Finite Group Databasebeta

← Q8×#6295 #13178740

Subgroups — Q8×#6295 #13178740

Order 1728.

1540 conjugacy classes of subgroups; only the 263 normal subgroups are shown.

normal subgroups only

This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 263 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×Q8×C3⋊D3C212yesmaximal, complemented
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C4×#6295C213yesmaximal
4#11723678C213yesmaximal, complemented
5#11723674C216yesmaximal, complemented
6Q8×#626—32nomaximal
7Q8×#625—31nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×Q8×C3⋊D3—28no—
10C2×C3×C4×C3⋊D3C2216yes—
11C2×C3×#634C2216yes—
12C2×#6295C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2213yes—
14C3×C4×C3⋊Dic3C2213yes—
15#88815C2216yes—
16#88815C2213yes—
17#88814C2216yes—
18C2×Q8×C3⋊D3D312yescomplemented
19C2×C3×D3×Q8—32no—
20C2×C3×D3×Q8—32no—
21C2×C3×D3×Q8—62no—
22Q8×C3⋊Dic3D311yescharacteristic, complemented
23C3×Q8×Dic3—32no—
24C3×Q8×Dic3—61no—
25C4×#626—36no—
26C4×#625—33no—
27#63873—33no—
28#63871—36no—
29#63866—36no—
30#63857—36no—
31#63856—36no—
32C22×C3×C3⋊D3C2312yes—
33C33×Q8D414yescomplemented
34C2×C33×C4C2313yes—
35C2×C3×C3⋊Dic3C2313yes—
36C2×C3×C3⋊Dic3C2316yes—
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3—212no—
39C3×#634—212no—
40#6295Q814yesdirect product factor
41Q8×#67—92no—
42Q8×D12—91no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D311yescharacteristic
47C2×C32×Q8C2×D312yes—
48Q8×C3⋊D3—28no—
49C2×C4×C3⋊D3C2×D316yes—
50C2×#634C2×D316yes—
51C3×D3×Q8—616no—
52C3×D3×Q8—68no—
53C3×D3×Q8—68no—
54C2×C3×C4×D3—36no—
55C2×C3×C4×D3—36no—
56C2×C3×C4×D3—66no—
57C2×C3×Dic6—36no—
58C2×C3×Dic6—36no—
59C2×C3×Dic6—66no—
60C2×#626—32no—
61C2×#625—31no—
62C4⋊(C3⋊Dic3)C2×D313yes—
63C4×C3⋊Dic3C2×D313yes—
64C3×C4⋊Dic3—36no—
65C3×C4⋊Dic3—63no—
66C3×C4×Dic3—36no—
67C3×C4×Dic3—63no—
68#3724—36no—
69#3724—33no—
70Dic3⋊Dic3—312no—
71Dic3⋊Dic3—36no—
72#3722—36no—
73(C2×D3)⋊Dic3—36no—
74(C2×D3)⋊Dic3—36no—
75C22×C33C2411yescharacteristic
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—22no—
78C2×C3×C3⋊D3C2×Q814yescomplemented
79C33×C4C2×D416yes—
80C3×C3⋊Dic3—26no—
81C3×C3⋊Dic3—23no—
82C3×C3⋊Dic3C2×Q812yescomplemented
83C22×C3×Q8—92no—
84C2×D3×Q8—32no—
85C2×D3×Q8—32no—
86C2×D3×Q8—62no—
87C3×C4×Q8—91no—
88#946—96no—
89Q8×Dic3—32no—
90Q8×Dic3—61no—
91#944—96no—
92C4×#67—96no—
93#924—96no—
94#897—96no—
95C4×D12—93no—
96#869—93no—
97C22×C3⋊D3C22×D312yes—
98C22×C3×D3—32no—
99C22×C3×D3—32no—
100C22×C3×D3—62no—
101C32×Q8—24no—
102C32×Q8—28no—
103C32×Q8—44no—
104C32×Q8D1214yescomplemented
105C32×Q8#6718yescomplemented
106C2×C32×C4—23no—
107C2×C32×C4—26no—
108C2×C32×C4—43no—
109C2×C32×C4C22×D313yes—
110C2×C32×C4C22×D316yes—
