Finite Group Databasebeta

← #11713968

Subgroups — #11713968

Order 768.

819 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 819 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#5421C212yesmaximal, complemented
3C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73065C212yesmaximal, complemented
5C4⋊#859C211yesmaximal, characteristic, complemented
6#32903—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C22×GL(2,𝔽3)C2212yes—
9C2×#930—41nomaximal
10C2×C4×SL(2,𝔽3)—22no—
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13#5421—24no—
14C2×#859C2212yes—
15C4⋊GL(2,𝔽3)—22no—
16C22×C4×Q8D311yescharacteristic, complemented
17C2×#460—31no—
18C2×#460—31no—
19C2×#448—32no—
20#2856—31no—
21#1880—32no—
22#1880—32no—
23#1874—32no—
24#1803—31no—
25#1803—31no—
26#1797—31no—
27C23×C3×C4—41no—
28C22×#67—42no—
29C22×D12—41no—
30C22×SL(2,𝔽3)—21no—
31C22×SL(2,𝔽3)—21no—
32C22×SL(2,𝔽3)D412yescomplemented
33C22×SL(2,𝔽3)C2311yescharacteristic
34C2×GL(2,𝔽3)—22no—
35C2×GL(2,𝔽3)—22no—
36C2×GL(2,𝔽3)—22no—
37#930—48no—
38C2×(C2×D3)⋊C4—42no—
39C2×C4⋊Dic3—41no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)D412yescomplemented
42#859—22no—
43C23×Q8C2×D311yescharacteristic
44C22×SD16—32no—
45C22×SD16—32no—
46C2×#127—31no—
47C2×#121—32no—
48C2×#121—61no—
49C2×#121—61no—
50C2×C4×Q8—22no—
51C2×C4×Q8—31no—
52C2×C4×Q8—31no—
53C2×C4×Q8—62no—
54C2×C4×Q8C2×D311yescharacteristic
55C2×C4×Q8C2×D311yescharacteristic
56C22×C4⋊C4—31no—
57C22×C42—31no—
58#460—62no—
59#460—64no—
60#460—62no—
61#448—64no—
62#448—64no—
63C2×C4⋊C8—31no—
64C2×C4⋊C8—31no—
65C2×#103—32no—
66C2×#103—32no—
67C2×#102—32no—
68C2×#102—32no—
69C2×C22⋊C8—32no—
70#390—32no—
71#390—61no—
72#390—61no—
73#384—31no—
74#340—32no—
75#340—32no—
76#336—32no—
77C24×C3—41no—
78C23×D3—42no—
79C22×C3×C4—44no—
80C22×C3×C4—41no—
81C22×C3×C4—41no—
82C22×C3×C4—82no—
83C22×C3×C4—82no—
84C2×#67—48no—
85C2×#67—84no—
86C2×#67—84no—
87C22×Dic3—42no—
88C2×D12—44no—
89C2×D12—82no—
90C2×D12—82no—
91C2×SL(2,𝔽3)—22no—
92C2×SL(2,𝔽3)C2×D411yescharacteristic
93C2×SL(2,𝔽3)C2×D411yescharacteristic
94C2×SL(2,𝔽3)#4111yescharacteristic
95GL(2,𝔽3)—42no—
96(C2×D3)⋊C4—48no—
97C4⋊Dic3—44no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—31no—
101C22×Q8—32no—
102C22×Q8—61no—
103C22×Q8—61no—
104C22×Q8D1212yescomplemented
105C22×Q8C22×D311yescharacteristic
106C22×D4—32no—
107C22×D4—61no—
108C22×D4—62no—
109C22×D4—61no—
110C23×C4—31no—
111C23×C4—31no—
112C23×C4S411yescharacteristic
113C2×SD16—62no—
114C2×SD16—64no—
115C2×SD16—64no—
116C2×SD16—62no—
117C2×SD16—62no—
118C2×SD16—64no—
119C2×SD16—62no—
120C2×SD16—62no—
121C2×SD16—62no—
122C22×C8—32no—
123C22×C8—32no—
124#127—62no—
125#127—62no—
126#121—64no—
127#121—64no—
128#121—124no—
129#121—124no—
130C4×Q8—21no—
131C4×Q8—32no—
132C4×Q8—34no—
133C4×Q8—62no—
134C4×Q8—62no—
135C4×Q8—61no—
136C4×Q8—62no—
137C4×Q8#6712yescomplemented
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—32no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—62no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—62no—
149C2×C42—31no—
150C2×C42—31no—
151C2×C42—31no—
152C2×C42—31no—
153C2×C42—62no—
154C2×C42—62no—
155C4⋊C8—62no—
156C4⋊C8—62no—
157#103—64no—
158#103—62no—
159#103—62no—
160#102—64no—
161#102—64no—
162C22⋊C8—64no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3—41no—
167C23×C3—81no—
168C23×C3—81no—
169C23×C3—81no—
170C23×C3—81no—
171C22×D3—44no—
172C22×D3—82no—
173C22×D3—82no—
174C22×D3—82no—
175C22×D3—82no—
176C22×D3—82no—
177C22×D3—82no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—88no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185#67—816no—
186C2×Dic3—44no—
187C2×Dic3—82no—
188C2×Dic3—82no—
189D12—88no—
190SL(2,𝔽3)#12111yescharacteristic, complemented
191C24—62no—
192C24C2×S411yescharacteristic
193C2×Q8—22no—
