Finite Group Databasebeta

← Q8×#6294 #13178718

Subgroups — Q8×#6294 #13178718

Order 1728.

1526 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C3×Q8×C3⋊Dic3C212yesmaximal, complemented
3#11723689C213yesmaximal
4C4×#6294C213yesmaximal
5#11723668C216yesmaximal, complemented
6Q8×#624—31nomaximal
7D3×Q8×Dic3—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×Q8×C3⋊D3C418yescomplemented
10C2×C3×C4×C3⋊D3C2213yes—
11C2×C3×#634C2213yes—
12C2×#6294C2213yes—
13C2×#6294C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2216yes—
15C3×C4×C3⋊Dic3C2216yes—
16#88815C2216yes—
17#88813C2216yes—
18C2×Q8×C3⋊D3D311yescharacteristic
19C2×C3×D3×Q8—32no—
20C2×C3×D3×Q8—61no—
21Q8×C3⋊Dic3D312yescomplemented
22C3×Q8×Dic3—32no—
23C3×Q8×Dic3—32no—
24C3×Q8×Dic3—62no—
25#63900—33no—
26D3×C4⋊Dic3—36no—
27C4×#624—33no—
28C4×D3×Dic3—36no—
29#63844—36no—
30#63843—36no—
31Dic3×Dic6—36no—
32C22×C3×C3⋊D3C2311yescharacteristic
33C33×Q8C2×C414yescomplemented
34C2×C33×C4C2313yes—
35C2×C3×C3⋊Dic3C2316yes—
36C2×C3×C3⋊Dic3C2313yes—
37C2×C3×C3⋊Dic3C2312yes—
38C3×C4×C3⋊D3C2×C4112yes—
39C3×#634C2×C4112yescomplemented
40#6294—26no—
41#6294Q814yesdirect product factor
42C2×Q8×Dic3—91no—
43C4×D3×Q8—92no—
44C2×C32×Q8—22no—
45C2×C32×Q8—21no—
46C2×C32×Q8—41no—
47C2×C32×Q8C2×D312yes—
48C2×C32×Q8C2×D311yescharacteristic
49Q8×C3⋊D3Dic318yescomplemented
50C2×C4×C3⋊D3C2×D313yes—
51C2×#634C2×D313yes—
52C3×D3×Q8—316no—
53C3×D3×Q8—68no—
54C2×C3×C4×D3—36no—
55C2×C3×C4×D3—63no—
56C2×C3×Dic6—36no—
57C2×C3×Dic6—63no—
58C2×#624—31no—
59C2×#624—33no—
60C2×D3×Dic3—32no—
61C2×D3×Dic3—36no—
62C4⋊(C3⋊Dic3)C2×D316yes—
63C4×C3⋊Dic3C2×D316yes—
64C3×C4⋊Dic3—36no—
65C3×C4⋊Dic3—36no—
66C3×C4⋊Dic3—66no—
67C3×C4×Dic3—36no—
68C3×C4×Dic3—36no—
69C3×C4×Dic3—66no—
70#3724—36no—
71Dic3⋊Dic3—312no—
72Dic32—36no—
73Dic32—36no—
74Dic32—36no—
75C22×C33C2411yescharacteristic
76C2×C3×C3⋊D3C22×C412yes—
77C2×C3×C3⋊D3C2×Q812yes—
78C2×C3×C3⋊D3#4112yes—
79C33×C4C22×C416yes—
80C3×C3⋊Dic3—26no—
81C3×C3⋊Dic3C22×C416yes—
82C3×C3⋊Dic3C2×Q814yescomplemented
83C22×C3×Q8—91no—
84C2×D3×Q8—32no—
85C2×D3×Q8—61no—
86C3×C4×Q8—92no—
87Q8×Dic3—32no—
88Q8×Dic3—32no—
89Q8×Dic3—62no—
90Q8×Dic3—96no—
91C2×C4⋊Dic3—93no—
92C2×C4×Dic3—93no—
93D3×C4⋊C4—96no—
94#887—96no—
95C42×D3—96no—
96C4×Dic6—96no—
97C22×C3⋊D3C22×D311yescharacteristic
98C22×C3×D3—32no—
99C22×C3×D3—61no—
100C32×Q8—28no—
101C32×Q8—24no—
102C32×Q8—44no—
103C32×Q8C4×D318yescomplemented
104C32×Q8C2×Dic314yescomplemented
105C2×C32×C4—26no—
106C2×C32×C4—23no—
107C2×C32×C4—43no—
108C2×C32×C4C22×D316yes—
109C2×C32×C4C22×D313yes—
110C2×C3⋊Dic3C22×D316yes—
111C2×C3⋊Dic3C22×D312yes—
112C2×C3⋊Dic3C22×D313yes—
113C4×C3⋊D3C2×Dic3112yes—
114#634C2×Dic3112yescomplemented
