Finite Group Databasebeta

← #72267883

Subgroups — #72267883

Order 1536.

1663 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 1663 conjugacy classes of subgroups — too many to draw. Only its 49 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
1C22⋊(Q8⋊SL(2,𝔽3))C211yesmaximal, characteristic, complemented
2Q8×#393C311yesmaximal, characteristic, complemented, Sylow
3C22×C22⋊SL(2,𝔽3)—41nomaximal
4C2×#393⋊C3—43nomaximal
5C2×#393⋊C3—41nomaximal
6C22×Q82C2×C311yescharacteristic, complemented
7C2×Q8×C22⋊C4—31no—
8#15201—31no—
9#15191—33no—
10#15125—33no—
11C4×#393—31no—
12Q8×#95—31no—
13C2×C22⋊SL(2,𝔽3)—42no—
14C2×C22⋊SL(2,𝔽3)—41no—
15C2×C22⋊SL(2,𝔽3)—41no—
16C2×C22⋊SL(2,𝔽3)—43no—
17Q8⋊SL(2,𝔽3)—88no—
18#393⋊C3—46no—
19#393⋊C3—42no—
20C24×Q8A411yescharacteristic
21C2×Q82—68no—
22C22×C4⋊Q8—33no—
23C22×C4×Q8—31no—
24C22×C4×Q8—31no—
25C22×C4×Q8—31no—
26C2×#393—31no—
27C2×#393A411yescharacteristic
28C2×#393A413yes—
29Q8×C22⋊C4—68no—
30C2×#391—33no—
31C2×#391—33no—
32C2×#380—33no—
33C2×C4×C22⋊C4—33no—
34#1446—33no—
35#1446—33no—
36#1446—31no—
37#1442—36no—
38#1442—33no—
39#1439—33no—
40C4⋊#95—36no—
41C4⋊#95—33no—
42C4×#95—33no—
43C23×A4—161no—
44C22⋊SL(2,𝔽3)—41no—
45C22⋊SL(2,𝔽3)—43no—
46C22⋊SL(2,𝔽3)—41no—
47C22×SL(2,𝔽3)—164no—
48C23×Q8—31no—
49C23×Q8—32no—
50C23×Q8—31no—
51C23×Q8—64no—
52C23×Q8—64no—
53C23×Q8SL(2,𝔽3)12yescomplemented
54C23×Q8C2×A411yescharacteristic
55C23×Q8C2×A411yescharacteristic
56C23×Q8C2×A413yes—
57C24×C4—31no—
58Q82—28no—
59Q82—68no—
60Q82—624no—
61Q82—68no—
62C2×C4⋊Q8—612no—
63C2×C4⋊Q8—624no—
64C2×C4×Q8—38no—
65C2×C4×Q8—612no—
66C2×C4×Q8—68no—
67C2×C4×Q8—64no—
68C2×C4×Q8—68no—
69C22×C4⋊C4—33no—
70C22×C4⋊C4—33no—
71C22×C4⋊C4—33no—
72C22×C22⋊C4—31no—
73C22×C22⋊C4—33no—
74C22×C42—33no—
75C22×C42—33no—
76#393—32no—
77#393—46no—
78#393—122no—
79#393SL(2,𝔽3)12yescomplemented
80#391—612no—
81#391—612no—
82#380—612no—
83C4×C22⋊C4—612no—
84C2×#95—31no—
85C2×#95—33no—
86C2×#95—33no—
87C2×#95—33no—
88C2×#95—36no—
89C2×#95—33no—
90C2×#95—33no—
91C2×#95—36no—
92C22×A4—162no—
93C22×A4—162no—
94C22×A4—162no—
95C22×A4—161no—
96C2×SL(2,𝔽3)—168no—
97C2×SL(2,𝔽3)—164no—
98C2×SL(2,𝔽3)—322no—
99C2×SL(2,𝔽3)—326no—
100C25C22⋊A411yescharacteristic
101C22×Q8—34no—
102C22×Q8—34no—
103C22×Q8—32no—
104C22×Q8—31no—
105C22×Q8—32no—
106C22×Q8—31no—
107C22×Q8—43no—
108C22×Q8—44no—
109C22×Q8—62no—
110C22×Q8—66no—
