Finite Group Databasebeta

← Q8×Dic3⋊Dic3 #11805570

Subgroups — Q8×Dic3⋊Dic3 #11805570

Order 1152.

1285 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×Q8×Dic3C211yesmaximal, characteristic
3C2×C3×Q8×Dic3C211yesmaximal, characteristic
4#10591357C213yesmaximal
5#10591352C213yesmaximal
6#10591349C213yesmaximal
7C4×Dic3⋊Dic3C213yesmaximal
8Q8×Dic3⋊C4—31nomaximal
9Q8×C4⋊Dic3—31nomaximal
10C22×C32×Q8C2211yescharacteristic
11Q8×C3⋊Dic3—24no—
12C2×C4⋊(C3⋊Dic3)C2213yes—
13C2×C4×C3⋊Dic3C2213yes—
14C3×Q8×Dic3—24no—
15C3×Q8×Dic3C418yescomplemented
16C2×C3×C4⋊Dic3C2213yes—
17C2×C3×C4⋊Dic3C2213yes—
18C2×C3×C4×Dic3C2213yes—
19C2×C3×C4×Dic3C2213yes—
20C2×Dic3⋊Dic3C2213yes—
21C2×Dic3⋊Dic3C2216yes—
22C2×Dic3⋊Dic3C2211yescharacteristic
23#63579C2213yes—
24#63579C2213yes—
25C2×C3×C4×Q8—31no—
26C2×C3×C4×Q8—31no—
27C2×Q8×Dic3—31no—
28C2×Q8×Dic3—31no—
29C2×Q8×Dic3—61no—
30C2×Q8×Dic3D311yescharacteristic
31C2×Q8×Dic3D311yescharacteristic
32#4984—33no—
33#4984—33no—
34#4975—33no—
35#4972—33no—
36#4935—33no—
37C4×C4⋊Dic3—33no—
38C4×Dic3⋊C4—33no—
39#4928—33no—
40C2×C32×Q8C2×C414yes—
41C2×C32×Q8D414yes—
42C2×C32×Q8Q814yes—
43C22×C32×C4C2313yes—
44C22×C3⋊Dic3C2313yes—
45C22×C3⋊Dic3C2311yescharacteristic
46C22×C3×Dic3C2313yes—
47C22×C3×Dic3C2313yes—
48C22×C3×Dic3C2311yescharacteristic
49C22×C3×Dic3C2311yescharacteristic
50C4⋊(C3⋊Dic3)—26no—
51C4×C3⋊Dic3—26no—
52C3×C4⋊Dic3—26no—
53C3×C4⋊Dic3C2×C4112yes—
54C3×C4×Dic3—26no—
55C3×C4×Dic3C2×C4112yes—
56Dic3⋊Dic3—26no—
57Dic3⋊Dic3—46no—
58Dic3⋊Dic3Q814yesdirect product factor
59Q8×C4⋊C4—91noSylow
60C22×C3×Q8—21no—
61C22×C3×Q8C2×D311yescharacteristic
62C22×C3×Q8C2×D311yescharacteristic
63C3×C4×Q8—38no—
64C3×C4×Q8—64no—
65C2×C3×C4⋊C4—33no—
66C2×C3×C4⋊C4—33no—
67C2×C3×C42—33no—
68C2×C3×C42—33no—
69Q8×Dic3—24no—
70Q8×Dic3—68no—
71Q8×Dic3—64no—
72Q8×Dic3—64no—
73Q8×Dic3Dic318yescomplemented
74C2×C4⋊Dic3—31no—
75C2×C4⋊Dic3—33no—
76C2×C4⋊Dic3—33no—
77C2×C4⋊Dic3—33no—
78C2×C4⋊Dic3—36no—
79C2×C4⋊Dic3—63no—
80C2×C4⋊Dic3C2×D313yes—
81C2×C4⋊Dic3C2×D313yes—
82C2×Dic3⋊C4—31no—
83C2×Dic3⋊C4—33no—
84C2×Dic3⋊C4—36no—
85C2×C4×Dic3—33no—
86C2×C4×Dic3—33no—
87C2×C4×Dic3—63no—
88C2×C4×Dic3C2×D313yes—
89C2×C4×Dic3C2×D313yes—
90C3⋊#95—33no—
91C3⋊#95—33no—
92C3⋊#95—33no—
93C3⋊#95—33no—
94C23×C32C2411yescharacteristic
95C32×Q8C4⋊C4116yescomplemented
96C2×C32×C4C22×C416yes—
97C2×C32×C4C2×D416yes—
98C2×C32×C4C2×Q816yes—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—23no—
102C2×C3⋊Dic3—23no—
103C2×C3⋊Dic3—23no—
104C2×C3⋊Dic3C2×Q812yes—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—23no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—23no—
109C2×C3×Dic3—23no—
110C2×C3×Dic3—23no—
111C2×C3×Dic3—23no—
112C2×C3×Dic3C22×C416yes—
113C2×C3×Dic3C22×C412yes—
114C2×C3×Dic3C2×Q812yes—
115C2×C3×Dic3C2×Q812yes—
116C2×C3×Dic3#4112yes—
117C2×C4×Q8—31no—
118C2×C4×Q8—31no—
119C2×C4×Q8—91no—
120#389—93no—
121#389—93no—
122#384—93no—
123C4×C4⋊C4—93no—
124C2×C3×Q8—24no—
125C2×C3×Q8—24no—
126C2×C3×Q8—24no—
127C2×C3×Q8Dic614yes—
128C2×C3×Q8Dic614yes—
129C2×C3×Q8C4×D314yes—
130C2×C3×Q8D1214yes—
131C2×C3×Q8C2×Dic314yes—
132C2×C3×Q8#6714yes—
