Finite Group Databasebeta

← #78887

Subgroups — #78887

Order 384.

712 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 149 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, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C2×#909C211yesmaximal, characteristic, complemented
4#5510C211yesmaximal, characteristic, complemented
5C3×#417C211yesmaximal, characteristic, complemented
6#5129C211yesmaximal, characteristic, complemented
7C2×#810C211yesmaximal, characteristic, complemented
8#4809C212yesmaximal, complemented
9#4806C212yesmaximal, complemented
10#4783C212yesmaximal, complemented
11#4781C212yesmaximal, complemented
12#3013—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14#1010—24no—
15#1010—24no—
16C2×D3×Q8C2212yes—
17C2×#237—22no—
18C2×#237C2212yes—
19C2×#235—21no—
20C2×#235C2211yescharacteristic
21C2×#235C2211yescharacteristic
22C22×Dic6—21no—
23C22×Dic6C2211yescharacteristic
24C2×C3×M16C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26#950—24no—
27C2×#216C2211yescharacteristic
28C2×#215C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#213C2211yescharacteristic
31#930—22no—
32C22×C3⋊C8C2211yescharacteristic
33#909—24no—
34C2×Dic12C2212yes—
35C2×#204C2212yes—
36#897—21no—
37#897C2212yes—
38#895—21no—
39#895C2212yes—
40C3×#103C2212yes—
41C3×#102C2212yes—
42(C2×D3)⋊C8C2212yes—
43(C2×D3)⋊C8C2212yes—
44#810C2212yes—
45#810C2212yes—
46C2×2−1+4—31no—
47C2×#137—31no—
48C2×#135—31no—
49#537—31no—
50#452—32no—
51#451—32no—
52#449—32no—
53#448—32no—
54#417D311yescharacteristic, complemented
55C2×#103—31no—
56#408—31no—
57C3×#41—22no—
58C3×#41D414yescomplemented
59C2×C3×Q8C2311yescharacteristic
60C2×C3×D4—21no—
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4—21no—
63C22×C3×C4—21no—
64C22×C3×C4C2311yescharacteristic
65C2×#67—22no—
66C2×#67—21no—
67C2×#67D412yes—
68C22×Dic3—21no—
69C22×Dic3—22no—
70D3×Q8—24no—
71D3×Q8—24no—
72#237—216no—
73#237—24no—
74#237—24no—
75#235—28no—
76#235—22no—
77#235—22no—
78#235—22no—
79#235—22no—
80C2×D12—22no—
81C2×D12D411yescharacteristic
82C2×D12C2311yescharacteristic
83C2×C4×D3—22no—
84C2×C4×D3—21no—
85C2×C4×D3C2312yes—
86C2×Dic6—28no—
87C2×Dic6—21no—
88C2×Dic6—22no—
89C2×Dic6—22no—
90C2×Dic6D411yescharacteristic
91C2×Dic6D414yescomplemented
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2312yes—
94C3×M16—22no—
95C2×C3×C8C2312yes—
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2312yes—
98#216—22no—
99#215—22no—
100#214—22no—
101#213—22no—
102(C2×D3)⋊C4—22no—
103(C2×D3)⋊C4—22no—
104C4⋊Dic3—22no—
105C2×C3⋊C8—22no—
106C2×C3⋊C8C2311yescharacteristic
107C2×C3⋊C8C2311yescharacteristic
108Dic12—24no—
109#204—24no—
1102−1+4—64no—
1112−1+4—64no—
112C2×#41—31no—
113C2×#41—32no—
114C2×#41—31no—
115C2×#41—61no—
116C2×#41—62no—
117C2×#41C2×D311yescharacteristic
118C22×Q8—32no—
119C22×Q8—31no—
120C22×Q8—61no—
121#137—64no—
122#135—64no—
123C2×Q16—31no—
124C2×Q16—32no—
125C2×SD16—31no—
126C2×SD16—31no—
127C2×SD16—32no—
128C2×D8—31no—
129C2×M16C2×D311yescharacteristic
130C22×C8—31no—
131C22⋊Q8—32no—
132C22⋊Q8—61no—
133#121—32no—
134#121—61no—
135#121—62no—
136C2×C4⋊C4C2×D311yescharacteristic
137#103—32no—
138#103—32no—
139#103C2×D312yes—
140#102C2×D312yes—
141C22⋊C8—32no—
142C22⋊C8—32no—
143C23×C3—21no—
144C23×C3C2×D411yescharacteristic
145C22×D3—21no—
146C22×D3C2×D412yes—
147C3×Q8—21no—
148C3×Q8C2×D412yes—
149C3×D4—24no—
150C3×D4—21no—
151C3×D4C2×D412yes—
152C2×C3×C4—21no—
153C2×C3×C4—24no—
154C2×C3×C4—22no—
155C2×C3×C4—81no—
156C2×C3×C4C2×D412yes—
157C2×C3×C4C2×D411yescharacteristic
158C2×C3×C4C2×D412yes—