115C2×C3×Dic3—32no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—36no—
118C2×C3×Dic3—36no—
119C2×C3×Dic3—36no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—63no—
122C2×C3×Dic3—66no—
123C3×C4×D3—324no—
124C3×C4×D3—612no—
125C3×Dic6—324no—
126C3×Dic6—612no—
127#624—34no—
128#624—66no—
129D3×Dic3—38no—
130D3×Dic3—612no—
131C2×C4×Q8—271noSylow
132C2×C33C23×C411yescharacteristic
133C2×C33C22×Q811yescharacteristic
134C2×C33C2×#4111yescharacteristic
135C3×C3⋊D3C4×Q814yescomplemented
136C2×C3×Q8—22no—
137C2×C3×Q8—21no—
138C2×C3×Q8—41no—
139C2×C3×Q8—98no—
140C2×C3×Q8—93no—
141C2×C3×Q8D3212yes—
142C2×C3×Q8D3211yescharacteristic
143C22×C3×C4—93no—
144C22×Dic3—91no—
145C22×Dic3—93no—
146D3×Q8—316no—
147D3×Q8—68no—
148C2×C4×D3—36no—
149C2×C4×D3—63no—
150C2×C4×D3—92no—
151C2×C4×D3—96no—
152C2×Dic6—36no—
153C2×Dic6—63no—
154C3×C4⋊C4—96no—
155C3×C42—96no—
156C4⋊Dic3—36no—
157C4⋊Dic3—36no—
158C4⋊Dic3—66no—
159C4⋊Dic3—96no—
160Dic3⋊C4—912no—
161C4×Dic3—36no—
162C4×Dic3—36no—
163C4×Dic3—66no—
164C4×Dic3—96no—
165C4×Dic3—96no—
166C4×Dic3—96no—
167C22×C32—22no—
168C22×C32—21no—
169C22×C32—41no—
170C22×C32C23×D312yes—
171C22×C32C23×D311yescharacteristic
172C2×C3⋊D3D3×Q812yes—
173C2×C3⋊D3#23912yes—
174C2×C3⋊D3C22×Dic312yes—
175C2×C3×D3—34no—
176C2×C3×D3—34no—
177C2×C3×D3—34no—
178C2×C3×D3—62no—
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C32×C4—212no—
182C32×C4—26no—
183C32×C4—46no—
184C32×C4C2×C4×D3112yes—
185C32×C4C22×Dic316yes—
186C3⋊Dic3—26no—
187C3⋊Dic3D3×Q814yescomplemented
188C3⋊Dic3C22×Dic316yes—
189C3×Dic3—312no—
190C3×Dic3—34no—
191C3×Dic3—34no—
192C3×Dic3—66no—
193C3×Dic3—64no—
194C3×Dic3—66no—
195C3×Dic3—66no—
196C3×Dic3—126no—
197C22×Q8—91no—
198C4×Q8—92no—
199C4×Q8—276no—
200C2×C4⋊C4—273no—
201C2×C42—273no—
202C33C2×C4×Q811yescharacteristic, complemented, Sylow
203C23×C3—91no—
204C22×D3—32no—
205C22×D3—61no—
206C3×Q8—28no—
207C3×Q8—24no—
208C3×Q8—44no—
209C3×Q8—912no—
210C3×Q8D3×Dic318yescomplemented
211C3×Q8#62414yescomplemented
212C2×C3×C4—26no—
213C2×C3×C4—23no—
214C2×C3×C4—43no—
215C2×C3×C4—92no—
216C2×C3×C4—93no—
217C2×C3×C4—912no—
218C2×C3×C4—96no—
219C2×C3×C4C2×D3216yes—
220C2×C3×C4C2×D3213yes—
221C2×Dic3—32no—
222C2×Dic3—32no—
223C2×Dic3—36no—
224C2×Dic3—36no—
225C2×Dic3—36no—
226C2×Dic3—62no—
227C2×Dic3—63no—
228C2×Dic3—66no—
229C2×Dic3—94no—
230C2×Dic3—186no—
231C4×D3—324no—
232C4×D3—612no—
233C4×D3—98no—
234C4×D3—1812no—
235Dic6—324no—
236Dic6—612no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—22no—
242C2×C32—21no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32C22×C4×D312yes—
247C2×C32C2×D3×Q811yescharacteristic
248C2×C32C2×D3×Q812yes—
249C2×C32C2×#23911yescharacteristic
250C2×C32D3×#4112yes—
