Finite Group Databasebeta

← #10595361

Subgroups — #10595361

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
1C2×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3C3×#946C211yesmaximal, characteristic, complemented
4C3×#944C211yesmaximal, characteristic, complemented
5C4×#626C211yesmaximal, characteristic
6#63896C212yesmaximal, complemented
7#63881C212yesmaximal
8#63871C211yesmaximal, characteristic, complemented
9#63865C212yesmaximal, complemented
10#63844C211yesmaximal, characteristic, complemented
11#63840C212yesmaximal, complemented
12#5727—31nomaximal
13#5632—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15Q8×C3⋊D3—24no—
16C2×C4×C3⋊D3C2212yes—
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×#634C2212yes—
19C2×#634C2211yescharacteristic
20C2×C3×C4×D3C2211yescharacteristic
21C2×C3×Dic6C2211yescharacteristic
22C2×#627C2212yes—
23C2×#626C2211yescharacteristic
24C2×D3×Dic3C2212yes—
25D3×Dic6—24no—
26C3×(C2×D3)⋊C4C2212yes—
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2212yes—
29C3×Dic3⋊C4C2212yes—
30C3×Dic3⋊C4C2212yes—
31C3×C4×Dic3C2211yescharacteristic
32#3724C2212yes—
33Dic3⋊Dic3C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35#3722C2211yescharacteristic
36#3722C2212yes—
37(C2×D3)⋊Dic3C2211yescharacteristic
38Dic32C2212yes—
39C2×D3×Q8—31no—
40C2×D3×Q8—31no—
41C2×D3×Q8—31no—
42C2×D3×Q8—61no—
43C22×Dic6—31no—
44C3×C4⋊Q8—31no—
45C3×C22⋊Q8—31no—
46#946—31no—
47#946D311yescharacteristic, complemented
48#944D311yescharacteristic, complemented
49C4×#67—31no—
50#897—31no—
51#896—32no—
52#896—32no—
53#896—32no—
54#893—32no—
55#888—32no—
56#888—32no—
57#887—31no—
58#67⋊C4—32no—
59#878—32no—
60C4×D12—31no—
61C4×Dic6—31no—
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C32×Q8—22no—
65C2×C32×C4C2312yes—
66C2×C32×C4C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2312yes—
69C4×C3⋊D3—22no—
70C4×C3⋊D3—24no—
71#634—24no—
72#634D414yescomplemented
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2311yescharacteristic
78C3×C4×D3—22no—
79C3×Dic6—22no—
80#627—24no—
81#626Q814yescomplemented
82D3×Dic3—24no—
83D4×Q8—91noSylow
84C2×C3×Q8—21no—
85C2×C3×Q8—31no—
86C2×C3×Q8C2×D311yescharacteristic
87C2×C3×Q8C2×D311yescharacteristic
88C22×C3×C4—31no—
89C2×#67—31no—
90C22×Dic3—32no—
91D3×Q8—64no—
92D3×Q8—64no—
93D3×Q8—64no—
94D3×Q8—68no—
95C2×D12—31no—
96C2×C4×D3—32no—
97C2×C4×D3—31no—
98C2×C4×D3—32no—
99C2×C4×D3—32no—
100C2×C4×D3—31no—
101C2×C4×D3—62no—
102C2×C4×D3—61no—
103C2×C4×D3C2×D311yescharacteristic
104C2×Dic6—31no—
105C2×Dic6—31no—
106C2×Dic6—32no—
107C2×Dic6—32no—
108C2×Dic6—32no—
109C2×Dic6—32no—
110C2×Dic6—61no—
111C2×Dic6—62no—
112C2×Dic6—64no—
113C2×Dic6C2×D311yescharacteristic
114C3×C4⋊C4—31no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—32no—
117C3×C4⋊C4—32no—
118C3×C22⋊C4—32no—
119C3×C42—31no—
120#217—31no—
121(C2×D3)⋊C4—31no—
122(C2×D3)⋊C4—31no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—31no—
126(C2×D3)⋊C4C2×D312yes—
127C4⋊Dic3—31no—
128C4⋊Dic3—32no—
129C4⋊Dic3C2×D311yescharacteristic
130Dic3⋊C4—32no—
131Dic3⋊C4—32no—
132Dic3⋊C4—31no—
133Dic3⋊C4—32no—
134Dic3⋊C4C2×D312yes—
135Dic3⋊C4C2×D312yes—
136Dic3⋊C4C2×D312yes—
137C4×Dic3—32no—
138C4×Dic3—32no—
139C4×Dic3C2×D311yescharacteristic
140C22×C32C2411yescharacteristic
141C2×C3⋊D3—21no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3C2×Q812yescomplemented
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3C2×Q812yescomplemented
147C32×C4—22no—
148C32×C4C2×D412yes—
149C3⋊Dic3—21no—
150C3⋊Dic3C2×D414yes—
151C3×Dic3—22no—
152C3×Dic3—21no—
153C3×Dic3—21no—
154C3×Dic3—22no—
155C3×Dic3C2×Q812yescomplemented
156C22×Q8—91no—
157C22×Q8—91no—
158C4⋊Q8—31no—
159C4⋊Q8—92no—
160C22⋊Q8—31no—
161C22⋊Q8—92no—
162C22⋊Q8—92no—
163C22⋊Q8—91no—
164C4×Q8—91no—
165C4×D4—91no—
166C4×D4—92no—
167C23×C3—31no—
168C22×D3—31no—
169C22×D3—31no—
170C22×D3—61no—
171C22×D3C22×D311yescharacteristic
172C3×Q8—24no—
173C3×Q8—22no—
174C3×Q8—22no—