159C2×C3×C4C2411yescharacteristic
160#67—28no—
161#67—24no—
162#67—42no—
163C2×Dic3—28no—
164C2×Dic3—24no—
165C2×Dic3—22no—
166C2×Dic3—21no—
167C2×Dic3C2×D412yes—
168D12—22no—
169D12—21no—
170D12—21no—
171D12—82no—
172C4×D3—28no—
173C4×D3—22no—
174C4×D3—22no—
175C4×D3—42no—
176Dic6—28no—
177Dic6—22no—
178Dic6—24no—
179Dic6—22no—
180Dic6—21no—
181Dic6—21no—
182Dic6C2×D414yes—
183C3×C8—22no—
184C3⋊C8—21no—
185C3⋊C8—21no—
186#41—22no—
187#41—68no—
188#41—64no—
189#41—616no—
190#41—64no—
191#41—62no—
192#41—62no—
193#41—62no—
194#41—62no—
195#41D1214yescomplemented
196C2×Q8—31no—
197C2×Q8—34no—
198C2×Q8—32no—
199C2×Q8—31no—
200C2×Q8—62no—
201C2×Q8—68no—
202C2×Q8—64no—
203C2×Q8—64no—
204C2×Q8—62no—
205C2×Q8—61no—
206C2×Q8C22×D311yescharacteristic
207C2×D4—21no—
208C2×D4—32no—
209C2×D4—31no—
210C2×D4—31no—
211C2×D4—62no—
212C2×D4—62no—
213C2×D4—61no—
214C2×D4C22×D311yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—32no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C22×D311yescharacteristic
223Q16—64no—
224Q16—62no—
225SD16—64no—
226SD16—62no—
227SD16—62no—
228D8—62no—
229M16—22no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—62no—
233C2×C8C22×D312yes—
234C4⋊C4—21no—
235C4⋊C4—62no—
236C4⋊C4C22×D312yes—
237C22⋊C4—62no—
238C22⋊C4—62no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3#12012yes—
245C22×C3C22×D411yescharacteristic
246C2×D3—22no—
247C2×D3—22no—
248C2×D3—22no—
249C2×D3—41no—
250C2×D3—41no—
251C2×D3—41no—
252C3×C4—22no—
253C3×C4—42no—
254C3×C4#12012yes—
255C3×C4C22×D411yescharacteristic
256C3×C4C22×D411yescharacteristic
257Dic3—22no—
258Dic3—24no—
259Dic3—41no—
260C23—21no—
261C23—32no—
262C23—61no—
263C23D3×D411yescharacteristic
264Q8—21no—
265Q8—34no—
266Q8—68no—
267Q8—62no—
268Q8—64no—
269Q8—62no—
270Q8—61no—
271Q8—61no—
272Q8C2×D1212yes—
273D4—24no—
274D4—21no—
275D4—68no—
276D4—64no—
277D4—62no—
278D4—61no—
279D4—61no—
280D4—122no—
281D4—242no—
282D4C2×D1212yes—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—32no—
287C2×C4—68no—
288C2×C4—68no—
289C2×C4—64no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—62no—
293C2×C4—61no—
294C2×C4—81no—
295C2×C4—122no—
296C2×C4C2×D1212yes—
297C2×C4D3×D411yescharacteristic
298C2×C4D3×D412yes—
299C2×C4C23×D311yescharacteristic
300C8—22no—
301C8—61no—
302C8—61no—
303C2×C3—22no—
304C2×C3—42no—
305C2×C3C2×#12011yescharacteristic
306C2×C3#57711yescharacteristic
307C2×C3#57811yescharacteristic
308D3—42no—
309D3—81no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—62no—
316C22—62no—
317C22—62no—
318C22—121no—
319C22—121no—
320C22—121no—
321C22#88212yes—
322C22C2×D3×D411yescharacteristic
323C4—22no—
324C4—42no—
325C4—62no—
326C4—64no—
327C4—121no—
328C4#88212yes—
329C4C22×D1211yescharacteristic
330C4C2×D3×D411yescharacteristic
331C3#301311yescharacteristic, complemented, Sylow
332C2—22no—
333C2—42no—
334C2—122no—
335C2—241no—
336C2C2×#88211yescharacteristic
337C2#577511yescharacteristic
338C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
hypercenter#417D3
hypocenterC3#3013
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C2×#950C2semidirect
3. C2×#909C2semidirect
4. #5510C2semidirect
5. C3×#417C2semidirect
6. #5129C2semidirect
7. C2×#810C2semidirect
8. #4809C2semidirect
9. #4806C2semidirect
10. #4783C2semidirect
11. #4781C2semidirect
54. #417D3semidirect
58. C3×#41D4semidirect
91. C2×Dic6D4semidirect
195. #41D12semidirect
331. C3#3013semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C2×#9502
C2×#9092
#55102
C3×#4172
#51292
C2×#8102
#48092
#48062
#47832
#47812
#30133