
193C3×Q8—62no—
194C3×Q8—121no—
195C3×Q8—121no—
196C3×Q8—121no—
197C3×D4—62no—
198C3×D4—121no—
199C3×D4—122no—
200C3×D4—122no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—122no—
210C2×C3×C4—121no—
211C2×C3×C4—122no—
212C2×C3×C4—121no—
213C2×C3×C4—121no—
214C2×C3×C4#21311yescharacteristic
215C2×C3×C4#21311yescharacteristic
216C2×C3×C4C2×#6711yescharacteristic
217C2×C3×C4C2×#6711yescharacteristic
218#67—42no—
219#67—42no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223D12—22no—
224D12—41no—
225C4×D3—42no—
226C4×D3—42no—
227Dic6—22no—
228Dic6—41no—
229Dic6—41no—
230Dic6—41no—
231C3×C8—21no—
232C3×C8—21no—
233C3×C8—22no—
234C3×C8—41no—
235C3×C8D3×D412yes—
236C3×C8C2×#6712yes—
237C2×C32—21no—
238C2×C32—41no—
239C2×C32#44711yescharacteristic
240C3×D3—82no—
241#41—62no—
242#41—62no—
243#41—121no—
244#41—121no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×D4—61no—
248C2×D4—121no—
249C22×C4—121no—
250C22×C4#62511yescharacteristic
251Q16—32no—
252Q16—61no—
253Q16—62no—
254SD16—62no—
255SD16—62no—
256D8—32no—
257D8—61no—
258M16—22no—
259C2×C8#62511yescharacteristic
260C2×C8C2×D3211yescharacteristic
261C4⋊C4—61no—
262C4⋊C4—121no—
263C22⋊C4—121no—
264C16—182no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—81no—
272C22×C3—122no—
273C22×C3—242no—
274C22×C3C2×#21311yescharacteristic
275C22×C3C2×#21311yescharacteristic
276C2×D3—42no—
277C2×D3—82no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—122no—
284C3×C4—241no—
285C3×C4—241no—
286C3×C4#93211yescharacteristic
287C3×C4#93711yescharacteristic
288C3×C4#94911yescharacteristic
289C3×C4#95311yescharacteristic
290Dic3—42no—
291Dic3—81no—
292Dic3—81no—
293C32#219111yescharacteristic, complemented, Sylow
294C23—121no—
295C23#371311yescharacteristic
296Q8—62no—
297Q8—121no—
298Q8—121no—
299Q8—121no—
300D4—62no—
301D4—121no—
302D4—122no—
303D4—122no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—122no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4#371311yescharacteristic
312C2×C4C2×#62511yescharacteristic
313C8—21no—
314C8D3×#6712yes—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—81no—
322C2×C3—242no—
323C2×C3#503611yescharacteristic
324C2×C3#521711yescharacteristic
325D3—82no—
326C22—21no—
327C22—41no—
328C22—122no—
329C22—242no—
330C22C2×#371311yescharacteristic
331C4—21no—
332C4—122no—
333C4—241no—
334C4—241no—
335C4#6382311yescharacteristic
336C4#6397611yescharacteristic
337C3—21no—
338C3#7109111yescharacteristic, complemented
339C3#7211311yescharacteristic, complemented
340C2—21no—
341C2—41no—
342C2—242no—
343C2#1059116011yescharacteristic
subgroupG/H
centerC2#10591160
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×M16C22
socleC2×C32#447
prime radicalC3×#5748C2
hypercenterC2×M16D32
hypocenterC32#2191
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C3×#5137C2semidirect
3. #10590636C2semidirect
4. #10589419C2semidirect
5. #10589123C2semidirect
13. C2×C3×Dic12C22semidirect
16. #63639C22semidirect
17. C3×#817C22semidirect
26. #5137D3semidirect
71. C2×Dic12C2×D3semidirect
80. #817C2×D3semidirect
293. C32#2191semidirect
338. C3#71091semidirect
339. C3#72113semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
C3×#51372
#105906362
#105894192
#105891232
#721133
#710913