Finite Group Databasebeta

← #11713992

Subgroups — #11713992

Order 768.

625 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 65 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×C8×A4C211yesmaximal, characteristic
2#85480C212yesmaximal
3C2×#5398C211yesmaximal, characteristic, complemented
4#73111C211yesmaximal, characteristic, complemented
5#73109C212yesmaximal, complemented
6#32931—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8C2×A4⋊Q8C2212yes—
9C2×C8×A4—22no—
10C2×C8×A4C2211yescharacteristic
11C2×C8×A4C2211yescharacteristic
12#5432—22no—
13C4⋊SL(2,ℤ/4)C2212yes—
14SL(2,ℤ/4)⋊C4—22no—
15#5398—22no—
16#5116—41nomaximal
17C24×C8D311yescharacteristic
18C2×#467—31no—
19C2×#467—31no—
20C2×#467—31no—
21#2848—32no—
22#1892—32no—
23#1892—32no—
24#1892—32no—
25#1808—31no—
26#1808—31no—
27#1808—31no—
28C23×A4D411yescharacteristic
29C2×C4×A4—21no—
30C2×C4×A4—21no—
31C2×C4×A4D411yescharacteristic
32C2×C4×A4C2311yescharacteristic
33C2×SL(2,ℤ/4)—22no—
34C2×SL(2,ℤ/4)—22no—
35A4⋊Q8—22no—
36A4⋊Q8—22no—
37C22×C3×C8—41no—
38#924—42no—
39C2×Dic12—41no—
40C8×A4—21no—
41C8×A4D412yes—
42#818—41no—
43#816—42no—
44C24×C4C2×D311yescharacteristic
45C22×Q16—31no—
46C23×C8—22no—
47C23×C8—31no—
48C23×C8—31no—
49C23×C8—31no—
50C23×C8—31no—
51C23×C8—62no—
52C23×C8C2×D311yescharacteristic
53C23×C8C2×D311yescharacteristic
54C2×C22⋊Q8—32no—
55C2×C22⋊Q8—32no—
56C2×C22⋊Q8—32no—
57#467—62no—
58#467—64no—
59#467—61no—
60#467—61no—
61#467—61no—
62#467—62no—
63#467—61no—
64C2×#107—31no—
65C2×#107—31no—
66C2×#107—31no—
67C2×#103—32no—
68C2×#103—32no—
69C2×#103—32no—
70#385—62no—
71#385—62no—
72#385—62no—
73#380—32no—
74#380—32no—
75#380—32no—
76#379—62no—
77#340—32no—
78#340—32no—
79#340—32no—
80C22×A4—21no—
81C22×A4—21no—
82C22×A4Q1612yescomplemented
83C22×A4C2×D411yescharacteristic
84C22×C3×C4—41no—
85C2×Dic6—42no—
86C4×A4—21no—
87C4×A4C2×D411yescharacteristic
88C4×A4#4111yescharacteristic
89SL(2,ℤ/4)—42no—
90SL(2,ℤ/4)—42no—
91C2×C3×C8—41no—
92C2×C3×C8—41no—
93C2×C3×C8—82no—
94#217—82no—
95C4⋊Dic3—42no—
96Dic3⋊C4—82no—
97Dic12—82no—
98C25D1211yescharacteristic
99C22×Q8—32no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—31no—
103C23×C4—31no—
104C23×C4—31no—
105C23×C4—31no—
106C23×C4—61no—
107C23×C4—61no—
108C23×C4D1211yescharacteristic
109C23×C4C22×D311yescharacteristic
110C2×Q16—61no—
111C2×Q16—62no—
112C2×Q16—62no—
113C2×Q16—61no—
114C22×C8—21no—
115C22×C8—31no—
116C22×C8—31no—
117C22×C8—31no—
118C22×C8—32no—
119C22×C8—34no—
120C22×C8—31no—
121C22×C8—31no—
122C22×C8—31no—
123C22×C8—31no—
124C22×C8—62no—
125C22×C8—62no—
126C22×C8—61no—
127C22×C8—61no—
128C22×C8—62no—
129C22×C8—62no—
130C22×C8—62no—
131C22×C8—62no—
132C22×C8—62no—
133C22×C8#6712yes—
134C22×C8S411yescharacteristic
135C22⋊Q8—62no—
136C22⋊Q8—64no—
137C22⋊Q8—62no—
138C22⋊Q8—62no—
139C22⋊Q8—62no—
140C22⋊Q8—64no—
141C22⋊Q8—62no—
142C22⋊Q8—62no—
143C22⋊Q8—62no—
144C22⋊Q8—62no—
145C2×C4⋊C4—32no—
146C2×C4⋊C4—32no—
147C2×C4⋊C4—32no—
148C2×C4⋊C4—62no—
149C2×C4⋊C4—62no—
150C2×C4⋊C4—62no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4—62no—
156C2×C22⋊C4—62no—
157#107—62no—
158#107—62no—
159#107—61no—
160#107—61no—
161#103—64no—
162#103—64no—
163#103—62no—
164#103—62no—
165#95—62no—
166#95—62no—
167#95—62no—
168C23×C3—41no—
169C2×A4—22no—
170C2×A4#12111yescharacteristic
171C2×A4C2×Q1611yescharacteristic
172C2×A4#13511yescharacteristic
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×Dic3—82no—
178C2×Dic3—82no—
179Dic6—82no—
