Finite Group Databasebeta

← #11714035

Subgroups — #11714035

Order 768.

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

normal subgroups only

This group has 587 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
1C2×A4×C4⋊C4C211yesmaximal, characteristic
2#85480C211yesmaximal, characteristic
3#73141C211yesmaximal, characteristic
4#73052C212yesmaximal, complemented
5#73051C212yesmaximal, complemented
6#32982—31nomaximal, Sylow
7C22×C4×A4—21no—
8C22×C4×A4C2211yescharacteristic
9C2×A4⋊Q8C2211yescharacteristic
10A4×C4⋊C4—21no—
11A4×C4⋊C4C2212yes—
12#5432—21no—
13C4⋊SL(2,ℤ/4)C2211yescharacteristic
14SL(2,ℤ/4)⋊C4—21no—
15C2×A4⋊C8C2212yes—
16#4971—41nomaximal
17C23×C4⋊C4D311yescharacteristic
18C2×#483—31no—
19C2×#483—31no—
20#2848—31no—
21#1872—32no—
22#1872—32no—
23#1853—32no—
24#1853—32no—
25#1823—31no—
26#1823—31no—
27#1779—31no—
28C23×A4D411yescharacteristic
29C2×C4×A4—22no—
30C2×C4×A4—22no—
31C2×C4×A4—42no—
32C2×C4×A4D411yescharacteristic
33C2×C4×A4C2311yescharacteristic
34C2×SL(2,ℤ/4)—21no—
35C2×SL(2,ℤ/4)—21no—
36A4⋊Q8—22no—
37C2×C3×C4⋊C4—41no—
38#924—41no—
39A4⋊C8—22no—
40#830—41no—
41#810—42no—
42#808—42no—
43C24×C4—21no—
44C24×C4C2×D311yescharacteristic
45C2×C22⋊Q8—31no—
46C2×C22⋊Q8—31no—
47C2×C22⋊Q8—31no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—31no—
50C22×C4⋊C4—31no—
51C22×C4⋊C4—32no—
52C22×C4⋊C4—61no—
53C22×C4⋊C4—61no—
54C22×C4⋊C4C2×D312yes—
55#483—62no—
56#483—62no—
57#483—62no—
58#483—61no—
59#483—61no—
60C2×#106—32no—
61C2×#106—32no—
62C2×#103—32no—
63C2×#103—32no—
64C2×C22⋊C8—31no—
65C2×C22⋊C8—31no—
66C2×C22⋊C8—32no—
67C2×C22⋊C8—32no—
68#385—61no—
69#385—61no—
70#385—61no—
71#380—31no—
72#380—31no—
73#380—31no—
74#379—61no—
75#340—32no—
76#340—32no—
77#340—32no—
78#340—32no—
79C22×A4—21no—
80C22×A4—21no—
81C22×A4SD1612yes—
82C22×A4C2×D411yescharacteristic
83C22×C3×C4—41no—
84C22×C3×C4—81no—
85C2×Dic6—41no—
86C4×A4—21no—
87C4×A4—42no—
88C4×A4#4112yes—
89SL(2,ℤ/4)—41no—
90SL(2,ℤ/4)—41no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—81no—
93#217—81no—
94C4⋊Dic3—41no—
95Dic3⋊C4—81no—
96C2×C3⋊C8—42no—
97C25#6711yescharacteristic
98C22×Q8—31no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—31no—
102C23×C4—31no—
103C23×C4—31no—
104C23×C4—31no—
105C23×C4—42no—
106C23×C4—61no—
107C23×C4—62no—
108C23×C4—62no—
109C23×C4—61no—
110C23×C4—122no—
111C23×C4#6711yescharacteristic
112C23×C4C22×D311yescharacteristic
113C22×C8—32no—
114C22×C8—32no—
115C22⋊Q8—62no—
116C22⋊Q8—64no—
117C22⋊Q8—61no—
118C22⋊Q8—61no—
119C22⋊Q8—61no—
120C22⋊Q8—62no—
121C22⋊Q8—61no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—32no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—64no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—62no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4S411yescharacteristic
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149#106—64no—
150#106—62no—
151#106—62no—
152#103—64no—
153#103—62no—
154#103—62no—
155C22⋊C8—62no—
156C22⋊C8—64no—
157C22⋊C8—61no—
158C22⋊C8—61no—
159C22⋊C8—61no—
160C22⋊C8—62no—
161C22⋊C8—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165C23×C3—41no—
166C2×A4—22no—
167C2×A4#12311yescharacteristic
168C2×A4C2×SD1611yescharacteristic
169C2×A4#13711yescharacteristic
