Finite Group Databasebeta

← #11715652

Subgroups — #11715652

Order 768.

957 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 957 conjugacy classes of subgroups — too many to draw. Only its 39 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2#85562C212yesmaximal, complemented
3#20599—31nomaximal, Sylow
4C2×C22⋊SL(2,𝔽3)C412yescomplemented
5C2×C22⋊SL(2,𝔽3)C2211yescharacteristic
6#5432—41nomaximal
7#5420—42nomaximal
8C24×Q8D311yescharacteristic
9C2×#419—32no—
10C2×#419—32no—
11C2×#407—31no—
12#2280—31no—
13#1787—32no—
14#1787—34no—
15#1787—32no—
16C23×A4—41no—
17C22⋊SL(2,𝔽3)C2×C411yescharacteristic
18C22×SL(2,𝔽3)—42no—
19C22×SL(2,𝔽3)—81no—
20C2×SL(2,ℤ/4)—42no—
21#859—44no—
22C23×Q8—32no—
23C23×Q8—31no—
24C23×Q8—32no—
25C23×Q8—34no—
26C23×Q8—32no—
27C23×Q8—61no—
28C23×Q8—61no—
29C23×Q8—62no—
30C23×Q8Dic312yescomplemented
31C23×Q8C2×D311yescharacteristic
32C24×C4—31no—
33C22×M16—32no—
34C2×#117—31no—
35C2×#117—31no—
36#419—62no—
37#419—64no—
38#419—62no—
39#419—64no—
40#419—62no—
41#419—62no—
42C2×#103—34no—
43C2×#103—34no—
44C2×#103—34no—
45C2×#103—34no—
46#407—62no—
47#407—62no—
48C2×C22⋊C8—32no—
49C2×C22⋊C8—32no—
50#381—61no—
51#380—32no—
52#380—32no—
53#379—61no—
54C4×C22⋊C4—61no—
55C4×C22⋊C4—61no—
56C22×A4—42no—
57C22×A4—41no—
58C22×A4—82no—
59C2×SL(2,𝔽3)—44no—
60C2×SL(2,𝔽3)—42no—
61C2×SL(2,𝔽3)—82no—
62C2×SL(2,𝔽3)—81no—
63SL(2,ℤ/4)—82no—
64#217—161no—
65C25S411yescharacteristic
66C22×Q8—21no—
67C22×Q8—32no—
68C22×Q8—31no—
69C22×Q8—34no—
70C22×Q8—32no—
71C22×Q8—32no—
72C22×Q8—34no—
73C22×Q8—32no—
74C22×Q8—34no—
75C22×Q8—34no—
76C22×Q8—32no—
77C22×Q8—32no—
78C22×Q8—34no—
79C22×Q8—32no—
80C22×Q8—34no—
81C22×Q8—32no—
82C22×Q8—61no—
83C22×Q8—61no—
84C22×Q8—62no—
85C22×Q8—61no—
86C22×Q8—62no—
87C22×Q8—61no—
88C22×Q8—62no—
89C22×Q8—62no—
90C22×Q8—64no—
91C22×Q8—64no—
92C22×Q8—62no—
93C22×Q8—64no—
94C22×Q8—64no—
95C22×Q8—64no—
96C22×Q8—64no—
97C22×Q8—61no—
98C22×Q8—61no—
99C22×Q8—62no—
100C22×Q8—62no—
101C22×Q8—62no—
102C22×Q8—61no—
103C22×Q8—62no—
104C22×Q8—61no—
105C22×Q8—62no—
106C22×Q8—64no—
107C22×Q8—62no—
108C22×Q8—62no—
109C22×Q8—64no—
110C22×Q8—62no—
111C22×Q8—64no—
112C22×Q8—62no—
113C22×Q8C2×Dic311yescharacteristic
114C22×Q8S412yes—
115C23×C4—31no—
116C23×C4—32no—
117C23×C4—31no—
118C23×C4—31no—
119C23×C4—31no—
120C23×C4—32no—
121C23×C4—32no—
122C23×C4—32no—
123C23×C4—31no—
124C23×C4—31no—
125C23×C4—62no—
126C23×C4—64no—
127C23×C4—62no—
128C2×M16—64no—
129C2×M16—64no—
130C2×M16—62no—
131C2×M16—62no—
132C22×C8—34no—
133#117—64no—
134#117—62no—
135#117—62no—
136C2×C4⋊C4—32no—
137C2×C4⋊C4—32no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C42—61no—
143C2×C42—61no—
144#103—64no—
145#103—68no—
146#103—64no—
147#103—64no—
148#103—64no—
149#103—64no—
150#103—64no—
151C22⋊C8—64no—
152C22⋊C8—62no—
153C22⋊C8—62no—
154#95—62no—
155#95—62no—
156C23×C3—161no—
157C2×A4—42no—
158C2×A4—41no—
159C2×A4—82no—
160C2×Dic3—162no—
161SL(2,𝔽3)—42no—
162SL(2,𝔽3)—81no—
163C24—22no—
164C24—31no—
165C24—31no—
166C24—32no—
167C24—62no—
168C24SL(2,ℤ/4)12yes—
169C24C2×S411yescharacteristic
170C2×Q8—22no—
171C2×Q8—21no—
172C2×Q8—32no—
173C2×Q8—34no—
174C2×Q8—34no—
175C2×Q8—32no—
176C2×Q8—32no—
177C2×Q8—32no—
178C2×Q8—34no—
179C2×Q8—32no—
180C2×Q8—34no—
181C2×Q8—34no—
182C2×Q8—34no—
183C2×Q8—34no—
184C2×Q8—34no—
185C2×Q8—32no—
186C2×Q8—32no—
187C2×Q8—34no—
188C2×Q8—62no—
189C2×Q8—64no—
190C2×Q8—62no—
191C2×Q8—61no—
192C2×Q8—62no—
193C2×Q8—61no—
194C2×Q8—61no—
195C2×Q8—64no—
196C2×Q8—62no—
197C2×Q8—62no—
198C2×Q8—64no—
199C2×Q8—64no—
200C2×Q8—64no—
201C2×Q8—64no—
202C2×Q8—64no—
203C2×Q8—62no—