175C3×Q8—62no—
176C2×C3×C4—22no—
177C2×C3×C4—21no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—32no—
183C2×C3×C4—62no—
184C2×C3×C4C22×D312yes—
185C2×C3×C4C22×D312yes—
186C2×C3×C4C22×D311yescharacteristic
187C2×C3×C4C22×D311yescharacteristic
188#67—34no—
189C2×Dic3—32no—
190C2×Dic3—32no—
191C2×Dic3—31no—
192C2×Dic3—31no—
193C2×Dic3—62no—
194C2×Dic3—61no—
195C2×Dic3—64no—
196C2×Dic3C22×D312yes—
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D311yescharacteristic
199C2×Dic3C22×D312yes—
200C2×Dic3C22×D311yescharacteristic
201D12—34no—
202C4×D3—22no—
203C4×D3—64no—
204C4×D3—64no—
205C4×D3—64no—
206C4×D3—68no—
207C4×D3—64no—
208C4×D3—62no—
209C4×D3—62no—
210Dic6—22no—
211Dic6—34no—
212Dic6—34no—
213Dic6—64no—
214Dic6—64no—
215Dic6—64no—
216Dic6—64no—
217Dic6—64no—
218Dic6—68no—
219C2×C32C22×D411yescharacteristic
220C2×C32C22×Q811yescharacteristic
221C2×C322−1+411yescharacteristic
222C3⋊D3—22no—
223C3×D3—22no—
224C2×Q8—31no—
225C2×Q8—92no—
226C2×Q8—91no—
227C2×Q8—92no—
228C2×Q8—184no—
229C2×Q8—184no—
230C2×Q8D3211yescharacteristic
231C2×D4—91no—
232C22×C4—31no—
233C22×C4—92no—
234C22×C4—92no—
235C22×C4—91no—
236C4⋊C4—32no—
237C4⋊C4—31no—
238C4⋊C4—32no—
239C4⋊C4—32no—
240C4⋊C4—91no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C22⋊C4—32no—
244C22⋊C4—91no—
245C22⋊C4—92no—
246C22⋊C4—91no—
247C42—31no—
248C42—92no—
249C22×C3—21no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3C23×D311yescharacteristic
254C22×C3C23×D311yescharacteristic
255C2×D3—21no—
256C2×D3—21no—
257C2×D3—32no—
258C2×D3—32no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—61no—
263C2×D3—61no—
264C2×D3—61no—
265C2×D3—61no—
266C2×D3D3×Q812yescomplemented
267C3×C4—24no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—32no—
272C3×C4—61no—
273C3×C4—62no—
274C3×C4—62no—
275C3×C4—61no—
276C3×C4D3×D412yes—
277C3×C4D3×D412yes—
278Dic3—22no—
279Dic3—21no—
280Dic3—21no—
281Dic3—22no—
282Dic3—34no—
283Dic3—34no—
284Dic3—64no—
285Dic3—62no—
286Dic3—61no—
287Dic3—61no—
288Dic3D3×Q812yescomplemented
289C32D4×Q811yescharacteristic, complemented, Sylow
290C23—31no—
291C23—91no—
292Q8—22no—
293Q8—62no—
294Q8—94no—
295Q8—184no—
296Q8—184no—
297D4—94no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—31no—
302C2×C4—32no—
303C2×C4—62no—
304C2×C4—92no—
305C2×C4—91no—
306C2×C4—184no—
307C2×C4—184no—
308C2×C4—182no—
309C2×C4C2×D3212yes—
310C2×C4C2×D3211yescharacteristic
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—62no—
315C2×C3C2×D3×D411yescharacteristic
316C2×C3C2×D3×D411yescharacteristic
317C2×C3C2×D3×Q811yescharacteristic
318C2×C3C2×D3×Q811yescharacteristic
319C2×C3#100711yescharacteristic
320C2×C3#101011yescharacteristic
321D3—22no—
322D3—64no—
323D3—62no—
324D3—62no—
325C22—32no—
326C22—61no—
327C22—61no—
328C22—92no—
329C22—181no—
330C22—181no—
331C22C22×D3211yescharacteristic
332C4—22no—
333C4—32no—
334C4—61no—
335C4—62no—
336C4—61no—
337C4—62no—
338C4—94no—
339C4—181no—
340C4#381112yes—
341C3—21no—
342C3#563211yescharacteristic, complemented
343C3#572711yescharacteristic, complemented
344C2—62no—
345C2—182no—
346C2#6431511yescharacteristic
347C2D32×Q811yescharacteristic
348C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC2×#626C22
hypercenterC2×Q8D32
hypocenterC32D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊D3C2semidirect
2. C2×D3×Dic6C2semidirect
3. C3×#946C2semidirect
4. C3×#944C2semidirect
6. #63896C2semidirect
8. #63871C2semidirect
9. #63865C2semidirect
10. #63844C2semidirect
11. #63840C2semidirect
47. #946D3semidirect
48. #944D3semidirect
72. #634D4semidirect
81. #626Q8semidirect
143. C2×C3⋊D3C2×Q8semidirect
146. C2×C3×D3C2×Q8semidirect
155. C3×Dic3C2×Q8semidirect
266. C2×D3D3×Q8semidirect
288. Dic3D3×Q8semidirect
289. C32D4×Q8semidirect
342. C3#5632semidirect
343. C3#5727semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D32
C2×D3×Dic62
C3×#9462
C3×#9442
C4×#6262
#638962
#638812
#638712
#638652
#638442
#638402
#57273
#56323