Finite Group Databasebeta

← #70415

Subgroups — #70415

Order 384.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#1010C211yesmaximal, characteristic
2C3×#421C211yesmaximal, characteristic, complemented
3#5236C211yesmaximal, characteristic, complemented
4#5132C211yesmaximal, characteristic, complemented
5#5120C211yesmaximal, characteristic, complemented
6#5006C211yesmaximal, characteristic, complemented
7#5004C211yesmaximal, characteristic, complemented
8#1796—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10#1010—22no—
11#1010—22no—
12#1010—22no—
13#1010—22no—
14C2×D3×Q8C412yescomplemented
15C2×#237—22no—
16C2×#237C412yescomplemented
17C2×#235—21no—
18C2×#235C2211yescharacteristic
19C22×Dic6—21no—
20C22×Dic6C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C3×#117C2211yescharacteristic
23#926C2211yescharacteristic
24C2×#208C2211yescharacteristic
25C3×#104—22no—
26#837—22no—
27#824—21no—
28#823—21no—
29#816—22no—
30#810—22no—
31#806—22no—
32C2×2−1+4—31no—
33#421—31no—
34#421D311yescharacteristic, complemented
35#419—31no—
36#419—31no—
37#413—31no—
38#410—31no—
39C3×#41—21no—
40C3×#41—21no—
41C3×#41D412yes—
42C3×#41D412yes—
43C2×C3×Q8C2×C411yescharacteristic
44C2×C3×D4—21no—
45C2×C3×D4C2×C411yescharacteristic
46C22×C3×C4—21no—
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49C2×#67D412yes—
50C22×Dic3—21no—
51C22×Dic3—22no—
52D3×Q8—24no—
53D3×Q8—24no—
54#237—28no—
55#237—28no—
56#237—24no—
57#237—24no—
58#235—24no—
59#235—24no—
60#235—21no—
61#235—21no—
62#235—21no—
63#235—21no—
64C2×D12D411yescharacteristic
65C2×C4×D3—22no—
66C2×C4×D3C2×C412yes—
67C2×Dic6—24no—
68C2×Dic6—24no—
69C2×Dic6—21no—
70C2×Dic6—21no—
71C2×Dic6—21no—
72C2×Dic6—22no—
73C2×Dic6C2×C412yes—
74C2×Dic6D411yescharacteristic
75C2×Dic6D412yes—
76C2×Dic6D412yes—
77C3×M16—21no—
78C3×M16—21no—
79C2×C3×C8—21no—
80C3×C4⋊C4—21no—
81C3×C22⋊C4—21no—
82C3×C42—21no—
83#217—21no—
84C4⋊Dic3—21no—
85C4×Dic3—21no—
86#208—21no—
87#208—21no—
88C2×C3⋊C8—21no—
892−1+4—62no—
902−1+4—62no—
912−1+4—62no—
922−1+4—62no—
93C2×#41—31no—
94C2×#41—32no—
95C2×#41—61no—
96C2×#41—62no—
97C2×#41C2×D311yescharacteristic
98C22×Q8—32no—
99C22×Q8—31no—
100C22×Q8—61no—
101C2×M16—31no—
102C2×M16C2×D311yescharacteristic
103#117—31no—
104#117C2×D311yescharacteristic
105#104—22no—
106#104—62no—
107#103—62no—
108#103—62no—
109#101—61no—
110#100—61no—
111#99—62no—
112C23×C3—21no—
113C23×C3C2×D411yescharacteristic
114C22×D3C22⋊C412yes—
115C3×Q8—21no—
116C3×Q8C22⋊C412yes—
117C3×D4—22no—
118C3×D4—22no—
119C3×D4—21no—
120C3×D4C22⋊C412yes—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—41no—
125C2×C3×C4C22×C411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131#67—28no—
132#67—22no—
133#67—22no—
134C2×Dic3—28no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—41no—
139C2×Dic3C22⋊C412yes—
140D12—21no—
141D12—21no—
142C4×D3—28no—
143C4×D3—22no—
144C4×D3—22no—
145Dic6—28no—
146Dic6—22no—
147Dic6—21no—
148Dic6—22no—
149Dic6—21no—
150Dic6—22no—
151Dic6C22⋊C414yes—
152C3×C8—41no—
153C3⋊C8—41no—