111C2×C3⋊Dic3C22×D313yes—
112C2×C3⋊Dic3C22×D316yes—
113C2×C3⋊Dic3C22×D311yescharacteristic
114C4×C3⋊D3—212no—
115#634—212no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—36no—
118C2×C3×Dic3—36no—
119C2×C3×Dic3—36no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—66no—
122C2×C3×Dic3—63no—
123C3×C4×D3—624no—
124C3×C4×D3—612no—
125C3×C4×D3—612no—
126C3×Dic6—624no—
127C3×Dic6—612no—
128C3×Dic6—612no—
129#626—38no—
130#625—34no—
131D4×Q8—271noSylow
132C2×C33C22×D411yescharacteristic
133C2×C33C22×Q811yescharacteristic
134C2×C332−1+411yescharacteristic
135C3×C3⋊D3—24no—
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—41no—
139C2×C3×Q8—96no—
140C2×C3×Q8—188no—
141C2×C3×Q8D3211yescharacteristic
142C2×C3×Q8D3212yes—
143C22×C3×C4—96no—
144C2×#67—92no—
145D3×Q8—616no—
146D3×Q8—68no—
147D3×Q8—68no—
148C2×D12—91no—
149C2×C4×D3—36no—
150C2×C4×D3—36no—
151C2×C4×D3—66no—
152C2×Dic6—36no—
153C2×Dic6—36no—
154C2×Dic6—66no—
155C3×C4⋊C4—93no—
156C3×C42—93no—
157#217—96no—
158(C2×D3)⋊C4—96no—
159(C2×D3)⋊C4—96no—
160C4⋊Dic3—36no—
161C4⋊Dic3—63no—
162C4⋊Dic3—93no—
163C4⋊Dic3—96no—
164Dic3⋊C4—912no—
165Dic3⋊C4—96no—
166C4×Dic3—36no—
167C4×Dic3—63no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—41no—
171C22×C32C23×D311yescharacteristic
172C22×C32C23×D312yes—
173C2×C3⋊D3—22no—
174C2×C3⋊D3—22no—
175C2×C3⋊D3D3×Q814yescomplemented
176C2×C3×D3—34no—
177C2×C3×D3—34no—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—64no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C32×C4—26no—
186C32×C4—212no—
187C32×C4—46no—
188C32×C4C2×D1216yes—
189C32×C4C2×#67112yes—
190C3⋊Dic3—26no—
191C3⋊Dic3—23no—
192C3⋊Dic3D3×Q812yescomplemented
193C3×Dic3—34no—
194C3×Dic3—62no—
195C3×Dic3—612no—
196C3×Dic3—66no—
197C3×Dic3—66no—
198C3×Dic3—66no—
199C3×Dic3—123no—
200C22×Q8—92no—
201C4⋊Q8—273no—
202C22⋊Q8—276no—
203C4×Q8—91no—
204C4×D4—273no—
205C33D4×Q811yescharacteristic, complemented, Sylow
206C23×C3—92no—
207C22×D3—32no—
208C22×D3—32no—
209C22×D3—62no—
210C3×Q8—24no—
211C3×Q8—28no—
212C3×Q8—44no—
213C3×Q8—1812no—
214C3×Q8#62514yescomplemented
215C3×Q8#62618yescomplemented
216C2×C3×C4—23no—
217C2×C3×C4—26no—
218C2×C3×C4—43no—
219C2×C3×C4—91no—
220C2×C3×C4—96no—
221C2×C3×C4—93no—
222C2×C3×C4—1812no—
223C2×C3×C4C2×D3213yes—
224C2×C3×C4C2×D3216yes—
225#67—98no—
226C2×Dic3—32no—
227C2×Dic3—36no—
228C2×Dic3—36no—
229C2×Dic3—36no—
230C2×Dic3—61no—
231C2×Dic3—66no—
232C2×Dic3—63no—
233D12—94no—
234C4×D3—624no—
235C4×D3—612no—
236C4×D3—612no—
237Dic6—624no—
238Dic6—612no—
239Dic6—612no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—22no—
244C2×C32—21no—
245C2×C32—22no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32C22×D1211yescharacteristic
