Finite Group Databasebeta

← #11713218

Subgroups — #11713218

Order 768.

572 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 59 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
1A4×C22⋊Q8C211yesmaximal, characteristic
2C2×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3#73147C212yesmaximal, complemented
4#73141C211yesmaximal, characteristic
5#73051C212yesmaximal, complemented
6#24003—31nomaximal, Sylow
7C2×Q8×A4C2211yescharacteristic
8C22×C4×A4C2211yescharacteristic
9C22×SL(2,ℤ/4)—21no—
10A4×C4⋊C4—21no—
11A4×C4⋊C4C2211yescharacteristic
12A4×C22⋊C4—21no—
13C4⋊SL(2,ℤ/4)—21no—
14C4⋊SL(2,ℤ/4)C2212yes—
15C2×A4⋊C8C2212yes—
16#5043—41nomaximal
17C22×C22⋊Q8D311yescharacteristic
18C2×#483—31no—
19C2×#483—31no—
20C2×#380—31no—
21#2035—32no—
22#2035—32no—
23#1872—31no—
24#1872—31no—
25#1853—32no—
26#1787—32no—
27#1779—31no—
28C23×A4D411yescharacteristic
29Q8×A4—22no—
30C2×C4×A4—22no—
31C2×C4×A4—21no—
32C2×C4×A4—21no—
33C2×C4×A4D411yescharacteristic
34C2×C4×A4C2311yescharacteristic
35C2×SL(2,ℤ/4)—22no—
36C2×SL(2,ℤ/4)—42no—
37C3×C22⋊Q8—41no—
38C2×C4⋊Dic3—41no—
39A4⋊C8—22no—
40#835—42no—
41#830—41no—
42#808—42no—
43C23×Q8C2×D311yescharacteristic
44C24×C4C2×D311yescharacteristic
45C2×C22⋊Q8—31no—
46C2×C22⋊Q8—31no—
47C2×C22⋊Q8—31no—
48C2×C22⋊Q8—31no—
49C2×C22⋊Q8—61no—
50C2×C22⋊Q8—61no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—31no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4C2×D311yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—61no—
57#483—62no—
58#483—62no—
59#483—61no—
60#483—61no—
61#483—61no—
62#483—61no—
63C2×#106—32no—
64C2×#106—32no—
65C2×#103—32no—
66C2×#103—32no—
67C2×#103—32no—
68C2×#103—32no—
69C2×C22⋊C8—31no—
70C2×C22⋊C8—31no—
71C2×C22⋊C8—32no—
72C2×C22⋊C8—32no—
73#380—32no—
74#380—32no—
75#380—61no—
76#380—61no—
77C2×#95—61no—
78#340—32no—
79#340—32no—
80C22×A4—21no—
81C22×A4—21no—
82C22×A4SD1612yes—
83C22×A4C2×D411yescharacteristic
84C2×C3×Q8—41no—
85C22×C3×C4—41no—
86C22×Dic3—81no—
87C4×A4—21no—
88C4×A4—41no—
89C4×A4—41no—
90C4×A4#4112yes—
91SL(2,ℤ/4)—42no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—81no—
94C3×C22⋊C4—81no—
95C4⋊Dic3—42no—
96C4⋊Dic3—81no—
97C2×C3⋊C8—42no—
98C25#6711yescharacteristic
99C22×Q8—22no—
100C22×Q8—31no—
101C22×Q8—31no—
102C22×Q8—61no—
103C22×Q8—62no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—31no—
108C23×C4—31no—
109C23×C4—31no—
110C23×C4—31no—
111C23×C4—62no—
112C23×C4—61no—
113C23×C4—61no—
114C23×C4#6711yescharacteristic
115C23×C4C22×D311yescharacteristic
116C22×C8—32no—
117C22×C8—32no—
118C22⋊Q8—32no—
119C22⋊Q8—32no—
120C22⋊Q8—31no—
121C22⋊Q8—31no—
122C22⋊Q8—31no—
123C22⋊Q8—61no—
124C22⋊Q8—61no—
125C22⋊Q8—61no—
126C22⋊Q8—62no—
127C22⋊Q8—62no—
128C22⋊Q8S411yescharacteristic
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—32no—
132C2×C4⋊C4—32no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—122no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—62no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4—122no—
152C2×C22⋊C4—122no—
153#106—64no—
154#106—62no—
155#106—62no—
156#103—64no—
157#103—62no—
158#103—62no—
159#103—122no—
160#103—122no—
161C22⋊C8—62no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164C22⋊C8—61no—
165C22⋊C8—62no—
166C22⋊C8—61no—
167C22⋊C8—122no—
168#95—62no—
169#95—62no—
170C23×C3—41no—
171C2×A4—22no—
172C2×A4#12311yescharacteristic
173C2×A4C2×SD1611yescharacteristic
174C2×A4#13711yescharacteristic
