Finite Group Databasebeta

← Q8×#638⋊C3 #13153661

Subgroups — Q8×#638⋊C3 #13153661

Order 1728.

1136 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#64155⋊C3C213yesmaximal, complemented
4Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#64153⋊C3C213yesmaximal, complemented
6C4×#638⋊C3C213yesmaximal
7#64133⋊C3C213yesmaximal, complemented
8Q8×#638C311yesmaximal, characteristic, complemented
9C3×Q8×#67—31nomaximal
10C2×Q8×H(3,𝔽3)—24no—
11C2×Q8×H(3,𝔽3)C2213yes—
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2213yes—
14C2×#638⋊C3C2211yescharacteristic
15Q8×(C3⋊D3)⋊C3—24no—
16C2×C4×(C3⋊D3)⋊C3C2213yes—
17C2×#634⋊C3C2213yes—
18#3757⋊C3C2213yes—
19#3752⋊C3C2213yes—
20C4⋊((C3⋊Dic3)⋊C3)C2213yes—
21#3750⋊C3C2216yes—
22#3750⋊C3C2213yes—
23C4×(C3⋊Dic3)⋊C3C2213yes—
24C22×C32×Q8—21no—
25C22×C32×Q8D311yescharacteristic, complemented
26C22×C32×Q8C2×C311yescharacteristic, complemented
27C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
28C2×C3×D3×Q8—31no—
29#64155C2×C313yescomplemented
30Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
31#64153C2×C313yescomplemented
32C4×#638C2×C313yes—
33#64133C2×C313yescomplemented
34C3×#946—33no—
35C3×Q8×Dic3—31no—
36C3×#944—33no—
37C3×C4×#67—33no—
38C3×#924—33no—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×(C3⋊D3)⋊C3C2311yescharacteristic
41Q8×H(3,𝔽3)—26no—
42Q8×H(3,𝔽3)D414yescomplemented
43C2×C4×H(3,𝔽3)—26no—
44C2×C4×H(3,𝔽3)C2313yes—
45C2×C4×H(3,𝔽3)C2313yes—
46#638⋊C3Q814yesdirect product factor
47C2×(C3⋊Dic3)⋊C3C2313yes—
48C2×(C3⋊Dic3)⋊C3C2313yes—
49C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
50C4×(C3⋊D3)⋊C3—26no—
51#634⋊C3—26no—
52C3×D4×Q8—91no—
53Q8×#67—31no—
54Q8×#67—91no—
55C2×C32×Q8—28no—
56C2×C32×Q8—23no—
57C2×C32×Q8—21no—
58C2×C32×Q8—24no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D313yes—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C22×C313yes—
63C2×C32×Q8C22×C311yescharacteristic
64C22×C32×C4—23no—
65C22×C32×C4C2×D313yes—
66C22×C32×C4C22×C313yes—
67C2×#638C22×C311yescharacteristic
68Q8×C3⋊D3—24no—
69C2×C4×C3⋊D3C22×C313yes—
70C2×#634C22×C313yes—
71C2×C3×#67—31no—
72C3×D3×Q8—64no—
73C2×C3×C4×D3—33no—
74C2×C3×Dic6—33no—
75#3757C22×C313yes—
76#3752C22×C313yes—
77C4⋊(C3⋊Dic3)C22×C313yes—
78#3750C22×C316yes—
79#3750C22×C313yes—
80C4×C3⋊Dic3C22×C313yes—
81C3×#217—33no—
82C3×(C2×D3)⋊C4—33no—
83C3×C4⋊Dic3—33no—
84C3×Dic3⋊C4—36no—
85C3×Dic3⋊C4—33no—
86C3×C4×Dic3—33no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)C2×Q812yescomplemented
