Finite Group Databasebeta

← #10594915

Subgroups — #10594915

Order 576.

638 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#634C211yesmaximal, characteristic, complemented
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3C3×C4×#67C211yesmaximal, characteristic
4C3×#924C211yesmaximal, characteristic, complemented
5#63982C212yesmaximal, complemented
6#67⋊Dic3C212yesmaximal
7#63896C212yesmaximal, complemented
8#63866C211yesmaximal, characteristic, complemented
9#63857C211yesmaximal, characteristic, complemented
10#63844C211yesmaximal, characteristic, complemented
11#63840C212yesmaximal, complemented
12#5632—31nomaximal
13D4×Dic6—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C22×C3⋊Dic3C2212yes—
16C2×#634—24no—
17C2×#634C2212yes—
18C2×#634C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×#627C2212yes—
24C2×D3×Dic3C2212yes—
25D3×Dic6—24no—
26C3×#217C2212yes—
27C3×#217C2211yescharacteristic
28C3×(C2×D3)⋊C4C2211yescharacteristic
29C3×C4⋊Dic3C2211yescharacteristic
30C3×Dic3⋊C4C2212yes—
31C3×Dic3⋊C4C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33#3724C2212yes—
34Dic3⋊Dic3C2212yes—
35Dic3⋊Dic3C2212yes—
36Dic3⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C2×D3×Q8—31no—
40C22×Dic6—31no—
41C22×Dic6—31no—
42C22×Dic6—31no—
43C22×Dic6—61no—
44C3×C22⋊Q8—31no—
45C3×C4×D4—31no—
46#944—31no—
47D4×Dic3—32no—
48C4×#67D311yescharacteristic
49#924—31no—
50#924D311yescharacteristic, complemented
51#897—31no—
52#896—32no—
53#888—32no—
54#888—32no—
55#887—31no—
56#67⋊C4—32no—
57#878—32no—
58#878—32no—
59#878—32no—
60#869—31no—
61C4×Dic6—31no—
62C23×C32C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C2×C32×C4—22no—
65C2×C32×C4C2311yescharacteristic
66C2×C32×C4C2311yescharacteristic
67C2×C3⋊Dic3—24no—
68C2×C3⋊Dic3C2312yes—
69C2×C3⋊Dic3C2312yes—
70#634—22no—
71#634—24no—
72#634D414yescomplemented
73C3×#67Q814yescomplemented
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2312yes—
77C2×C3×Dic3C2311yescharacteristic
78C2×C3×Dic3C2311yescharacteristic
79C3×C4×D3—22no—
80C3×Dic6—22no—
81#627—24no—
82D3×Dic3—24no—
83D4×Q8—91noSylow
84C2×C3×Q8—31no—
85C2×C3×D4—31no—
86C22×C3×C4—21no—
87C22×C3×C4—31no—
88C22×C3×C4C2×D311yescharacteristic
89C22×C3×C4C2×D311yescharacteristic
90C2×#67C2×D311yescharacteristic
91C22×Dic3—32no—
92C22×Dic3—32no—
93C22×Dic3—32no—
94C22×Dic3—62no—
95D3×Q8—64no—
96C2×C4×D3—32no—
97C2×C4×D3C2×D311yescharacteristic
98C2×Dic6—31no—
99C2×Dic6—31no—
100C2×Dic6—31no—
101C2×Dic6—32no—
102C2×Dic6—32no—
103C2×Dic6—31no—
104C2×Dic6—32no—
105C2×Dic6—32no—
106C2×Dic6—61no—
107C2×Dic6—62no—
108C2×Dic6—61no—
109C2×Dic6—64no—
110C2×Dic6—64no—
111C2×Dic6—64no—
112C2×Dic6—68no—
113C2×Dic6C2×D311yescharacteristic
114C3×C4⋊C4—31no—
115C3×C4⋊C4—31no—
116C3×C4⋊C4—32no—
117C3×C22⋊C4—31no—
118C3×C22⋊C4—32no—
119C3×C22⋊C4—31no—
120C3×C42—31no—
121#217—32no—
122#217C2×D311yescharacteristic
123#217C2×D312yes—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4C2×D311yescharacteristic
126C4⋊Dic3—32no—
127C4⋊Dic3—32no—
128C4⋊Dic3—32no—
129C4⋊Dic3C2×D311yescharacteristic
130Dic3⋊C4—32no—
131Dic3⋊C4—32no—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4—32no—
135Dic3⋊C4C2×D312yes—
136Dic3⋊C4C2×D311yescharacteristic
137C4×Dic3—32no—
138C4×Dic3—32no—
139C4×Dic3C2×D311yescharacteristic
140C22×C32—21no—
141C22×C32—21no—
142C22×C32C2×Q812yescomplemented
143C22×C32C2411yescharacteristic
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3C2×Q812yescomplemented
147C32×C4—21no—
148C32×C4C2×D412yes—
149C3⋊Dic3—22no—
150C3⋊Dic3C2×D414yes—
151C3×Dic3—22no—
152C3×Dic3—21no—
153C3×Dic3—22no—
154C3×Dic3—21no—
155C3×Dic3C2×Q812yescomplemented
156C22×Q8—91no—
157C22×Q8—91no—
158C4⋊Q8—91no—
159C4⋊Q8—92no—
160C22⋊Q8—31no—
161C22⋊Q8—92no—
162C22⋊Q8—92no—
163C22⋊Q8—91no—
164C4×Q8—91no—
165C4×D4—31no—
166C4×D4—92no—
167C23×C3—21no—
168C23×C3—31no—
169C23×C3C22×D311yescharacteristic
170C23×C3C22×D311yescharacteristic
171C22×D3C22×D311yescharacteristic
172C3×Q8—62no—
173C3×D4—34no—
174C2×C3×C4—24no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—62no—
