Finite Group Databasebeta

← #82592

Subgroups — #82592

Order 384.

824 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 157 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×C3×#120C211yesmaximal, characteristic, complemented
2C2×#953C211yesmaximal, characteristic, complemented
3C22×#217C211yesmaximal, characteristic, complemented
4C2×#939C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5222C212yesmaximal, complemented
7#5220C211yesmaximal, characteristic, complemented
8#4959C212yesmaximal, complemented
9#4958C211yesmaximal, characteristic, complemented
10#4952C212yesmaximal, complemented
11#4942C211yesmaximal, characteristic, complemented
12#2617—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×C3×D4C2212yes—
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C22×#67C2211yescharacteristic, complemented
19C23×Dic3C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C3×#120—24no—
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2212yes—
24#953—24no—
25C2×#217—24no—
26C2×#217C2211yescharacteristic
27C2×#217C2212yes—
28C2×#217C2212yes—
29C2×#217C2212yes—
30C2×#217C2211yescharacteristic, complemented
31C2×#217C2212yescomplemented
32#939—24no—
33C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
34C2×(C2×D3)⋊C4C2212yescomplemented
35C2×Dic3⋊C4C2211yescharacteristic, complemented
36C2×Dic3⋊C4C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#121—31no—
41C2×#120—31no—
42C2×#120D311yescharacteristic, complemented
43C22×C22⋊C4—31no—
44#397—32no—
45#394—32no—
46#394—32no—
47#392—32no—
48#386—31no—
49#386—31no—
50#380—31no—
51C24×C3—22no—
52C24×C3—22no—
53C24×C3—42no—
54C24×C3—42no—
55C24×C3D414yescomplemented
56C24×C3C2311yescharacteristic
57C24×C3C2312yes—
58C24×C3C2311yescharacteristic
59C23×D3C2311yescharacteristic
60C2×C3×D4—22no—
61C2×C3×D4—24no—
62C2×C3×D4—22no—
63C2×C3×D4—22no—
64C2×C3×D4—42no—
65C2×C3×D4—42no—
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2312yes—
68C2×#67—22no—
69C2×#67—24no—
70C2×#67—22no—
71C2×#67—22no—
72C2×#67—22no—
73C2×#67—42no—
74C2×#67—42no—
75C2×#67—42no—
76C22×Dic3—22no—
77C22×Dic3—42no—
78C22×Dic3—42no—
79C22×Dic3D414yescomplemented
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2312yes—
82C22×Dic3C2312yes—
83C22×Dic3C2311yescharacteristic
84C22×Dic3C2311yescharacteristic
85C3×C22⋊C4—24no—
86C3×C22⋊C4—22no—
87#217—24no—
88#217—22no—
89#217—22no—
90#217—24no—
91#217—24no—
92#217—42no—
93(C2×D3)⋊C4—22no—
94(C2×D3)⋊C4—42no—
95Dic3⋊C4—22no—
96Dic3⋊C4—42no—
97C25C2×D311yescharacteristic
98C22×D4—31no—
99C22×D4—32no—
100C22×D4—31no—
101C22×D4C2×D311yescharacteristic
102C22×D4C2×D312yes—
103C23×C4—31no—
104C23×C4—31no—
105#121—64no—
106#120—24no—
107#120—64no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4—31no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—32no—
118C2×C22⋊C4—64no—
119C2×C22⋊C4C2×D311yescharacteristic
120C2×C22⋊C4C2×D312yes—
121#95—32no—
122#95—32no—
123#95—32no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—28no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—48no—
139C23×C3—44no—
140C23×C3—44no—
141C23×C3—48no—
142C23×C3C2×D412yes—
143C23×C3C2×D414yes—
144C23×C3C2×D412yes—
145C23×C3#4112yes—
146C23×C3C2411yescharacteristic
147C22×D3—21no—
148C22×D3—21no—
149C22×D3—21no—
150C22×D3—22no—
151C22×D3—22no—
152C3×D4—48no—
153C3×D4—44no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160#67—44no—
161#67—48no—
162#67—44no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—21no—
166C2×Dic3—22no—
167C2×Dic3—21no—
168C2×Dic3—22no—
169C2×Dic3—21no—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3—22no—
