Finite Group Databasebeta

← #74701

Subgroups — #74701

Order 384.

852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×Dic6C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic
3C2×#885C212yesmaximal, complemented
4C3×#386C211yesmaximal, characteristic, complemented
5#5220C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4958C211yesmaximal, characteristic, complemented
8#4947C212yesmaximal, complemented
9Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
10#4673C212yesmaximal, complemented
11#4646C212yesmaximal, complemented
12#2436—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C22×Dic6—24no—
20C22×Dic6—28no—
21C22×Dic6C2212yes—
22C22×Dic6C2212yes—
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2212yes—
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2212yes—
28#939—24no—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32C2×Dic3⋊C4C2211yescharacteristic
33C2×C4×Dic3C2212yes—
34#885—28no—
35C3×#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C23×Q8—31no—
41C2×#124—32no—
42C2×#121—31no—
43#393—32no—
44#393—32no—
45#391—32no—
46#386—31no—
47#386—31no—
48#386—31no—
49#386D311yescharacteristic, complemented
50C4×C22⋊C4—31no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—24no—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2312yes—
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—24no—
66C2×#67—22no—
67C22×Dic3—22no—
68C22×Dic3—24no—
69C22×Dic3—24no—
70C22×Dic3D414yes—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C22×Dic3C2312yes—
75C22×Dic3C2311yescharacteristic
76C2×Dic6—232no—
77C2×Dic6—24no—
78C2×Dic6—24no—
79C2×Dic6—24no—
80C2×Dic6—24no—
81C2×Dic6—24no—
82C2×Dic6D418yescomplemented
83C3×C22⋊C4—22no—
84C3×C22⋊C4—24no—
85#217—22no—
86#217—24no—
87#217—22no—
88(C2×D3)⋊C4—24no—
89(C2×D3)⋊C4—24no—
90(C2×D3)⋊C4—22no—
91Dic3⋊C4—22no—
92C4×Dic3—24no—
93C22×Q8—32no—
94C22×Q8—32no—
95C22×Q8—64no—
96C22×Q8—68no—
97C22×D4—31no—
98C22×D4—31no—
99C22×D4C2×D311yescharacteristic
100C23×C4—31no—
101C23×C4—31no—
102C23×C4C2×D311yescharacteristic
103#124—68no—
104#121—64no—
105C2×C4⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—31no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4C2×D312yes—
113C2×C22⋊C4C2×D311yescharacteristic
114C2×C42—32no—
115#95—32no—
116#95—32no—
117#95—32no—
118#95—32no—
119#95C2×D312yes—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3C2×D412yes—
128C23×C3#4112yes—
129C23×C3#4112yes—
130C23×C3C2411yescharacteristic
131C22×D3—22no—
132C22×D3—21no—
133C22×D3—21no—
134C22×D3—21no—
135C22×D3—22no—
136C3×D4—44no—
137C2×C3×C4—28no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—22no—
146C2×C3×C4C2×D414yes—
147#67—44no—
148#67—44no—
149C2×Dic3—28no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—28no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—22no—
160C2×Dic3C2×D412yes—
161C2×Dic3C2×D414yes—
162C2×Dic3C2×D414yes—
163C2×Dic3C2×D412yes—
164Dic6—216no—
165Dic6—216no—
166C24—31no—
167C24C22×D311yescharacteristic
168C24C22×D311yescharacteristic
169C2×Q8—38no—
170C2×Q8—64no—
171C2×Q8—64no—
172C2×Q8—64no—
173C2×Q8—64no—
174C2×Q8—632no—
175C2×Q8—64no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—62no—
180C2×D4—64no—
181C2×D4—62no—
182C2×D4—62no—
183C2×D4—62no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—31no—
187C22×C4—34no—
188C22×C4—31no—
189C22×C4—32no—
190C22×C4—31no—
191C22×C4—32no—
192C22×C4—64no—
193C22×C4—64no—
194C22×C4—62no—
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D312yes—
197C22×C4C22×D312yes—
198C4⋊C4—62no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—62no—
202C22⋊C4—64no—
203C22⋊C4—62no—
204C22⋊C4—64no—
205C22⋊C4—62no—
206C22⋊C4—64no—
207C42—64no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3#12414yescomplemented
218C22×C3C22×D411yescharacteristic
219C22×C3C22×D412yes—
220C22×C3C2×#4111yescharacteristic
221C22×C3C2×#4111yescharacteristic
222C22×C32−1+412yes—
223C2×D3—41no—
224C2×D3—41no—
225C2×D3—41no—
226C2×D3—42no—
227C2×D3—42no—
228C3×C4—24no—
229C3×C4—41no—
230C3×C4—42no—
231Dic3—22no—
232Dic3—24no—
233Dic3—41no—
234Dic3—42no—
235Dic3#12014yes—
236C23—22no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—61no—
244C23—62no—
245C23—61no—
246C23—61no—
247C23—62no—
248C23#23512yes—
249C23D3×D412yes—
250C23#23712yes—
251C23C23×D311yescharacteristic
252Q8—616no—
253Q8—616no—
254D4—44no—
255D4—124no—
256D4—124no—
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—34no—
267C2×C4—32no—
268C2×C4—32no—
269C2×C4—34no—
270C2×C4—68no—
271C2×C4—68no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—61no—
277C2×C4—62no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4D3×D414yes—
282C2×C3—24no—
283C2×C3—81no—
284C2×C3C2×#12011yescharacteristic
285C2×C3C2×#12411yescharacteristic
286C2×C3#53611yescharacteristic
287C2×C3#54812yes—
288C2×C3#54812yes—
289D3—81no—
290C22—22no—
291C22—24no—
292C22—22no—
293C22—24no—
294C22—41no—
295C22—41no—
296C22—42no—
297C22—41no—
298C22—42no—
299C22—121no—
300C22—121no—
301C22—122no—
302C22—121no—
303C22—122no—
304C22#88514yescomplemented
305C22C2×#23511yescharacteristic
306C22C2×D3×D411yescharacteristic
307C22C2×D3×D412yes—
308C22C2×#23711yescharacteristic
309C22#101012yes—
310C4—24no—
311C4—34no—
312C4—41no—
313C4—42no—
314C4—62no—
315C4—64no—
316C4—121no—
317C4—122no—
318C3#243611yescharacteristic, complemented, Sylow
319C2—24no—
320C2—81no—
321C2—241no—
322C2C2×#88511yescharacteristic
323C2#555212yes—
324C2D3×#12011yescharacteristic
325C2#559812yes—
326C2#564811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC2×#939C2
hypercenter#386D3
hypocenterC3#2436
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×Dic6C2semidirect
3. C2×#885C2semidirect
4. C3×#386C2semidirect
5. #5220C2semidirect
6. #5213C2semidirect
7. #4958C2semidirect
8. #4947C2semidirect
9. Dic3×C22⋊C4C2semidirect
10. #4673C2semidirect
11. #4646C2semidirect
49. #386D3semidirect
82. C2×Dic6D4semidirect
217. C22×C3#124semidirect
304. C22#885semidirect
318. C3#2436semidirect

Maximal subgroups

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

subgroupindex
C23×Dic62
C2×#9392
C2×#8852
C3×#3862
#52202
#52132
#49582
#49472
Dic3×C22⋊C42
#46732
#46462
#24363