Finite Group Databasebeta

← #69499

Subgroups — #69499

Order 384.

578 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 83 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×#953C211yesmaximal, characteristic
2#5486C211yesmaximal, characteristic, complemented
3C3×#379C211yesmaximal, characteristic, complemented
4#5247C211yesmaximal, characteristic, complemented
5C2×#806C211yesmaximal, characteristic, complemented
6#4527C211yesmaximal, characteristic, complemented
7#4526C211yesmaximal, characteristic, complemented
8#1889—31nomaximal, Sylow
9C25×C3C2211yescharacteristic
10C22×#67—21no—
11C22×#67C2211yescharacteristic
12C2×C3×C22⋊C4—22no—
13C2×C3×C22⋊C4C2211yescharacteristic
14C2×C3×C22⋊C4C2211yescharacteristic
15#953—22no—
16#953C414yescomplemented
17C2×#217—21no—
18C2×#217—22no—
19C2×#217C2211yescharacteristic
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C4×#67—22no—
23#67⋊C4—22no—
24D3×C22⋊C4—21no—
25#877—21no—
26#806—22no—
27C2×#120—31no—
28#518—31no—
29C2×#99—31no—
30#379—31no—
31#379D311yescharacteristic, complemented
32#342—31no—
33#342—31no—
34C24×C3—21no—
35C24×C3—21no—
36C24×C3—21no—
37C24×C3—21no—
38C24×C3—21no—
39C24×C3—21no—
40C24×C3—44no—
41C24×C3C2×C412yescomplemented
42C24×C3C2311yescharacteristic
43C23×D3D411yescharacteristic
44C22×C3×C4—21no—
45C22×C3×C4D412yescomplemented
46C2×#67—21no—
47C2×#67—21no—
48C2×#67—21no—
49C2×#67—21no—
50C2×#67—24no—
51C2×#67—44no—
52C2×#67C2×C412yes—
53C2×#67D412yes—
54C22×Dic3—21no—
55C22×Dic3—21no—
56C22×Dic3D411yescharacteristic
57C22×Dic3D412yescomplemented
58C2×C4×D3—22no—
59C3×C22⋊C4—21no—
60C3×C22⋊C4—21no—
61C3×C22⋊C4—21no—
62C3×C22⋊C4—21no—
63C3×C22⋊C4—44no—
64#217—21no—
65#217—21no—
66#217—21no—
67#217—24no—
68#217—21no—
69#217—21no—
70#217—44no—
71#217C2×C412yescomplemented
72(C2×D3)⋊C4—22no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75C25C2×D311yescharacteristic
76C22×D4—31no—
77C22×D4—61no—
78#120—34no—
79#120—62no—
80C4×D4—62no—
81C4×D4—62no—
82#117—61no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—31no—
87C2×C22⋊C4—61no—
88C2×C22⋊C4—61no—
89C2×C22⋊C4—62no—
90C2×C22⋊C4C2×D311yescharacteristic
91C2×C22⋊C4C2×D311yescharacteristic
92#99—62no—
93C23×C3—24no—
94C23×C3—24no—
95C23×C3—24no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—44no—
100C23×C3—42no—
101C23×C3—42no—
102C23×C3—42no—
103C23×C3—42no—
104C23×C3—44no—
105C23×C3—44no—
106C23×C3—44no—
107C23×C3—41no—
108C23×C3—41no—
109C23×C3—41no—
110C23×C3—41no—
111C23×C3C22×C411yescharacteristic
112C23×C3C2×D411yescharacteristic
113C23×C3C2×D411yescharacteristic
114C23×C3C2×D411yescharacteristic
115C23×C3C2×D411yescharacteristic
116C23×C3#4111yescharacteristic
117C23×C3#4111yescharacteristic
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—42no—
121C22×D3C22⋊C412yescomplemented
122C2×C3×C4—22no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
125C2×C3×C4—41no—
126#67—22no—
127#67—22no—
128#67—22no—
129#67—22no—
130#67—48no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—41no—
139C2×Dic3C22⋊C412yescomplemented
140C4×D3—82no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—31no—
148C24—44no—
149C24C4×D312yescomplemented
150C24C22×D311yescharacteristic
151C2×D4—32no—
152C2×D4—32no—
153C2×D4—61no—
154C2×D4—61no—