173C2×Dic3—22no—
174C2×Dic3—44no—
175C2×Dic3—44no—
176C2×Dic3C2×D412yes—
177C2×Dic3C2×D412yes—
178C24—22no—
179C24—22no—
180C24—31no—
181C24—42no—
182C24—42no—
183C24#6714yescomplemented
184C24C22×D311yescharacteristic
185C24C22×D312yes—
186C24C22×D311yescharacteristic
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—62no—
194C2×D4—62no—
195C2×D4—62no—
196C2×D4—64no—
197C2×D4—62no—
198C2×D4—122no—
199C2×D4—122no—
200C2×D4—122no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—34no—
204C22×C4—32no—
205C22×C4—31no—
206C22×C4—32no—
207C22×C4—62no—
208C22×C4—122no—
209C22×C4—122no—
210C22×C4C22×D311yescharacteristic
211C22×C4C22×D312yes—
212C4⋊C4—62no—
213C4⋊C4—122no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—62no—
217C22⋊C4—64no—
218C22⋊C4—64no—
219C22⋊C4—62no—
220C22⋊C4—64no—
221C22⋊C4—62no—
222C22⋊C4—122no—
223C22⋊C4—122no—
224C22×C3—22no—
225C22×C3—24no—
226C22×C3—22no—
227C22×C3—24no—
228C22×C3—24no—
229C22×C3—24no—
230C22×C3—24no—
231C22×C3—416no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—42no—
239C22×C3—41no—
240C22×C3—42no—
241C22×C3#12114yes—
242C22×C3C22×D411yescharacteristic
243C22×C3C22×D411yescharacteristic
244C22×C3C2×#4111yescharacteristic
245C22×C3H(5,𝔽2)12yes—
246C22×C3H(5,𝔽2)12yes—
247C2×D3—41no—
248C2×D3—42no—
249C2×D3—41no—
250C2×D3—41no—
251C2×D3—42no—
252C3×C4—41no—
253C3×C4—42no—
254Dic3—22no—
255Dic3—22no—
256Dic3—41no—
257Dic3—42no—
258Dic3—42no—
259C23—22no—
260C23—22no—
261C23—28no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—48no—
274C23—44no—
275C23—44no—
276C23—48no—
277C23—61no—
278C23—61no—
279C23—62no—
280C23—62no—
281C23—61no—
282C23D3×D412yes—
283C23#23712yes—
284C23C2×#6714yes—
285C23C2×#6712yes—
286C23C23×D311yescharacteristic
287D4—48no—
288D4—44no—
289D4—124no—
290D4—128no—
291D4—124no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—62no—
311C2×C4—124no—
312C2×C4—124no—
313C2×C3—24no—
314C2×C3—44no—
315C2×C3—81no—
316C2×C3C2×#12111yescharacteristic
317C2×C3#53412yes—
318C2×C3#53411yescharacteristic
319C2×C3#53511yescharacteristic
320C2×C3#53812yes—
321D3—81no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—24no—
329C22—416no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—121no—
340C22—122no—
341C22—121no—
342C22—121no—
343C22—122no—
344C22#93914yes—
345C22C2×D3×D411yescharacteristic
346C22C2×#23711yescharacteristic
347C22#100412yes—
348C22#100412yes—
349C22C22×#6711yescharacteristic
350C4—41no—
351C4—42no—
352C4—62no—
353C4—62no—
354C4—121no—
355C4—122no—
356C4—122no—
357C3#261711yescharacteristic, complemented, Sylow
358C2—24no—
359C2—44no—
360C2—81no—
361C2—241no—
362C2#558911yescharacteristic
363C2#559211yescharacteristic
364C2#559312yes—
365C2C2×#93911yescharacteristic
366C2#580412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2617
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×C3×#120C2semidirect
2. C2×#953C2semidirect
3. C22×#217C2semidirect
4. C2×#939C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #5220C2semidirect
8. #4959C2semidirect
9. #4958C2semidirect
10. #4952C2semidirect
11. #4942C2semidirect
18. C22×#67C22semidirect
30. C2×#217C22semidirect
31. C2×#217C22semidirect
33. C2×(C2×D3)⋊C4C22semidirect
34. C2×(C2×D3)⋊C4C22semidirect
35. C2×Dic3⋊C4C22semidirect
36. C2×Dic3⋊C4C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
42. C2×#120D3semidirect
55. C24×C3D4semidirect
79. C22×Dic3D4semidirect
183. C24#67semidirect
357. C3#2617semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C2×#9532
C22×#2172
C2×#9392
#52232
#52222
#52202
#49592
#49582
#49522
#49422
#26173