Finite Group Databasebeta

← #82647

Subgroups — #82647

Order 384.

660 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic
2C2×#936C211yesmaximal, characteristic, complemented
3C2×D4×Dic3C211yesmaximal, characteristic, complemented
4C2×#924C211yesmaximal, characteristic, complemented
5#5222C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7C4×#217C211yesmaximal, characteristic, complemented
8#4973C211yesmaximal, characteristic, complemented
9#4972C211yesmaximal, characteristic, complemented
10#4952C212yesmaximal, complemented
11#4942C212yesmaximal, complemented
12#2427—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×Dic3C2212yes—
17C22×Dic6C2211yescharacteristic
18C3×#121—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2212yes—
27#936—24no—
28D4×Dic3—24no—
29C2×C4⋊Dic3C2211yescharacteristic
30#924—24no—
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2211yescharacteristic
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#124—31no—
41C2×C22⋊Q8—31no—
42C2×#121D311yescharacteristic
43C2×C4×D4—31no—
44#397—32no—
45#394—32no—
46#391—31no—
47#389—31no—
48#386—32no—
49#380—32no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C2×C3×D4—24no—
54C2×C3×D4—24no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4—42no—
58C2×C3×D4D414yes—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×Dic3—24no—
66C22×Dic3—24no—
67C22×Dic3—42no—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2312yes—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C2×Dic6—22no—
75C2×Dic6—22no—
76C2×Dic6—22no—
77C3×C4⋊C4—22no—
78C3×C22⋊C4—24no—
79#217—24no—
80#217—24no—
81#217—22no—
82#217—24no—
83#217—24no—
84#217—22no—
85C4⋊Dic3D414yescomplemented
86Dic3⋊C4—24no—
87Dic3⋊C4—42no—
88C4×Dic3—22no—
89C4×Dic3—22no—
90C22×Q8—31no—
91C22×D4C2×D312yes—
92C22×D4C2×D311yescharacteristic
93C23×C4—32no—
94C23×C4C2×D311yescharacteristic
95#124—64no—
96C22⋊Q8—64no—
97#121—24no—
98C4×D4—64no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4C2×D312yes—
110C2×C42—31no—
111C2×C42—31no—
112#95—31no—
113#95—31no—
114#95—32no—
115#95—32no—
116#95—32no—
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—22no—
122C23×C3—21no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—21no—
127C23×C3—22no—
128C23×C3—21no—
129C23×C3C2×D414yes—
130C23×C3#4112yes—
131C23×C3C2411yescharacteristic
132C3×D4—28no—
133C3×D4—48no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—44no—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4#4112yes—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—21no—
160C2×Dic3—48no—
161C2×Dic3C2×D414yes—
162C2×Dic3#4112yes—
163Dic6—44no—
164C24C22×D311yescharacteristic
165C24C22×D312yes—
166C2×Q8—62no—
167C2×Q8—62no—
168C2×Q8—62no—
169C2×D4—24no—
170C2×D4—24no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—42no—
174C2×D4#6714yes—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—31no—
184C22×C4—64no—
185C22×C4—64no—
186C22×C4—122no—
187C22×C4C22×D312yes—
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—34no—
192C4⋊C4—64no—
193C4⋊C4—122no—
194C22⋊C4—24no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—64no—
200C22⋊C4—64no—
201C42—62no—
202C42—62no—
203C22×C3—24no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—22no—
209C22×C3—41no—
210C22×C3—42no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3C22×D411yescharacteristic
218C22×C3C22×D411yescharacteristic
219C22×C3C2×#4111yescharacteristic
220C22×C3C2×#4111yescharacteristic
221C22×C3C2×#4111yescharacteristic
222C22×C3H(5,𝔽2)11yescharacteristic
223C22×C32−1+411yescharacteristic
224C3×C4—41no—
225C3×C4—42no—
226C3×C4#12114yes—
227Dic3—22no—
228Dic3—24no—
229Dic3—42no—
230Dic3—41no—
231Dic3—41no—
232Dic3—41no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—21no—
245C23#23712yes—
246C23C2×#6714yes—
247C23C23×D311yescharacteristic
248Q8—124no—
249D4—28no—
250D4—48no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—34no—
258C2×C4—32no—
259C2×C4—44no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—62no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—128no—
277C2×C4D3×D412yes—
278C2×C4#23712yes—
279C2×C4C2×#6712yes—
280C2×C3—44no—
281C2×C3—42no—
282C2×C3C2×#12111yescharacteristic
283C2×C3#52911yescharacteristic
284C2×C3#54211yescharacteristic
285C2×C3#54611yescharacteristic
286C2×C3#54611yescharacteristic
287C2×C3#54711yescharacteristic
288C2×C3#54811yescharacteristic
289C22—24no—
290C22—22no—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22C2×D3×D411yescharacteristic
304C22C2×#23711yescharacteristic
305C22C2×#23711yescharacteristic
306C22#100411yescharacteristic
307C22D3×#4111yescharacteristic
308C22#101011yescharacteristic
309C22C22×#6711yescharacteristic
310C4—41no—
311C4—42no—
312C4—62no—
313C4—64no—
314C4—122no—
315C4—121no—
316C4—121no—
317C4—121no—
318C4#93914yes—
319C3#242711yescharacteristic, complemented, Sylow
320C2—44no—
321C2—42no—
322C2#559811yescharacteristic
323C2#560011yescharacteristic
324C2#560311yescharacteristic
325C2#561211yescharacteristic
326C2C2×#93911yescharacteristic
327C2#580611yescharacteristic
328C2#582411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2427
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#936C2semidirect
3. C2×D4×Dic3C2semidirect
4. C2×#924C2semidirect
5. #5222C2semidirect
6. #5220C2semidirect
7. C4×#217C2semidirect
8. #4973C2semidirect
9. #4972C2semidirect
10. #4952C2semidirect
11. #4942C2semidirect
85. C4⋊Dic3D4semidirect
319. C3#2427semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9362
C2×D4×Dic32
C2×#9242
#52222
#52202
C4×#2172
#49732
#49722
#49522
#49422
#24273