250C2×C32C2×D3×Q812yes—
251C2×C32C2×D3×Q811yescharacteristic
252C2×C32#100712yes—
253C2×C32#101011yescharacteristic
254C2×C32C22×#6712yes—
255C3⋊D3—24no—
256C3×D3—68no—
257C3×D3—64no—
258C3×D3—64no—
259C2×Q8—96no—
260C2×Q8—188no—
261C2×Q8#111311yescharacteristic
262C2×D4—271no—
263C22×C4—96no—
264C4⋊C4—93no—
265C4⋊C4—273no—
266C4⋊C4—276no—
267C22⋊C4—276no—
268C42—93no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—41no—
272C22×C3—94no—
273C22×C3—182no—
274C22×C3—182no—
275C22×C3C22×D3212yes—
276C22×C3C22×D3211yescharacteristic
277C2×D3—34no—
278C2×D3—34no—
279C2×D3—64no—
280C2×D3—64no—
281C2×D3—64no—
282C2×D3—62no—
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C3×C4—212no—
287C3×C4—26no—
288C3×C4—46no—
289C3×C4—92no—
290C3×C4—183no—
291C3×C4—186no—
292C3×C4C2×#62516yes—
293C3×C4C2×#626112yes—
294Dic3—34no—
295Dic3—62no—
296Dic3—612no—
297Dic3—66no—
298Dic3—66no—
299Dic3—66no—
300Dic3—123no—
301C32—21no—
302C32—22no—
303C32—41no—
304C32Q8×D1211yescharacteristic, complemented
305C32Q8×#6712yescomplemented
306C23—92no—
307Q8—1812no—
308Q8#629514yesdirect product factor
309D4—274no—
310C2×C4—91no—
311C2×C4—93no—
312C2×C4—96no—
313C2×C4—1812no—
314C2×C4C2×#111313yes—
315C2×C3—21no—
316C2×C3—22no—
317C2×C3—21no—
318C2×C3—22no—
319C2×C3—22no—
320C2×C3—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—184no—
325C2×C3#6431512yes—
326C2×C3#6431611yescharacteristic
327C2×C3D32×Q811yescharacteristic
328C2×C3D32×Q812yes—
329C2×C3C22×#62511yescharacteristic
330C2×C3C22×#62612yes—
331D3—68no—
332D3—64no—
333D3—64no—
334C22—94no—
335C22—182no—
336C22—182no—
337C22C22×#111311yescharacteristic
338C4—92no—
339C4—183no—
340C4—186no—
341C4C2×#629516yes—
342C3—22no—
343C3—21no—
344C3—41no—
345C3Q8×#62511yescharacteristic, complemented
346C3Q8×#62612yescomplemented
347C2—184no—
348C2#1172462411yescharacteristic
349C2Q8×#111311yescharacteristic
350C2C22×#629511yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×#6295C22
hypercenterC2×Q8#1113
hypocenterC33D4×Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
2. C3×Q8×C3⋊Dic3C2semidirect
4. #11723678C2semidirect
5. #11723674C2semidirect
18. C2×Q8×C3⋊D3D3semidirect
22. Q8×C3⋊Dic3D3semidirect
33. C33×Q8D4semidirect
40. #6295Q8direct (N × H)
78. C2×C3×C3⋊D3C2×Q8semidirect
82. C3×C3⋊Dic3C2×Q8semidirect
104. C32×Q8D12semidirect
105. C32×Q8#67semidirect
175. C2×C3⋊D3D3×Q8semidirect
192. C3⋊Dic3D3×Q8semidirect
205. C33D4×Q8semidirect
214. C3×Q8#625semidirect
215. C3×Q8#626semidirect
304. C32Q8×D12semidirect
305. C32Q8×#67semidirect
308. Q8#6295direct (N × H)
345. C3Q8×#625semidirect
346. C3Q8×#626semidirect

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
C4×#62952
#117236782
#117236742
Q8×#6263
Q8×#6253