Finite Group Databasebeta

← #84114

Subgroups — #84114

Order 384.

712 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×2−1+4C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic, complemented
3C2×#950C211yesmaximal, characteristic, complemented
4C2×#942C211yesmaximal, characteristic, complemented
5#5239C212yesmaximal, complemented
6#5238C212yesmaximal, complemented
7#5234C211yesmaximal, characteristic, complemented
8#5232C211yesmaximal, characteristic, complemented
9#5228C212yesmaximal, complemented
10#5227C212yesmaximal, complemented
11C2×#835C211yesmaximal, characteristic, complemented
12#3013—31nomaximal, Sylow
13C3×2−1+4—24no—
14C3×2−1+4—24no—
15C2×C3×#41—21no—
16C2×C3×#41—22no—
17C2×C3×#41C2212yes—
18C2×C3×#41C2211yescharacteristic
19C2×C3×#41C2211yescharacteristic
20C22×C3×Q8—21no—
21C22×C3×Q8C2212yes—
22C22×C3×Q8C2211yescharacteristic
23C2×#235C2211yescharacteristic
24#950—24no—
25#946—21no—
26C2×#216C2211yescharacteristic
27C2×#216C2212yes—
28#942—24no—
29C2×#215C2211yescharacteristic
30C2×#215C2212yes—
31#939—22no—
32#938—21no—
33C2×#214C2211yescharacteristic
34C2×#213C2211yescharacteristic
35#930C2212yes—
36C2×C4⋊Dic3C2211yescharacteristic
37#924C2212yes—
38C2×#208C2211yescharacteristic
39C22×C3⋊C8C2211yescharacteristic
40#835C2212yes—
41#835C2212yes—
42#835C2212yes—
43#832C2212yes—
44#830C2212yes—
45#830C2212yes—
46C2×2−1+4D311yescharacteristic, complemented
47C2×#137—31no—
48C2×#135—31no—
49#537—31no—
50#452—32no—
51#451—32no—
52#449—32no—
53#448—32no—
54#417—31no—
55C2×#103—31no—
56#408—31no—
57C3×#41—216no—
58C3×#41—28no—
59C3×#41—24no—
60C3×#41—24no—
61C3×#41—22no—
62C3×#41—22no—
63C3×#41—22no—
64C3×#41D414yescomplemented
65C2×C3×Q8—28no—
66C2×C3×Q8—24no—
67C2×C3×Q8—24no—
68C2×C3×Q8—22no—
69C2×C3×Q8—21no—
70C2×C3×Q8—22no—
71C2×C3×Q8D411yescharacteristic
72C2×C3×Q8D414yescomplemented
73C2×C3×Q8C2311yescharacteristic
74C2×C3×Q8C2312yes—
75C2×C3×D4—22no—
76C2×C3×D4—21no—
77C2×C3×D4D411yescharacteristic
78C2×C3×D4D412yes—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—22no—
81C22×C3×C4—21no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4C2312yes—
85C22×C3×C4C2311yescharacteristic
86C2×#67—21no—
87C2×#67—22no—
88C22×Dic3—21no—
89#235—22no—
90#235—22no—
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×Dic6C2311yescharacteristic
94#217—22no—
95#216—24no—
96#216—22no—
97#215—24no—
98#215—22no—
99#214—22no—
100#213—22no—
101(C2×D3)⋊C4—22no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2312yes—
104Dic3⋊C4—22no—
105#208—22no—
106C2×C3⋊C8—22no—
107C2×C3⋊C8C2312yes—
108C2×C3⋊C8C2311yescharacteristic
109C2×C3⋊C8C2311yescharacteristic
1102−1+4—24no—
1112−1+4—24no—
112C2×#41—21no—
113C2×#41—22no—
114C2×#41—31no—
115C2×#41C2×D312yes—
116C2×#41C2×D311yescharacteristic
117C2×#41C2×D311yescharacteristic
118C22×Q8—21no—
119C22×Q8C2×D312yes—
120C22×Q8C2×D311yescharacteristic
121#137—64no—
122#135—64no—
123C2×Q16—31no—
124C2×Q16—32no—
125C2×SD16—31no—
126C2×SD16—32no—
127C2×SD16—31no—
128C2×D8—31no—
129C2×M16—31no—
130C22×C8—31no—
131C22⋊Q8—32no—
132C22⋊Q8—61no—
133#121—32no—
134#121—61no—
135#121—62no—
136C2×C4⋊C4—31no—
137#103—32no—
138#103—32no—
139#103—32no—
140#102—32no—
141C22⋊C8—32no—
142C22⋊C8—32no—
143C23×C3—21no—
144C23×C3C2×D412yes—
145C23×C3C2×D411yescharacteristic
146C22×D3—21no—
147C3×Q8—28no—
148C3×Q8—22no—
149C3×Q8—24no—
150C3×Q8—21no—
151C3×Q8—22no—
152C3×Q8C2×D412yes—
153C3×Q8C2×D414yes—
154C3×D4—28no—
155C3×D4—24no—
156C3×D4—24no—
157C3×D4—22no—
158C3×D4—21no—
159C3×D4C2×D412yes—
160C2×C3×C4—28no—
161C2×C3×C4—28no—
162C2×C3×C4—24no—