90C22×H(3,𝔽3)C2411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3C2×Q812yescomplemented
94C4×H(3,𝔽3)—23no—
95C4×H(3,𝔽3)C2×D416yes—
96(C3⋊Dic3)⋊C3—23no—
97(C3⋊Dic3)⋊C3—23no—
98(C3⋊Dic3)⋊C3C2×Q812yescomplemented
99C22×C3×Q8—31no—
100C22×C3×Q8—31no—
101C22×C3×Q8—61no—
102C22×C3×Q8—91no—
103C22×C3×Q8C3×D311yescharacteristic
104C2×D3×Q8—31no—
105C2×D3×Q8—91no—
106C3×C4⋊Q8—93no—
107C3×C22⋊Q8—93no—
108C3×C22⋊Q8—93no—
109C3×C4×Q8—91no—
110C3×C4×D4—93no—
111#946—33no—
112#946—93no—
113Q8×Dic3—31no—
114Q8×Dic3—91no—
115#944—33no—
116#944—93no—
117C4×#67—33no—
118C4×#67—93no—
119#924—33no—
120#924—93no—
121C23×C32—21no—
122C23×C32C22×D311yescharacteristic
123C23×C32C23×C311yescharacteristic
124C22×C3⋊D3C23×C311yescharacteristic
125C22×C3×D3—31no—
126C32×Q8—212no—
127C32×Q8—26no—
128C32×Q8—26no—
129C32×Q8—24no—
130C32×Q8#6714yescomplemented
131C32×Q8C3×D414yescomplemented
132C2×C32×C4—212no—
133C2×C32×C4—26no—
134C2×C32×C4—26no—
135C2×C32×C4—23no—
136C2×C32×C4—23no—
137C2×C32×C4C22×D313yes—
138C2×C32×C4C22×D313yes—
139C2×C32×C4C23×C313yes—
140C2×C32×C4C23×C313yes—
141#638C3×Q814yescomplemented
142C2×C3⋊Dic3C23×C313yes—
143C2×C3⋊Dic3C23×C313yes—
144C2×C3⋊Dic3C23×C311yescharacteristic
145C4×C3⋊D3—26no—
146#634—26no—
147C3×#67—34no—
148C2×C3×Dic3—31no—
149C2×C3×Dic3—33no—
150C2×C3×Dic3—33no—
151C3×C4×D3—66no—
152C3×Dic6—66no—
153D4×Q8—91noSylow
154C2×H(3,𝔽3)—22no—
155C2×H(3,𝔽3)C22×D411yescharacteristic
156C2×H(3,𝔽3)C22×Q811yescharacteristic
157C2×H(3,𝔽3)2−1+411yescharacteristic
158(C3⋊D3)⋊C3—22no—
159C2×C3×Q8—24no—
160C2×C3×Q8—31no—
161C2×C3×Q8—31no—
162C2×C3×Q8—33no—
163C2×C3×Q8—33no—
164C2×C3×Q8—64no—
165C2×C3×Q8—64no—
166C2×C3×Q8—68no—
167C2×C3×Q8—63no—
168C2×C3×Q8—61no—
169C2×C3×Q8—93no—
170C2×C3×Q8—184no—
171C2×C3×Q8C2×C3×D313yes—
172C2×C3×Q8C2×C3×D311yescharacteristic
173C2×C3×D4—91no—
174C22×C3×C4—33no—
175C22×C3×C4—33no—
176C22×C3×C4—63no—
177C22×C3×C4—93no—
178C22×C3×C4C2×C3×D313yes—
179C2×#67—31no—
180C2×#67—91no—
181D3×Q8—64no—
182D3×Q8—184no—
183C2×C4×D3—33no—
184C2×C4×D3—93no—
185C2×Dic6—33no—
186C2×Dic6—93no—
187C3×C4⋊C4—93no—
188C3×C4⋊C4—96no—
189C3×C4⋊C4—93no—
190C3×C22⋊C4—93no—
191C3×C22⋊C4—93no—
192C3×C42—93no—
193#217—33no—
194#217—93no—
195(C2×D3)⋊C4—33no—
196(C2×D3)⋊C4—93no—
197C4⋊Dic3—33no—
198C4⋊Dic3—93no—
199Dic3⋊C4—36no—
200Dic3⋊C4—33no—
201Dic3⋊C4—96no—
202Dic3⋊C4—93no—
203C4×Dic3—33no—
204C4×Dic3—93no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—22no—
213C22×C32D3×Q812yescomplemented
214C22×C32C2×C3×Q812yescomplemented
215C22×C32C23×D311yescharacteristic
216C22×C32C24×C311yescharacteristic