194C2×Q8—32no—
195C2×Q8—31no—
196C2×Q8—31no—
197C2×Q8—31no—
198C2×Q8—32no—
199C2×Q8—32no—
200C2×Q8—61no—
201C2×Q8—61no—
202C2×Q8—61no—
203C2×Q8—64no—
204C2×Q8—62no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8C2×D1211yescharacteristic
208C2×Q8#23511yescharacteristic
209C2×Q8C2×#6711yescharacteristic
210C2×D4—62no—
211C2×D4—62no—
212C2×D4—62no—
213C2×D4—62no—
214C2×D4—62no—
215C2×D4—62no—
216C2×D4—122no—
217C2×D4—122no—
218C2×D4—122no—
219C2×D4—124no—
220C2×D4—124no—
221C2×D4—124no—
222C2×D4—122no—
223C2×D4—122no—
224C2×D4—122no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—31no—
228C22×C4—32no—
229C22×C4—32no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—121no—
241C22×C4GL(2,𝔽3)14yescomplemented
242C22×C4C2×S411yescharacteristic
243C22×C4C2×S411yescharacteristic
244SD16—122no—
245SD16—122no—
246SD16—122no—
247SD16—128no—
248SD16—122no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8—62no—
253C2×C8—62no—
254C2×C8—62no—
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—64no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—122no—
263C4⋊C4—122no—
264C22⋊C4—124no—
265C22⋊C4—124no—
266C22⋊C4—124no—
267C42—32no—
268C42—32no—
269C42—61no—
270C42—61no—
271C42—62no—
272C42—62no—
273C22×C3—44no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—82no—
282C22×C3—82no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3—82no—
287C2×D3—84no—
288C2×D3—84no—
289C2×D3—84no—
290C2×D3—162no—
291C2×D3—162no—
292C2×D3—162no—
293C2×D3—162no—
294C3×C4—44no—
295C3×C4—82no—
296Dic3—84no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—122no—
302C23—122no—
303C23—122no—
304C23—122no—
305C23—122no—
306C23—122no—
307C23—122no—
308C23C4⋊S412yes—
309C23C2×GL(2,𝔽3)12yes—
310C23#98612yes—
311C23C22×S411yescharacteristic
312Q8—31no—
313Q8—31no—
314Q8—31no—
315Q8—34no—
316Q8—61no—
317Q8—62no—
318Q8—62no—
319Q8—62no—
320Q8#93011yescharacteristic, complemented
321D4—122no—
322D4—122no—
323D4—122no—
324D4—122no—
325D4—244no—
326D4—248no—
327D4—244no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—28no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—124no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4C2×GL(2,𝔽3)12yes—
355C2×C4C2×GL(2,𝔽3)12yes—
356C2×C4#98312yes—
357C2×C4GL(2,ℤ/4)12yes—
358C8—122no—
359C8—122no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—84no—
368D3—164no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—242no—
376C22—242no—
377C22—242no—
378C22—242no—
379C22—242no—
380C22—242no—
381C22—242no—
382C22C4⋊GL(2,𝔽3)14yescomplemented
383C22C2×C4⋊S411yescharacteristic
384C22#591211yescharacteristic
385C22C22×GL(2,𝔽3)11yescharacteristic
386C22C2×#98311yescharacteristic
387C22C2×#98611yescharacteristic
388C22#592311yescharacteristic
389C22C2×GL(2,ℤ/4)11yescharacteristic
390C4—22no—
391C4—31no—
392C4—31no—
393C4—31no—
394C4—31no—
395C4—62no—
396C4—62no—
397C4—121no—
398C4—242no—
399C4#542114yescomplemented
400C3—41noSylow
401C2—24no—
402C2—482no—
403C2C2×C4⋊GL(2,𝔽3)11yescharacteristic
404C2#8532511yescharacteristic
405C2#8533111yescharacteristic
406C2#8533211yescharacteristic
407C2#8548611yescharacteristic
408C2C2×#542111yescharacteristic
409C2#8552311yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC22×C4×Q8D3
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×SL(2,𝔽3)C2semidirect
2. C2×#5421C2semidirect
3. C2×C4⋊GL(2,𝔽3)C2semidirect
4. #73065C2semidirect
5. C4⋊#859C2semidirect
16. C22×C4×Q8D3semidirect
32. C22×SL(2,𝔽3)D4semidirect
41. C4×SL(2,𝔽3)D4semidirect
104. C22×Q8D12semidirect
137. C4×Q8#67semidirect
190. SL(2,𝔽3)#121semidirect
241. C22×C4GL(2,𝔽3)semidirect
320. Q8#930semidirect
382. C22C4⋊GL(2,𝔽3)semidirect
399. C4#5421semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SL(2,𝔽3)2
C2×#54212
C2×C4⋊GL(2,𝔽3)2
#730652
C4⋊#8592
#329033
C2×#9304