251C2×C32C23×Dic311yescharacteristic
252C3⋊D3Q8×Dic314yescomplemented
253C3×D3—38no—
254C3×D3—64no—
255C2×Q8—98no—
256C2×Q8—93no—
257C2×Q8#111311yescharacteristic
258C22×C4—93no—
259C22×C4—271no—
260C22×C4—273no—
261C4⋊C4—96no—
262C4⋊C4—276no—
263C42—96no—
264C42—276no—
265C22×C3—22no—
266C22×C3—21no—
267C22×C3—41no—
268C22×C3—92no—
269C22×C3—92no—
270C22×C3—92no—
271C22×C3C22×D3212yes—
272C22×C3C22×D3211yescharacteristic
273C2×D3—34no—
274C2×D3—34no—
275C2×D3—34no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C3×C4—26no—
280C3×C4—212no—
281C3×C4—46no—
282C3×C4—94no—
283C3×C4—96no—
284C3×C4—186no—
285C3×C4C2×D3×Dic3112yes—
286C3×C4C2×#62416yes—
287Dic3—34no—
288Dic3—34no—
289Dic3—312no—
290Dic3—66no—
291Dic3—64no—
292Dic3—66no—
293Dic3—66no—
294Dic3—126no—
295C32—22no—
296C32—21no—
297C32—41no—
298C32C4×D3×Q812yescomplemented
299C32C2×Q8×Dic311yescharacteristic, complemented
300C23—91no—
301Q8—912no—
302Q8#629414yesdirect product factor
303C2×C4—92no—
304C2×C4—93no—
305C2×C4—912no—
306C2×C4—96no—
307C2×C4—274no—
308C2×C4—546no—
309C2×C4C2×#111313yes—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—22no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—94no—
320C2×C3D32×Q812yes—
321C2×C3D32×Q811yescharacteristic
322C2×C3D3×#23912yes—
323C2×C3#6431911yescharacteristic
324C2×C3C22×D3×Dic312yes—
325C2×C3C22×#62411yescharacteristic
326D3—38no—
327D3—64no—
328C22—92no—
329C22—92no—
330C22—92no—
331C22C22×#111311yescharacteristic
332C4—94no—
333C4—96no—
334C4—186no—
335C4C2×#629416yes—
336C3—21no—
337C3—22no—
338C3—41no—
339C3D3×Q8×Dic312yescomplemented
340C3Q8×#62411yescharacteristic, complemented
341C2—94no—
342C2Q8×#111311yescharacteristic
343C2#1172462611yescharacteristic
344C2C22×#629411yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutatorC2×C33C23×C4
FittingC2×C33×Q8C22
socle, cosocleC22×C33C24
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×Q8#1113
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×Q8×C3⋊Dic3C2semidirect
5. #11723668C2semidirect
9. C3×Q8×C3⋊D3C4semidirect
21. Q8×C3⋊Dic3D3semidirect
33. C33×Q8C2×C4semidirect
39. C3×#634C2×C4semidirect
41. #6294Q8direct (N × H)
49. Q8×C3⋊D3Dic3semidirect
82. C3×C3⋊Dic3C2×Q8semidirect
103. C32×Q8C4×D3semidirect
104. C32×Q8C2×Dic3semidirect
114. #634C2×Dic3semidirect
135. C3×C3⋊D3C4×Q8semidirect
187. C3⋊Dic3D3×Q8semidirect
202. C33C2×C4×Q8semidirect
210. C3×Q8D3×Dic3semidirect
211. C3×Q8#624semidirect
252. C3⋊D3Q8×Dic3semidirect
298. C32C4×D3×Q8semidirect
299. C32C2×Q8×Dic3semidirect
302. Q8#6294direct (N × H)
339. C3D3×Q8×Dic3semidirect
340. C3Q8×#624semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C3×Q8×C3⋊Dic32
#117236892
C4×#62942
#117236682
Q8×#6243
D3×Q8×Dic33