133C22×C3×C4—23no—
134C22×C3×C4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—33no—
137C22×C3×C4—33no—
138C22×C3×C4C22×D313yes—
139C22×C3×C4C22×D313yes—
140C22×Dic3—31no—
141C22×Dic3—31no—
142C22×Dic3—33no—
143C22×Dic3—33no—
144C22×Dic3—61no—
145C22×Dic3—63no—
146C22×Dic3C22×D313yes—
147C22×Dic3C22×D311yescharacteristic
148C22×Dic3C22×D313yes—
149C22×Dic3C22×D311yescharacteristic
150C3×C4⋊C4—312no—
151C3×C4⋊C4—66no—
152C3×C42—312no—
153C3×C42—66no—
154C4⋊Dic3—26no—
155C4⋊Dic3—34no—
156C4⋊Dic3—66no—
157C4⋊Dic3—612no—
158C4⋊Dic3—66no—
159C4⋊Dic3—66no—
160C4⋊Dic3—126no—
161C4⋊Dic3C2×Dic3112yes—
162Dic3⋊C4—34no—
163Dic3⋊C4—66no—
164Dic3⋊C4—126no—
165C4×Dic3—26no—
166C4×Dic3—612no—
167C4×Dic3—66no—
168C4×Dic3—66no—
169C4×Dic3C2×Dic3112yes—
170C22×C32C23×C411yescharacteristic
171C22×C32C22×D411yescharacteristic
172C22×C32C22×Q811yescharacteristic
173C22×C32C22×Q811yescharacteristic
174C22×C32C2×#4111yescharacteristic
175C22×C32H(5,𝔽2)11yescharacteristic
176C22×C322−1+411yescharacteristic
177C32×C4C2×C4⋊C4112yes—
178C3⋊Dic3—22no—
179C3⋊Dic3—43no—
180C3×Dic3—26no—
181C3×Dic3—22no—
182C3×Dic3—43no—
183C3×Dic3C4×Q814yescomplemented
184C22×Q8D3211yescharacteristic
185C4×Q8—38no—
186C4×Q8—64no—
187C4×Q8—184no—
188C2×C4⋊C4—33no—
189C2×C4⋊C4—33no—
190C2×C4⋊C4—91no—
191C2×C4⋊C4—93no—
192C2×C4⋊C4—93no—
193C2×C4⋊C4—96no—
194C2×C42—33no—
195C2×C42—33no—
196C2×C42—93no—
197#95—93no—
198#95—93no—
199C23×C3—21no—
200C23×C3C23×D311yescharacteristic
201C23×C3C23×D311yescharacteristic
202C3×Q8—216no—
203C3×Q8Dic3⋊C4116yescomplemented
204C3×Q8C4⋊Dic3116yescomplemented
205C2×C3×C4—26no—
206C2×C3×C4—26no—
207C2×C3×C4—26no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—32no—
211C2×C3×C4—36no—
212C2×C3×C4—32no—
213C2×C3×C4—61no—
214C2×C3×C4—63no—
215C2×C3×C4—63no—
216C2×C3×C4—61no—
217C2×C3×C4—63no—
218C2×C3×C4—63no—
219C2×C3×C4—63no—
220C2×C3×C4C2×Dic616yes—
221C2×C3×C4C2×Dic616yes—
222C2×C3×C4C2×C4×D316yes—
223C2×C3×C4C2×D1216yes—
224C2×C3×C4C22×Dic316yes—
225C2×C3×C4C2×#6716yes—
226C2×Dic3—21no—
227C2×Dic3—23no—
228C2×Dic3—23no—
229C2×Dic3—21no—
230C2×Dic3—23no—
231C2×Dic3—23no—
232C2×Dic3—23no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—66no—
243C2×Dic3—63no—
244C2×Dic3—63no—
245C2×Dic3—63no—
246C2×Dic3—63no—
247C2×Dic3—63no—
248C2×Dic3—63no—
249C2×Dic3—123no—
250C2×Dic3—123no—
251C2×Dic3D3×Q812yes—
252C2×Dic3D3×Q812yes—
253C2×Dic3#23912yes—
254C2×Dic3C22×Dic316yes—
255C2×Dic3C22×Dic312yes—
256C2×C32C22×C4⋊C411yescharacteristic
257C2×C32C2×C4×Q811yescharacteristic
258C2×C32#52011yescharacteristic
259C2×C32D4×Q811yescharacteristic
260C2×C32#55011yescharacteristic
261C2×C32#55711yescharacteristic
262C2×C32Q8211yescharacteristic
263C2×Q8D3×Dic314yes—
264C2×Q8#62614yes—
265C2×Q8#62714yes—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—33no—
269C22×C4—33no—
270C22×C4—91no—
271C22×C4—93no—
272C22×C4C2×D3213yes—
273C4⋊C4—312no—
274C4⋊C4—66no—
275C4⋊C4—94no—
276C4⋊C4—186no—
277C4⋊C4—186no—
278C4⋊C4—366no—
279C42—312no—
280C42—66no—
281C42—186no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3C22×Dic611yescharacteristic