204C2×Q8—64no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—64no—
208C2×Q8—64no—
209C2×Q8—64no—
210C2×Q8—64no—
211C2×Q8—64no—
212C2×Q8—62no—
213C2×Q8—62no—
214C2×Q8—61no—
215C2×Q8—64no—
216C2×Q8—62no—
217C2×Q8—62no—
218C2×Q8—61no—
219C2×Q8—62no—
220C2×Q8—61no—
221C2×Q8—64no—
222C2×Q8—62no—
223C2×Q8—64no—
224C2×Q8—64no—
225C2×Q8—64no—
226C2×Q8—64no—
227C2×Q8—62no—
228C2×Q8—64no—
229C2×Q8—64no—
230C2×Q8—62no—
231C2×Q8—62no—
232C2×Q8—64no—
233C2×Q8—62no—
234C2×Q8—62no—
235C2×Q8—61no—
236C2×Q8SL(2,ℤ/4)14yescomplemented
237C2×Q8C2×S412yes—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—34no—
241C22×C4—32no—
242C22×C4—32no—
243C22×C4—32no—
244C22×C4—32no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—32no—
248C22×C4—31no—
249C22×C4—32no—
250C22×C4—31no—
251C22×C4—32no—
252C22×C4—32no—
253C22×C4—32no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—32no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—64no—
262C22×C4—64no—
263C22×C4—64no—
264C22×C4—64no—
265C22×C4—64no—
266C22×C4—64no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—64no—
274C22×C4—64no—
275C22×C4—62no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—62no—
279C22×C4—62no—
280M16—124no—
281M16—124no—
282C2×C8—64no—
283C2×C8—64no—
284C2×C8—64no—
285C4⋊C4—64no—
286C4⋊C4—64no—
287C22⋊C4—122no—
288C22⋊C4—124no—
289C22⋊C4—122no—
290C22⋊C4—121no—
291C22⋊C4—122no—
292C22⋊C4—121no—
293C42—122no—
294C42—122no—
295C22×C3—162no—
296C22×C3—161no—
297C22×C3—322no—
298A4—41no—
299Dic3—322no—
300C23—22no—
301C23—31no—
302C23—32no—
303C23—32no—
304C23—31no—
305C23—34no—
306C23—32no—
307C23—32no—
308C23—31no—
309C23—62no—
310C23—62no—
311C23—62no—
312C23—61no—
313C23—64no—
314C23—62no—
315C23—64no—
316C23#98312yes—
317C23C2×SL(2,ℤ/4)11yescharacteristic
318C23#102011yescharacteristic
319Q8—21no—
320Q8—34no—
321Q8—32no—
322Q8—32no—
323Q8—34no—
324Q8—32no—
325Q8—62no—
326Q8—61no—
327Q8—62no—
328Q8—64no—
329Q8—61no—
330Q8—64no—
331Q8—62no—
332Q8—62no—
333Q8—62no—
334Q8—62no—
335Q8—64no—
336Q8—64no—
337Q8—62no—
338Q8—61no—
339Q8—62no—
340Q8C2×SL(2,ℤ/4)12yes—
341C2×C4—32no—
342C2×C4—32no—
343C2×C4—31no—
344C2×C4—34no—
345C2×C4—31no—
346C2×C4—32no—
347C2×C4—34no—
348C2×C4—34no—
349C2×C4—34no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—31no—
353C2×C4—32no—
354C2×C4—31no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—64no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—64no—
361C2×C4—64no—
362C2×C4—64no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4—61no—
366C2×C4—62no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C8—124no—
380C2×C3—162no—
381C2×C3—161no—
382C2×C3—322no—
383C22—22no—
384C22—32no—
385C22—31no—
386C22—32no—
387C22—32no—
388C22—31no—
389C22—31no—
390C22—34no—
391C22—32no—
392C22—62no—
393C22—64no—
394C22—62no—
395C22—61no—
396C22—62no—
397C22—64no—
398C22—62no—
399C22#542011yescharacteristic, complemented
400C22#593612yes—
401C22C2×#102011yescharacteristic
402C4—32no—
403C4—32no—
404C4—34no—
405C4—61no—
406C4—62no—
407C4—61no—
408C4—242no—
409C4—242no—
410C3—161noSylow
411C2—22no—
412C2—32no—
413C2—31no—
414C2—31no—
415C2—62no—
416C2#8750712yes—
417C2C2×#593611yescharacteristic
subgroupG/H
center, FrattiniC22C2×#1020
commutator, hypocenterC22⋊SL(2,𝔽3)C2×C4
FittingC24×Q8D3
socleC24C2×S4
prime radicalC22×C22⋊SL(2,𝔽3)C2
cosocleC2×C22⋊SL(2,𝔽3)C22
hypercenterC23#1020
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #85562C2semidirect
4. C2×C22⋊SL(2,𝔽3)C4semidirect
30. C23×Q8Dic3semidirect
236. C2×Q8SL(2,ℤ/4)semidirect
399. C22#5420semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊SL(2,𝔽3)2
#855622
#205993
#54324
#54204