175C3×Q8—82no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—82no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×Dic3—82no—
182C2×Dic3—162no—
183C3⋊C8—82no—
184C24—21no—
185C24—21no—
186C24—31no—
187C24—31no—
188C24—32no—
189C24—61no—
190C24—61no—
191C24#21512yes—
192C24C2×#6711yescharacteristic
193C2×Q8—32no—
194C2×Q8—31no—
195C2×Q8—61no—
196C2×Q8—62no—
197C2×Q8—62no—
198C2×Q8—62no—
199C2×Q8—62no—
200C2×Q8—122no—
201C2×Q8—122no—
202C2×Q8C2×S411yescharacteristic
203C22×C4—21no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—32no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—62no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4—62no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—121no—
229C22×C4—121no—
230C22×C4—121no—
231C22×C4—121no—
232C22×C4—122no—
233C22×C4—122no—
234C22×C4—121no—
235C22×C4—122no—
236C22×C4—122no—
237C22×C4—122no—
238C22×C4—121no—
239C22×C4#23712yes—
240C22×C4C2×S411yescharacteristic
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—62no—
246C2×C8—62no—
247C4⋊C4—21no—
248C4⋊C4—32no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—62no—
257C4⋊C4—61no—
258C4⋊C4—62no—
259C4⋊C4—121no—
260C4⋊C4—122no—
261C4⋊C4—121no—
262C4⋊C4C2×S411yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—62no—
267C22⋊C4—61no—
268C22⋊C4—122no—
269C22⋊C4—124no—
270C22⋊C4—122no—
271C22×C3—42no—
272C22×C3—41no—
273C22×C3—81no—
274C22×C3—81no—
275A4#48311yescharacteristic, complemented
276C3×C4—42no—
277C3×C4—81no—
278C3×C4—161no—
279C3×C4—161no—
280Dic3—162no—
281C23—22no—
282C23—31no—
283C23—31no—
284C23—31no—
285C23—31no—
286C23—32no—
287C23—32no—
288C23—32no—
289C23—31no—
290C23—61no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23—62no—
296C23—61no—
297C23—61no—
298C23—62no—
299C23—62no—
300C23—61no—
301C23—61no—
302C23—122no—
303C23#93511yescharacteristic
304C23C2×#21511yescharacteristic
305C23#95111yescharacteristic
306C23GL(2,ℤ/4)11yescharacteristic
307Q8—22no—
308Q8—64no—
309Q8—62no—
310Q8—122no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—31no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—32no—
318C2×C4—31no—
319C2×C4—62no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—122no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—242no—
345C2×C4—244no—
346C2×C4—242no—
347C2×C4GL(2,ℤ/4)11yescharacteristic
348C2×C4C22×S411yescharacteristic
349C8—122no—
350C8—122no—
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—32no—
367C22—62no—
368C22—62no—
369C22—62no—
370C22—62no—
371C22—61no—
372C22—62no—
373C22—62no—
374C22—61no—
375C22—61no—
376C22—121no—
377C22—121no—
378C22#504311yescharacteristic, complemented
379C22#541712yes—
380C22C2×GL(2,ℤ/4)11yescharacteristic
381C4—21no—
382C4—32no—
383C4—41no—
384C4—41no—
385C4—61no—
386C4—121no—
387C4—121no—
388C4—242no—
389C4—242no—
390C4#591412yes—
391C3—41noSylow
392C2—22no—
393C2—31no—
394C2—31no—
395C2—31no—
396C2—31no—
397C2—62no—
398C2#8549111yescharacteristic
399C2C2×#541711yescharacteristic
400C2#8551211yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C22⋊Q8D3
socleC24C2×#67
prime radicalC23×A4D4
hypercenterC22⋊Q8S4
hypocenterA4#483
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #73147C2semidirect
5. #73051C2semidirect
275. A4#483semidirect
378. C22#5043semidirect

Maximal subgroups

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

subgroupindex
A4×C22⋊Q82
C2×C4⋊SL(2,ℤ/4)2
#731472
#731412
#730512
#240033
#50434