180Dic6—82no—
181C3×C8—42no—
182C3×C8—81no—
183C24—21no—
184C24—21no—
185C24—31no—
186C24—31no—
187C24—32no—
188C24—61no—
189C24—61no—
190C24Dic1212yescomplemented
191C24C2×D1211yescharacteristic
192C2×Q8—62no—
193C2×Q8—62no—
194C2×Q8—62no—
195C2×Q8—62no—
196C2×Q8—62no—
197C2×Q8—62no—
198C22×C4—21no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—32no—
205C22×C4—32no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—62no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—62no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4#23511yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×S411yescharacteristic
231Q16—122no—
232Q16—122no—
233C2×C8—22no—
234C2×C8—31no—
235C2×C8—32no—
236C2×C8—32no—
237C2×C8—32no—
238C2×C8—32no—
239C2×C8—31no—
240C2×C8—62no—
241C2×C8—61no—
242C2×C8—61no—
243C2×C8—61no—
244C2×C8—62no—
245C2×C8—64no—
246C2×C8—62no—
247C2×C8—61no—
248C2×C8C2×S411yescharacteristic
249C2×C8C2×S411yescharacteristic
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—122no—
257C4⋊C4—122no—
258C4⋊C4—124no—
259C4⋊C4—122no—
260C4⋊C4—122no—
261C22⋊C4—122no—
262C22⋊C4—124no—
263C22⋊C4—124no—
264C22⋊C4—122no—
265C22⋊C4—124no—
266C22⋊C4—122no—
267C22⋊C4—122no—
268C22⋊C4—122no—
269C22⋊C4—122no—
270C22×C3—42no—
271C22×C3—41no—
272C22×C3—81no—
273C22×C3—81no—
274A4#46711yescharacteristic, complemented
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—81no—
278Dic3—162no—
279Dic3—162no—
280C23—22no—
281C23—31no—
282C23—31no—
283C23—31no—
284C23—32no—
285C23—31no—
286C23—32no—
287C23—32no—
288C23—32no—
289C23—32no—
290C23—31no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23—62no—
296C23—61no—
297C23—61no—
298C23—62no—
299C23—62no—
300C23—61no—
301C23—64no—
302C23#90411yescharacteristic
303C23C2×Dic1211yescharacteristic
304C23#93011yescharacteristic
305C23C4⋊S411yescharacteristic
306Q8—122no—
307Q8—122no—
308Q8—122no—
309Q8—122no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—31no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4C4⋊S411yescharacteristic
346C2×C4C22×S411yescharacteristic
347C8—21no—
348C8—32no—
349C8—61no—
350C8GL(2,ℤ/4)12yes—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—82no—
355C22—21no—
356C22—21no—
357C22—31no—
358C22—31no—
359C22—31no—
360C22—31no—
361C22—31no—
362C22—31no—
363C22—31no—
364C22—31no—
365C22—31no—
366C22—34no—
367C22—32no—
368C22—62no—
369C22—62no—
370C22—62no—
371C22—62no—
372C22—61no—
373C22—62no—
374C22—62no—
375C22—62no—
376C22—61no—
377C22—61no—
378C22#511611yescharacteristic, complemented
379C22#539812yescomplemented
380C22C2×C4⋊S411yescharacteristic
381C4—21no—
382C4—31no—
383C4—31no—
384C4—61no—
385C4—242no—
386C4—242no—
387C4—242no—
388C4—242no—
389C4#591211yescharacteristic
390C4C2×GL(2,ℤ/4)11yescharacteristic
391C3—41noSylow
392C2—22no—
393C2—31no—
394C2—31no—
395C2—31no—
396C2—31no—
397C2—62no—
398C2C2×#539811yescharacteristic
399C2#8544011yescharacteristic
400C2#8548611yescharacteristic
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC24×C8D3
socleC24C2×D12
prime radicalC23×A4D4
hypercenterC22×C8S4
hypocenterA4#467
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. C2×#5398C2semidirect
4. #73111C2semidirect
5. #73109C2semidirect
82. C22×A4Q16semidirect
190. C24Dic12semidirect
274. A4#467semidirect
378. C22#5116semidirect
379. C22#5398semidirect

Maximal subgroups

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

subgroupindex
C22×C8×A42
#854802
C2×#53982
#731112
#731092
#329313
#51164