290C22×C3C22×Dic611yescharacteristic
291C22×C3C22×C4×D311yescharacteristic
292C22×C3C22×D1211yescharacteristic
293C22×C3C2×D3×Q811yescharacteristic
294C22×C3C2×D3×Q811yescharacteristic
295C22×C3C2×#23911yescharacteristic
296C22×C3#100711yescharacteristic
297C22×C3D3×#4111yescharacteristic
298C22×C3#100911yescharacteristic
299C22×C3#100911yescharacteristic
300C22×C3#101011yescharacteristic
301C22×C3C23×Dic311yescharacteristic
302C22×C3C22×#6711yescharacteristic
303C3×C4—212no—
304C3×C4—34no—
305C3×C4—66no—
306C3×C4—62no—
307C3×C4—123no—
308C3×C4C2×Dic3⋊C4112yes—
309C3×C4C2×C4⋊Dic3112yes—
310Dic3—22no—
311Dic3—26no—
312Dic3—43no—
313Dic3—64no—
314Dic3—62no—
315Dic3—62no—
316Dic3—126no—
317Dic3—123no—
318Dic3—123no—
319Dic3Q8×Dic314yescomplemented
320C32Q8×C4⋊C411yescharacteristic, complemented, Sylow
321C23C22×D3211yescharacteristic
322Q8Dic3⋊Dic3116yesdirect product factor
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—36no—
328C2×C4—63no—
329C2×C4—61no—
330C2×C4—63no—
331C2×C4—61no—
332C2×C4—63no—
333C2×C4—63no—
334C2×C4—63no—
335C2×C4—92no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—183no—
339C2×C4—183no—
340C2×C4—183no—
341C2×C4C2×D3×Dic316yes—
342C2×C4C2×#62616yes—
343C2×C4C2×#62716yes—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3Q8×Dic611yescharacteristic
352C2×C3Q8×Dic611yescharacteristic
353C2×C3#556911yescharacteristic
354C2×C3#557011yescharacteristic
355C2×C3C4×D3×Q811yescharacteristic
356C2×C3#557411yescharacteristic
357C2×C3Q8×D1211yescharacteristic
358C2×C3#557711yescharacteristic
359C2×C3C22×Dic3⋊C411yescharacteristic
360C2×C3C22×C4⋊Dic311yescharacteristic
361C2×C3C2×Q8×Dic311yescharacteristic
362C2×C3Q8×#6711yescharacteristic
363C2×C3#582711yescharacteristic
364C2×C3#583411yescharacteristic
365C22#6431511yescharacteristic
366C22D32×Q811yescharacteristic
367C22D3×#23911yescharacteristic
368C22#6432011yescharacteristic
369C22C22×D3×Dic311yescharacteristic
370C22C22×#62611yescharacteristic
371C22C22×#62711yescharacteristic
372C4—34no—
373C4—66no—
374C4—62no—
375C4—123no—
376C4—182no—
377C4—363no—
378C4C2×Dic3⋊Dic3112yes—
379C3—21no—
380C3Q8×C4⋊Dic311yescharacteristic, complemented
381C3Q8×Dic3⋊C411yescharacteristic, complemented
382C2D3×Q8×Dic311yescharacteristic
383C2#1059539411yescharacteristic
384C2Q8×#62611yescharacteristic
385C2#1059541011yescharacteristic
386C2Q8×#62711yescharacteristic
387C2#1059542211yescharacteristic
388C2C22×Dic3⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC22×C32×Q8C22
socle, prime radical, cosocleC23×C32C24
hypercenterC22×Q8D32
hypocenterC32Q8×C4⋊C4
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
15. C3×Q8×Dic3C4semidirect
58. Dic3⋊Dic3Q8direct (N × H)
73. Q8×Dic3Dic3semidirect
95. C32×Q8C4⋊C4semidirect
183. C3×Dic3C4×Q8semidirect
203. C3×Q8Dic3⋊C4semidirect
204. C3×Q8C4⋊Dic3semidirect
319. Dic3Q8×Dic3semidirect
320. C32Q8×C4⋊C4semidirect
322. Q8Dic3⋊Dic3direct (N × H)
380. C3Q8×C4⋊Dic3semidirect
381. C3Q8×Dic3⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊Dic32
C2×C3×Q8×Dic32
#105913572
#105913522
#105913492
C4×Dic3⋊Dic32
Q8×Dic3⋊C43
Q8×C4⋊Dic33