163C2×C3×C4—22no—
164C2×C3×C4—21no—
165C2×C3×C4—24no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4C2×D412yes—
169C2×C3×C4C2×D411yescharacteristic
170C2×C3×C4C2×D412yes—
171C2×C3×C4C2×D412yes—
172C2×C3×C4C2411yescharacteristic
173#67—42no—
174#67—82no—
175C2×Dic3—21no—
176C2×Dic3—22no—
177C2×Dic3—81no—
178D12—21no—
179D12—21no—
180C4×D3—42no—
181Dic6—21no—
182Dic6—21no—
183C3⋊C8—21no—
184C3⋊C8—21no—
185C3⋊C8—22no—
186#41—216no—
187#41—28no—
188#41—24no—
189#41—24no—
190#41—22no—
191#41—22no—
192#41—22no—
193#41—62no—
194#41—62no—
195#41#6714yescomplemented
196C2×Q8—28no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—22no—
200C2×Q8—21no—
201C2×Q8—22no—
202C2×Q8—31no—
203C2×Q8#6711yescharacteristic
204C2×Q8#6714yescomplemented
205C2×Q8C22×D311yescharacteristic
206C2×Q8C22×D312yes—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—31no—
210C2×D4—61no—
211C2×D4—62no—
212C2×D4#6712yes—
213C2×D4#6711yescharacteristic
214C2×D4C22×D311yescharacteristic
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4C22×D312yes—
222C22×C4C22×D311yescharacteristic
223Q16—62no—
224Q16—64no—
225SD16—62no—
226SD16—64no—
227SD16—62no—
228D8—62no—
229M16—62no—
230C2×C8—32no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8—62no—
234C4⋊C4—32no—
235C4⋊C4—61no—
236C4⋊C4—62no—
237C22⋊C4—62no—
238C22⋊C4—62no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3#12012yes—
248C22×C3C22×D411yescharacteristic
249C2×D3—41no—
250C2×D3—41no—
251C2×D3—41no—
252C3×C4—24no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4#12012yes—
256C3×C4C22×D411yescharacteristic
257C3×C4C22×D411yescharacteristic
258Dic3—41no—
259Dic3—42no—
260C23—21no—
261C23—61no—
262C23C2×#6712yes—
263C23C2×#6711yescharacteristic
264Q8—28no—
265Q8—22no—
266Q8—24no—
267Q8—21no—
268Q8—22no—
269Q8—61no—
270Q8—61no—
271Q8C2×#6712yes—
272Q8C2×#6714yes—
273D4—28no—
274D4—24no—
275D4—24no—
276D4—22no—
277D4—21no—
278D4—61no—
279D4—61no—
280D4—122no—
281D4—242no—
282D4C2×#6712yes—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—62no—
292C2×C4—61no—
293C2×C4—122no—
294C2×C4—241no—
295C2×C4C2×#6712yes—
296C2×C4C2×#6711yescharacteristic
297C2×C4C2×#6712yes—
298C2×C4C2×#6712yes—
299C2×C4C23×D311yescharacteristic
300C8—61no—
301C8—61no—
302C8—62no—
303C2×C3—22no—
304C2×C3—42no—
305C2×C3—42no—
306C2×C3C2×#12011yescharacteristic
307C2×C3#57711yescharacteristic
308C2×C3#57811yescharacteristic
309D3—81no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—121no—
319C22—121no—
320C22—121no—
321C22#95312yes—
322C22C22×#6711yescharacteristic
323C4—24no—
324C4—22no—
325C4—22no—
326C4—122no—
327C4—121no—
328C4#95312yes—
329C4C22×#6711yescharacteristic
330C4C22×#6711yescharacteristic
331C3#301311yescharacteristic, complemented, Sylow
332C2—22no—
333C2—42no—
334C2—42no—
335C2—241no—
336C2#583711yescharacteristic
337C2#583811yescharacteristic
338C2C2×#95311yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×2−1+4C2
socleC22×C3C22×D4
hypercenterC2×2−1+4D3
hypocenterC3#3013
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C3×2−1+4C2semidirect
2. #5833C2semidirect
3. C2×#950C2semidirect
4. C2×#942C2semidirect
5. #5239C2semidirect
6. #5238C2semidirect
7. #5234C2semidirect
8. #5232C2semidirect
9. #5228C2semidirect
10. #5227C2semidirect
11. C2×#835C2semidirect
46. C2×2−1+4D3semidirect
64. C3×#41D4semidirect
72. C2×C3×Q8D4semidirect
195. #41#67semidirect
204. C2×Q8#67semidirect
331. C3#3013semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+42
#58332
C2×#9502
C2×#9422
#52392
#52382
#52342
#52322
#52282
#52272
C2×#8352
#30133