155C2×D4—61no—
156C2×D4—61no—
157C2×D4—64no—
158C2×D4—124no—
159C22×C4—21no—
160C22×C4—31no—
161C22×C4—32no—
162C22×C4—62no—
163C22×C4—61no—
164C22×C4—61no—
165C22×C4#6712yescomplemented
166C4⋊C4—62no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—32no—
172C22⋊C4—44no—
173C22⋊C4—62no—
174C22⋊C4—64no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—124no—
181C42—62no—
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—24no—
186C22×C3—24no—
187C22×C3—24no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C22×C3—44no—
194C22×C3—44no—
195C22×C3—44no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—42no—
199C22×C3—44no—
200C22×C3C2×C22⋊C411yescharacteristic
201C22×C3C4×D411yescharacteristic
202C22×C3C4×D411yescharacteristic
203C22×C3#12011yescharacteristic
204C22×C3#12111yescharacteristic
205C22×C3#12111yescharacteristic
206C22×C3#12311yescharacteristic
207C2×D3—22no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—81no—
211C3×C4—42no—
212C3×C4—81no—
213Dic3—22no—
214Dic3—42no—
215Dic3—42no—
216Dic3—81no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—32no—
224C23—44no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—61no—
237C23—61no—
238C23—122no—
239C23C2×C4×D311yescharacteristic
240C23#23511yescharacteristic
241C23D3×D411yescharacteristic
242C23D3×D411yescharacteristic
243C23D3×D411yescharacteristic
244C23#23711yescharacteristic
245C23C2×#6711yescharacteristic
246D4—62no—
247D4—62no—
248D4—62no—
249D4—62no—
250D4—128no—
251C2×C4—22no—
252C2×C4—32no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—121no—
264C2×C4—242no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—44no—
272C2×C3#38611yescharacteristic
273C2×C3B(4,𝔽2)12yes—
274D3—42no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—24no—
280C22—24no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—44no—
286C22—42no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—42no—
292C22—44no—
293C22—62no—
294C22—122no—
295C22—122no—
296C22—241no—
297C22D3×C22⋊C411yescharacteristic
298C22#67⋊C411yescharacteristic
299C22#88311yescharacteristic
300C22#88411yescharacteristic
301C22C4×#6711yescharacteristic
302C22#93711yescharacteristic
303C22#93811yescharacteristic
304C4—42no—
305C4—62no—
306C4—81no—
307C4—122no—
308C4—122no—
309C4—241no—
310C3#188911yescharacteristic, complemented, Sylow
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—44no—
318C2—122no—
319C2#495611yescharacteristic
320C2#503212yes—
subgroupG/H
centerC22#937
commutatorC23×C3C22×C4
FrattiniC24C22×D3
FittingC3×#379C2
socleC22×C3#120
prime radicalC2×#953C2
cosocleC24×C3C23
hypercenter#379D3
hypocenterC3#1889
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #5486C2semidirect
3. C3×#379C2semidirect
4. #5247C2semidirect
5. C2×#806C2semidirect
6. #4527C2semidirect
7. #4526C2semidirect
16. #953C4semidirect
31. #379D3semidirect
41. C24×C3C2×C4semidirect
45. C22×C3×C4D4semidirect
57. C22×Dic3D4semidirect
71. #217C2×C4semidirect
121. C22×D3C22⋊C4semidirect
139. C2×Dic3C22⋊C4semidirect
149. C24C4×D3semidirect
165. C22×C4#67semidirect
310. C3#1889semidirect

Maximal subgroups

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

subgroupindex
C2×#9532
#54862
C3×#3792
#52472
C2×#8062
#45272
#45262
#18893