154#41—21no—
155#41—21no—
156#41—64no—
157#41—64no—
158#41—64no—
159#41—68no—
160#41—68no—
161#41—64no—
162#41—61no—
163#41—61no—
164#41—61no—
165#41—61no—
166#41D1212yes—
167#41#6712yes—
168C2×Q8—31no—
169C2×Q8—32no—
170C2×Q8—32no—
171C2×Q8—32no—
172C2×Q8—62no—
173C2×Q8—64no—
174C2×Q8—64no—
175C2×Q8—64no—
176C2×Q8—64no—
177C2×Q8—61no—
178C2×Q8—61no—
179C2×Q8—61no—
180C2×Q8C4×D311yescharacteristic
181C2×D4—21no—
182C2×D4—32no—
183C2×D4—31no—
184C2×D4—62no—
185C2×D4C4×D311yescharacteristic
186C22×C4—21no—
187C22×C4—32no—
188C22×C4—62no—
189C22×C4—62no—
190C22×C4—61no—
191C22×C4C22×D311yescharacteristic
192M16—21no—
193M16—21no—
194M16—61no—
195M16—61no—
196C2×C8—21no—
197C2×C8—61no—
198C4⋊C4—21no—
199C4⋊C4—61no—
200C22⋊C4—21no—
201C22⋊C4—61no—
202C42—21no—
203C42—61no—
204C22×C3—22no—
205C22×C3—41no—
206C22×C3—42no—
207C22×C3C2×C22⋊C411yescharacteristic
208C22×C3#12011yescharacteristic
209C22×C3#12011yescharacteristic
210C2×D3—22no—
211C2×D3—42no—
212C3×C4—22no—
213C3×C4—81no—
214C3×C4C2×C22⋊C411yescharacteristic
215C3×C4C2×C22⋊C411yescharacteristic
216C3×C4#12011yescharacteristic
217C3×C4#12011yescharacteristic
218Dic3—22no—
219Dic3—24no—
220Dic3—81no—
221C23—21no—
222C23—32no—
223C23D3×D411yescharacteristic
224Q8—21no—
225Q8—34no—
226Q8—68no—
227Q8—61no—
228Q8—62no—
229Q8—62no—
230Q8—61no—
231Q8—62no—
232Q8(C2×D3)⋊C412yes—
233D4—22no—
234D4—22no—
235D4—21no—
236D4—68no—
237D4—62no—
238D4—62no—
239D4—61no—
240D4—61no—
241D4(C2×D3)⋊C412yes—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—32no—
246C2×C4—41no—
247C2×C4—68no—
248C2×C4—68no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—121no—
255C2×C4C2×C4×D311yescharacteristic
256C2×C4C2×D1211yescharacteristic
257C2×C4D3×D411yescharacteristic
258C2×C4D3×D411yescharacteristic
259C2×C4D3×D411yescharacteristic
260C2×C4C2×#6711yescharacteristic
261C8—41no—
262C8—121no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—42no—
267C2×C3#37911yescharacteristic
268D3—42no—
269C22—22no—
270C22—41no—
271C22—42no—
272C22—62no—
273C22—122no—
274C22D3×C22⋊C411yescharacteristic
275C22#88211yescharacteristic
276C22#93711yescharacteristic
277C4—22no—
278C4—62no—
279C4—64no—
280C4—81no—
281C4—241no—
282C4D3×C22⋊C411yescharacteristic
283C4#88211yescharacteristic
284C4C2×(C2×D3)⋊C411yescharacteristic
285C4#93711yescharacteristic
286C3#179611yescharacteristic, complemented, Sylow
287C2—21no—
288C2—21no—
289C2—21no—
290C2—42no—
291C2—122no—
292C2#495511yescharacteristic
subgroupG/H
centerC2#4955
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#421C2
socleC2×C3#379
prime radicalC2×#1010C2
cosocleC22×C3×C4C23
hypercenter#421D3
hypocenterC3#1796
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C3×#421C2semidirect
3. #5236C2semidirect
4. #5132C2semidirect
5. #5120C2semidirect
6. #5006C2semidirect
7. #5004C2semidirect
14. C2×D3×Q8C4semidirect
16. C2×#237C4semidirect
34. #421D3semidirect
286. C3#1796semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C3×#4212
#52362
#51322
#51202
#50062
#50042
#17963