111C22×Q8—62no—
112C22×Q8—66no—
113C22×Q8—66no—
114C22×Q8—66no—
115C22×Q8—68no—
116C22×Q8—624no—
117C22×Q8—123no—
118C22×Q8—123no—
119C22×Q8—121no—
120C22×Q8—124no—
121C22×Q8—124no—
122C22×Q8—1212no—
123C22×Q8C2×SL(2,𝔽3)11yescharacteristic
124C22×Q8C2×SL(2,𝔽3)11yescharacteristic, complemented
125C23×C4—31no—
126C23×C4—31no—
127C23×C4—32no—
128C23×C4—32no—
129C23×C4—33no—
130C23×C4—31no—
131C23×C4—31no—
132C23×C4—33no—
133C23×C4—66no—
134C23×C4—66no—
135C4⋊Q8—624no—
136C4⋊Q8—672no—
137C4×Q8—68no—
138C4×Q8—624no—
139C4×Q8—624no—
140C4×Q8—68no—
141C4×Q8—1212no—
142C4×Q8—124no—
143C2×C4⋊C4—312no—
144C2×C4⋊C4—612no—
145C2×C4⋊C4—66no—
146C2×C4⋊C4—612no—
147C2×C4⋊C4—612no—
148C2×C4⋊C4—66no—
149C2×C22⋊C4—32no—
150C2×C22⋊C4—32no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—66no—
154C2×C22⋊C4—66no—
155C2×C22⋊C4—123no—
156C2×C42—312no—
157C2×C42—612no—
158C2×C42—66no—
159C2×C42—612no—
160#95—32no—
161#95—32no—
162#95—66no—
163#95—66no—
164#95—66no—
165#95—612no—
166#95—123no—
167#95—126no—
168#95—123no—
169C23×C3—641no—
170C2×A4—161no—
171C2×A4—161no—
172C2×A4—161no—
173C2×A4—164no—
174SL(2,𝔽3)—164no—
175SL(2,𝔽3)—322no—
176SL(2,𝔽3)—326no—
177C24—61no—
178C24—61no—
179C24—61no—
180C24—61no—
181C24C22⋊SL(2,𝔽3)12yes—
182C24C22⋊SL(2,𝔽3)12yes—
183C24#99712yes—
184C24C2×C22⋊A411yescharacteristic
185C2×Q8—22no—
186C2×Q8—26no—
187C2×Q8—34no—
188C2×Q8—312no—
189C2×Q8—312no—
190C2×Q8—34no—
191C2×Q8—34no—
192C2×Q8—48no—
193C2×Q8—62no—
194C2×Q8—62no—
195C2×Q8—64no—
196C2×Q8—612no—
197C2×Q8—612no—
198C2×Q8—62no—
199C2×Q8—62no—
200C2×Q8—64no—
201C2×Q8—68no—
202C2×Q8—62no—
203C2×Q8—66no—
204C2×Q8—1212no—
205C2×Q8—124no—
206C2×Q8—128no—
207C2×Q8—1224no—
208C2×Q8—128no—
209C2×Q8—246no—
210C2×Q8—246no—
211C2×Q8—246no—
212C2×Q8—246no—
213C2×Q8#120⋊C314yes—
214C22×C4—36no—
215C22×C4—36no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—32no—
219C22×C4—32no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—36no—
223C22×C4—32no—
224C22×C4—32no—
225C22×C4—612no—
226C22×C4—612no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—63no—
230C22×C4—61no—
231C22×C4—62no—
232C22×C4—66no—
233C22×C4—62no—
234C22×C4—66no—
235C22×C4—61no—
236C22×C4—63no—
237C22×C4—63no—
238C22×C4—63no—
239C22×C4—63no—
240C22×C4—66no—
241C22×C4—63no—
242C22×C4—66no—
243C22×C4—63no—
244C22×C4—124no—
245C22×C4—1212no—
246C4⋊C4—624no—
247C4⋊C4—624no—
