Finite Group Databasebeta

← #82666

Subgroups — #82666

Order 384.

732 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×#938C211yesmaximal, characteristic, complemented
3C2×D4×Dic3C211yesmaximal, characteristic, complemented
4C2×C4×#67C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#5212C211yesmaximal, characteristic, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4977C211yesmaximal, characteristic, complemented
10#4958C212yesmaximal, complemented
11#4952C212yesmaximal, complemented
12#2535—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C23×Dic3C2212yes—
19C22×C4×D3C2211yescharacteristic
20C3×#121—24no—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2211yescharacteristic
26C2×#217C2212yes—
27#938—24no—
28D4×Dic3—24no—
29C4×#67—24no—
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2212yes—
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#121—31no—
41C2×#121D311yescharacteristic, complemented
42C2×C4×D4—31no—
43C2×C4×D4—31no—
44#397—32no—
45#392—32no—
46#386—32no—
47#386—32no—
48#382—31no—
49#380—31no—
50#380—31no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C2×C3×D4—24no—
55C2×C3×D4—24no—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—42no—
59C2×C3×D4D414yescomplemented
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4—22no—
63C22×C3×C4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—24no—
69C2×#67—24no—
70C2×#67—22no—
71C2×#67—42no—
72C22×Dic3—24no—
73C22×Dic3—42no—
74C22×Dic3—42no—
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2312yes—
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C2×C4×D3—22no—
81C2×C4×D3—22no—
82C2×C4×D3—22no—
83C3×C4⋊C4—22no—
84C3×C22⋊C4—24no—
85#217—24no—
86#217—22no—
87#217—24no—
88#217—42no—
89(C2×D3)⋊C4—22no—
90(C2×D3)⋊C4—42no—
91C4⋊Dic3—22no—
92C4⋊Dic3—22no—
93Dic3⋊C4—22no—
94C4×Dic3D414yescomplemented
95C22×D4—31no—
96C22×D4—32no—
97C22×D4C2×D312yes—
98C22×D4C2×D311yescharacteristic
99C23×C4—32no—
100C23×C4—31no—
101C23×C4C2×D311yescharacteristic
102#121—24no—
103#121—64no—
104C4×D4—64no—
105C4×D4—64no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4C2×D311yescharacteristic
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4C2×D312yes—
117C2×C42—31no—
118#95—32no—
119#95—32no—
120#95—32no—
121#95—32no—
122C23×C3—22no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3—21no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—21no—
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—21no—
134C23×C3C2×D414yes—
135C23×C3#4112yes—
136C23×C3C2411yescharacteristic
137C22×D3—21no—
138C22×D3—21no—
139C22×D3—21no—
140C22×D3—21no—
141C22×D3—21no—
142C22×D3—21no—
143C22×D3#4112yes—
144C3×D4—28no—
145C3×D4—48no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—22no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—44no—
153C2×C3×C4C2×D412yes—
154C2×C3×C4C2×D412yes—
155C2×C3×C4#4112yes—
156#67—28no—
157#67—48no—
158C2×Dic3—21no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—22no—
171C2×Dic3—48no—
172C2×Dic3C2×D412yes—
173C2×Dic3C2×D412yes—
174C4×D3—44no—
175C24—31no—
176C24C22×D311yescharacteristic
177C24C22×D312yes—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—42no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—64no—
187C2×D4—64no—
188C2×D4—122no—
189C2×D4#6714yescomplemented
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—32no—
196C22×C4—31no—
197C22×C4—32no—
198C22×C4—64no—
199C22×C4—62no—
200C22×C4—62no—
201C22×C4—62no—
202C22×C4—122no—
203C22×C4—122no—
204C22×C4C22×D312yes—
205C22×C4C22×D311yescharacteristic
206C22×C4C22×D311yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4—62no—
211C22⋊C4—24no—
212C22⋊C4—62no—
213C22⋊C4—64no—
214C22⋊C4—62no—
215C22⋊C4—64no—
216C22⋊C4—122no—
217C22⋊C4—122no—
218C42—34no—
219C22×C3—24no—
220C22×C3—22no—
221C22×C3—24no—
222C22×C3—22no—
223C22×C3—24no—
224C22×C3—22no—
225C22×C3—41no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—42no—
232C22×C3—42no—
233C22×C3C22×D411yescharacteristic
234C22×C3C22×D411yescharacteristic
235C22×C3C2×#4111yescharacteristic
236C22×C3C2×#4111yescharacteristic
237C22×C3C2×#4111yescharacteristic
238C22×C3H(5,𝔽2)11yescharacteristic
239C22×C3H(5,𝔽2)11yescharacteristic
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—41no—
244C2×D3—41no—
245C2×D3—41no—
246C2×D3—41no—
247C3×C4—41no—
248C3×C4—42no—
249C3×C4#12114yes—
250Dic3—22no—
251Dic3—22no—
252Dic3—41no—
253Dic3—42no—
254Dic3—42no—
255C23—21no—
256C23—22no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—32no—
268C23—61no—
269C23—61no—
270C23—61no—
271C23—61no—
272C23—61no—
273C23—61no—
274C23#23712yes—
275C23C2×#6714yes—
276C23C23×D311yescharacteristic
277D4—28no—
278D4—48no—
279D4—68no—
280D4—128no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—44no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—62no—
296C2×C4—61no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—61no—
302C2×C4—62no—
303C2×C4—128no—
304C2×C4—124no—
305C2×C4D3×D412yes—
306C2×C4#23712yes—
307C2×C4C2×#6712yes—
308C2×C3—44no—
309C2×C3—42no—
310C2×C3C2×#12111yescharacteristic
311C2×C3#52511yescharacteristic
312C2×C3#53511yescharacteristic
313C2×C3#54611yescharacteristic
314C2×C3#54611yescharacteristic
315C2×C3#55311yescharacteristic
316C2×C3#55311yescharacteristic
317D3—42no—
318C22—24no—
319C22—22no—
320C22—24no—
321C22—24no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—42no—
331C22—42no—
332C22—62no—
333C22—62no—
334C22—62no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22—121no—
339C22C2×D3×D411yescharacteristic
340C22C2×#23711yescharacteristic
341C22C2×#23711yescharacteristic
342C22#100411yescharacteristic
343C22D3×#4111yescharacteristic
344C22#100911yescharacteristic
345C22C22×#6711yescharacteristic
346C4—41no—
347C4—42no—
348C4—62no—
349C4—62no—
350C4—121no—
351C4—122no—
352C4—122no—
353C4#93914yes—
354C3#253511yescharacteristic, complemented, Sylow
355C2—44no—
356C2—42no—
357C2—122no—
358C2#560611yescharacteristic
359C2#560711yescharacteristic
360C2#561411yescharacteristic
361C2#561811yescharacteristic
362C2C2×#93911yescharacteristic
363C2#580611yescharacteristic
364C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2535
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#938C2semidirect
3. C2×D4×Dic3C2semidirect
4. C2×C4×#67C2semidirect
5. #5223C2semidirect
6. #5220C2semidirect
7. #5212C2semidirect
8. #4987C2semidirect
9. #4977C2semidirect
10. #4958C2semidirect
11. #4952C2semidirect
41. C2×#121D3semidirect
59. C2×C3×D4D4semidirect
94. C4×Dic3D4semidirect
189. C2×D4#67semidirect
354. C3#2535semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9382
C2×D4×Dic32
C2×C4×#672
#52232
#52202
#52122
#49872
#49772
#49582
#49522
#25353