217C2×C3⋊D3—21no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3C2×C3×Q812yescomplemented
220C2×C3×D3—32no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C32×C4—26no—
224C32×C4—23no—
225C32×C4—23no—
226C32×C4—26no—
227C32×C4C2×#6716yes—
228C32×C4C2×C3×D416yes—
229C3⋊Dic3—23no—
230C3⋊Dic3—23no—
231C3⋊Dic3C2×C3×Q812yescomplemented
232C3×Dic3—32no—
233C3×Dic3—63no—
234C3×Dic3—63no—
235C22×Q8—91no—
236C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
237C4⋊Q8—93no—
238C22⋊Q8—93no—
239C22⋊Q8—93no—
240C4×Q8—91no—
241C4×D4—93no—
242H(3,𝔽3)D4×Q811yescharacteristic, complemented, Sylow
243C23×C3—31no—
244C23×C3—31no—
245C23×C3—61no—
246C23×C3—91no—
247C23×C3C22×C3×D311yescharacteristic
248C22×D3—31no—
249C22×D3—91no—
250C3×Q8—26no—
251C3×Q8—34no—
252C3×Q8—34no—
253C3×Q8—66no—
254C3×Q8—66no—
255C3×Q8—612no—
256C3×Q8—64no—
257C3×Q8—186no—
258C3×Q8C3×#6714yes—
259C3×D4—94no—
260C2×C3×C4—26no—
261C2×C3×C4—33no—
262C2×C3×C4—33no—
263C2×C3×C4—33no—
264C2×C3×C4—33no—
265C2×C3×C4—66no—
266C2×C3×C4—66no—
267C2×C3×C4—612no—
268C2×C3×C4—63no—
269C2×C3×C4—63no—
270C2×C3×C4—91no—
271C2×C3×C4—93no—
272C2×C3×C4—93no—
273C2×C3×C4—186no—
274C2×C3×C4C22×C3×D313yes—
275C2×C3×C4C22×C3×D313yes—
276#67—34no—
277#67—94no—
278C2×Dic3—31no—
279C2×Dic3—33no—
280C2×Dic3—33no—
281C2×Dic3—91no—
282C2×Dic3—93no—
283C2×Dic3—93no—
284C4×D3—66no—
285C4×D3—186no—
286Dic6—66no—
287Dic6—186no—
288C2×C32—24no—
289C2×C32—22no—
290C2×C32—22no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32C2×D3×Q811yescharacteristic
295C2×C32#100711yescharacteristic
296C2×C32C22×#6711yescharacteristic
297C2×C32C22×C3×D411yescharacteristic
298C2×C32C22×C3×Q811yescharacteristic
299C2×C32C3×2−1+411yescharacteristic
300C3⋊D3—22no—
301C3×D3—62no—
302C2×Q8—24no—
303C2×Q8—93no—
304C2×Q8—184no—
305C2×Q8C2×(C3⋊D3)⋊C313yes—
306C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
307C2×D4—91no—
308C22×C4—93no—
309C22×C4C2×(C3⋊D3)⋊C313yes—
310C4⋊C4—93no—
311C4⋊C4—93no—
312C4⋊C4—96no—
313C22⋊C4—93no—
314C22⋊C4—93no—
315C42—93no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—32no—
321C22×C3—32no—
322C22×C3—61no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3—62no—
327C22×C3—62no—
328C22×C3—61no—
329C22×C3—62no—
330C22×C3—92no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3C3×D3×Q812yes—
334C22×C3C23×C3×D311yescharacteristic
335C2×D3—32no—
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—92no—
339C2×D3—181no—
340C2×D3—181no—
341C3×C4—23no—
342C3×C4—36no—
343C3×C4—36no—
344C3×C4—63no—