248C4⋊C4—1212no—
249C22⋊C4—64no—
250C22⋊C4—64no—
251C22⋊C4—1212no—
252C42—624no—
253C42—1212no—
254C22×C3—642no—
255C22×C3—642no—
256C22×C3—642no—
257C22×C3—641no—
258A4—161no—
259C23—31no—
260C23—31no—
261C23—31no—
262C23—31no—
263C23—41no—
264C23—62no—
265C23—62no—
266C23—62no—
267C23—62no—
268C23—62no—
269C23—62no—
270C23—121no—
271C23—121no—
272C23—121no—
273C23—122no—
274C23Q8⋊SL(2,𝔽3)14yescomplemented
275C23C2×C22⋊SL(2,𝔽3)11yescharacteristic
276C23C2×C22⋊SL(2,𝔽3)11yescharacteristic
277C23C2×#99711yescharacteristic
278Q8—22no—
279Q8—34no—
280Q8—312no—
281Q8—34no—
282Q8—62no—
283Q8—62no—
284Q8—62no—
285Q8—64no—
286Q8—86no—
287Q8—128no—
288Q8—128no—
289Q8—246no—
290Q8—248no—
291Q8#393⋊C314yescomplemented
292C2×C4—312no—
293C2×C4—36no—
294C2×C4—36no—
295C2×C4—34no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—64no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—66no—
305C2×C4—64no—
306C2×C4—66no—
307C2×C4—66no—
308C2×C4—66no—
309C2×C4—128no—
310C2×C4—128no—
311C2×C4—121no—
312C2×C4—122no—
313C2×C4—121no—
314C2×C4—123no—
315C2×C4—126no—
316C2×C4—126no—
317C2×C4—121no—
318C2×C4—123no—
319C2×C4—123no—
320C2×C4—126no—
321C2×C4—126no—
322C2×C4—123no—
323C2×C3—644no—
324C2×C3—641no—
325C2×C3—641no—
326C2×C3—641no—
327C22—31no—
328C22—31no—
329C22—31no—
330C22—32no—
331C22—31no—
332C22—31no—
333C22—31no—
334C22—31no—
335C22—31no—
336C22—31no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—64no—
341C22—122no—
342C22—122no—
343C22—122no—
344C22#8564511yescharacteristic
345C22C2×Q8⋊SL(2,𝔽3)11yescharacteristic
346C4—34no—
347C4—34no—
348C4—62no—
349C4—62no—
350C4—122no—
351C4—122no—
352C4—126no—
353C4—243no—
354C3—641noSylow
355C2—31no—
356C2—31no—
357C2—31no—
358C2—31no—
359C2—44no—
360C2#1170984411yescharacteristic
361C2#1170994511yescharacteristic
362C2#1170998211yescharacteristic
subgroupG/H
center, hypercenterC22#85645
commutator, cosocle, hypocenterC22×Q82C2×C3
Frattini, socleC24C2×C22⋊A4
FittingQ8×#393C3
LCSG ▷ C22×Q82
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22⋊(Q8⋊SL(2,𝔽3))C2semidirect
2. Q8×#393C3semidirect
6. C22×Q82C2×C3semidirect
53. C23×Q8SL(2,𝔽3)semidirect
79. #393SL(2,𝔽3)semidirect
124. C22×Q8C2×SL(2,𝔽3)semidirect
274. C23Q8⋊SL(2,𝔽3)semidirect
291. Q8#393⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22⋊(Q8⋊SL(2,𝔽3))2
Q8×#3933
C22×C22⋊SL(2,𝔽3)4
C2×#393⋊C34