185C2×C3×C4C22×D311yescharacteristic
186C2×C3×C4C22×D311yescharacteristic
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189#67Dic614yescomplemented
190C2×Dic3—32no—
191C2×Dic3—32no—
192C2×Dic3—32no—
193C2×Dic3—32no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—64no—
197C2×Dic3—64no—
198C2×Dic3—64no—
199C2×Dic3—68no—
200C2×Dic3C22×D312yes—
201C2×Dic3C22×D311yescharacteristic
202C2×Dic3C22×D311yescharacteristic
203C2×Dic3C22×D312yes—
204C2×Dic3C22×D311yescharacteristic
205C4×D3—22no—
206C4×D3—64no—
207Dic6—22no—
208Dic6—34no—
209Dic6—34no—
210Dic6—64no—
211Dic6—64no—
212Dic6—64no—
213Dic6—64no—
214Dic6—64no—
215Dic6—68no—
216Dic6—64no—
217Dic6—62no—
218Dic6—62no—
219C2×C32—22no—
220C2×C32C22×D411yescharacteristic
221C2×C32C22×Q811yescharacteristic
222C2×C322−1+411yescharacteristic
223C3×D3—22no—
224C2×Q8—31no—
225C2×Q8—92no—
226C2×Q8—91no—
227C2×Q8—92no—
228C2×Q8—91no—
229C2×Q8—184no—
230C2×Q8—184no—
231C2×D4—31no—
232C22×C4—31no—
233C22×C4—92no—
234C22×C4—92no—
235C22×C4D3211yescharacteristic
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—32no—
239C4⋊C4—92no—
240C4⋊C4—92no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C22⋊C4—31no—
244C22⋊C4—32no—
245C22⋊C4—31no—
246C22⋊C4—92no—
247C42—31no—
248C42—92no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—32no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3C2×Dic612yescomplemented
261C22×C3D3×Q812yescomplemented
262C22×C3C23×D311yescharacteristic
263C22×C3C23×D311yescharacteristic
264C2×D3—21no—
265C2×D3—21no—
266C2×D3C2×Dic612yescomplemented
267C3×C4—22no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—32no—
272C3×C4—61no—
273C3×C4—62no—
274C3×C4—61no—
275C3×C4—62no—
276C3×C4D3×D412yes—
277C3×C4D3×D412yes—
278Dic3—22no—
279Dic3—21no—
280Dic3—22no—
281Dic3—21no—
282Dic3—34no—
283Dic3—34no—
284Dic3—64no—
285Dic3—64no—
286Dic3—62no—
287Dic3—62no—
288Dic3C2×Dic612yescomplemented
289C32D4×Q811yescharacteristic, complemented, Sylow
290C23—31no—
291C23C2×D3211yescharacteristic
292Q8—62no—
293Q8—94no—
294Q8—184no—
295Q8—184no—
296Q8—182no—
297D4—34no—
298C2×C4—22no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—32no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—62no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4—184no—
308C2×C4—184no—
309C2×C4C2×D3211yescharacteristic
310C2×C4C2×D3211yescharacteristic
311C2×C3—24no—
312C2×C3—22no—
313C2×C3—22no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—62no—
318C2×C3C22×Dic611yescharacteristic
319C2×C3C2×D3×D411yescharacteristic
320C2×C3C2×D3×D411yescharacteristic
321C2×C3C2×D3×Q811yescharacteristic
322C2×C3#101011yescharacteristic
323C2×C3#101011yescharacteristic
324D3—22no—
325C22—21no—
326C22—21no—
327C22—32no—
328C22—61no—
329C22—61no—
330C22D3×Dic612yescomplemented
331C22C22×D3211yescharacteristic
332C4—21no—
333C4—32no—
334C4—61no—
335C4—62no—
336C4—61no—
337C4—62no—
338C4—94no—
339C4—182no—
340C4#381112yes—
341C3—21no—
342C3D4×Dic611yescharacteristic, complemented
343C3#563211yescharacteristic, complemented
344C2—22no—
345C2—62no—
346C2C2×D3×Dic611yescharacteristic
347C2#6429811yescharacteristic
348C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×C3×#67C22
hypercenterC22×C4D32
hypocenterC32D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#634C2semidirect
2. C2×D3×Dic6C2semidirect
4. C3×#924C2semidirect
5. #63982C2semidirect
7. #63896C2semidirect
8. #63866C2semidirect
9. #63857C2semidirect
10. #63844C2semidirect
11. #63840C2semidirect
50. #924D3semidirect
72. #634D4semidirect
73. C3×#67Q8semidirect
142. C22×C32C2×Q8semidirect
146. C2×C3×D3C2×Q8semidirect
155. C3×Dic3C2×Q8semidirect
189. #67Dic6semidirect
260. C22×C3C2×Dic6semidirect
261. C22×C3D3×Q8semidirect
266. C2×D3C2×Dic6semidirect
288. Dic3C2×Dic6semidirect
289. C32D4×Q8semidirect
330. C22D3×Dic6semidirect
342. C3D4×Dic6semidirect
343. C3#5632semidirect

Maximal subgroups

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

subgroupindex
C22×#6342
C2×D3×Dic62
C3×C4×#672
C3×#9242
#639822
#67⋊Dic32
#638962
#638662
#638572
#638442
#638402
#56323
D4×Dic63