345C3×C4—63no—
346C3×C4—66no—
347C3×C4—66no—
348C3×C4—92no—
349C3×C4—183no—
350C3×C4—183no—
351C3×C4C2×C3×#6716yes—
352Dic3—32no—
353Dic3—63no—
354Dic3—63no—
355Dic3—92no—
356Dic3—183no—
357Dic3—183no—
358C32—21no—
359C32Q8×#6711yescharacteristic, complemented
360C32C3×D4×Q811yescharacteristic, complemented
361C23—91no—
362C23C22×(C3⋊D3)⋊C311yescharacteristic
363Q8—26no—
364Q8—186no—
365Q8#638⋊C314yesdirect product factor
366D4—94no—
367C2×C4—26no—
368C2×C4—91no—
369C2×C4—93no—
370C2×C4—93no—
371C2×C4—186no—
372C2×C4C22×(C3⋊D3)⋊C313yes—
373C2×C4C22×(C3⋊D3)⋊C313yes—
374C2×C3—22no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—62no—
382C2×C3—62no—
383C2×C3—64no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—61no—
387C2×C3—182no—
388C2×C3C2×C3×D3×Q811yescharacteristic
389C2×C3C3×#100711yescharacteristic
390C2×C3C22×C3×#6711yescharacteristic
391D3—62no—
392D3—182no—
393C22—21no—
394C22—21no—
395C22—92no—
396C22—181no—
397C22—181no—
398C22Q8×(C3⋊D3)⋊C312yescomplemented
399C22C23×(C3⋊D3)⋊C311yescharacteristic
400C4—23no—
401C4—92no—
402C4—183no—
403C4—183no—
404C4C2×#638⋊C316yes—
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×Q8×#6711yescharacteristic
409C2—22no—
410C2—182no—
411C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
412C2#1172262711yescharacteristic
413C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
prime radicalC2×#638⋊C3C22
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C3×D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
2. C2×Q8×(C3⋊D3)⋊C3C2semidirect
3. #64155⋊C3C2semidirect
4. Q8×(C3⋊Dic3)⋊C3C2semidirect
5. #64153⋊C3C2semidirect
7. #64133⋊C3C2semidirect
8. Q8×#638C3semidirect
25. C22×C32×Q8D3semidirect
26. C22×C32×Q8C2×C3semidirect
27. C2×Q8×C3⋊D3C2×C3semidirect
29. #64155C2×C3semidirect
30. Q8×C3⋊Dic3C2×C3semidirect
31. #64153C2×C3semidirect
33. #64133C2×C3semidirect
42. Q8×H(3,𝔽3)D4semidirect
46. #638⋊C3Q8direct (N × H)
89. C22×H(3,𝔽3)C2×Q8semidirect
93. C2×(C3⋊D3)⋊C3C2×Q8semidirect
98. (C3⋊Dic3)⋊C3C2×Q8semidirect
130. C32×Q8#67semidirect
131. C32×Q8C3×D4semidirect
141. #638C3×Q8semidirect
213. C22×C32D3×Q8semidirect
214. C22×C32C2×C3×Q8semidirect
219. C2×C3⋊D3C2×C3×Q8semidirect
231. C3⋊Dic3C2×C3×Q8semidirect
236. C22×Q8(C3⋊D3)⋊C3semidirect
242. H(3,𝔽3)D4×Q8semidirect
359. C32Q8×#67semidirect
360. C32C3×D4×Q8semidirect
365. Q8#638⋊C3direct (N × H)
398. C22Q8×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×Q8×(C3⋊D3)⋊C32
#64155⋊C32
Q8×(C3⋊Dic3)⋊C32
#64153⋊C32
C4×#638⋊C32
#64133⋊C32
Q8×#6383
C3×Q8×#673