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—82no—
173C2×C3×C4—82no—
174C2×C3×C4—162no—
175C2×Dic3—81no—
176C2×Dic3—81no—
177Dic6—82no—
178C3⋊C8—82no—
179C24—21no—
180C24—21no—
181C24—31no—
182C24—31no—
183C24—32no—
184C24—61no—
185C24—61no—
186C24#21412yes—
187C24C2×#6711yescharacteristic
188C2×Q8—61no—
189C2×Q8—62no—
190C2×Q8—62no—
191C2×Q8—61no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—32no—
198C22×C4—34no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—42no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—62no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—62no—
211C22×C4—62no—
212C22×C4—62no—
213C22×C4—62no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4—61no—
217C22×C4—62no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—122no—
223C22×C4—122no—
224C22×C4—124no—
225C22×C4—124no—
226C22×C4—122no—
227C22×C4—122no—
228C22×C4—122no—
229C22×C4#23512yes—
230C22×C4C2×S411yescharacteristic
231C2×C8—62no—
232C2×C8—62no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—62no—
236C2×C8—62no—
237C4⋊C4—21no—
238C4⋊C4—34no—
239C4⋊C4—32no—
240C4⋊C4—62no—
241C4⋊C4—62no—
242C4⋊C4—62no—
243C4⋊C4—61no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4C2×S412yes—
253C22⋊C4—121no—
254C22⋊C4—122no—
255C22⋊C4—122no—
256C22⋊C4—121no—
257C22⋊C4—122no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C22⋊C4—121no—
261C22⋊C4—121no—
262C22×C3—42no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3—81no—
266A4#48311yescharacteristic, complemented
267C3×C4—42no—
268C3×C4—81no—
269C3×C4—162no—
270Dic3—161no—
271Dic3—161no—
272C23—22no—
273C23—31no—
274C23—31no—
275C23—31no—
276C23—32no—
277C23—31no—
278C23—32no—
279C23—32no—
280C23—32no—
281C23—32no—
282C23—31no—
283C23—61no—
284C23—61no—
285C23—61no—
286C23—61no—
287C23—62no—
288C23—61no—
289C23—61no—
290C23—62no—
291C23—62no—
292C23—61no—
293C23—64no—
294C23#92911yescharacteristic
295C23C2×#21411yescharacteristic
296C23#94211yescharacteristic
297C23GL(2,ℤ/4)11yescharacteristic
298Q8—122no—
299Q8—122no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—32no—
307C2×C4—31no—
308C2×C4—42no—
309C2×C4—62no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—62no—
314C2×C4—64no—
315C2×C4—62no—
316C2×C4—61no—
317C2×C4—62no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—124no—
321C2×C4—124no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4GL(2,ℤ/4)11yescharacteristic
338C2×C4C22×S411yescharacteristic
339C8—122no—
340C8—122no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—82no—
345C22—21no—
346C22—21no—
347C22—31no—
348C22—31no—
349C22—31no—
350C22—31no—
351C22—31no—
352C22—31no—
353C22—31no—
354C22—31no—
355C22—31no—
356C22—34no—
357C22—32no—
358C22—62no—
359C22—62no—
360C22—62no—
361C22—62no—
362C22—61no—
363C22—62no—
364C22—62no—
365C22—62no—
366C22—61no—
367C22—61no—
368C22#497111yescharacteristic, complemented
369C22#541412yes—
370C22C2×GL(2,ℤ/4)11yescharacteristic
371C4—21no—
372C4—32no—
373C4—42no—
374C4—61no—
375C4—122no—
376C4—241no—
377C4—241no—
378C4—241no—
379C4—241no—
380C4#591212yes—
381C3—41noSylow
382C2—22no—
383C2—31no—
384C2—31no—
385C2—31no—
386C2—31no—
387C2—62no—
388C2#8548511yescharacteristic
389C2C2×#541411yescharacteristic
390C2#8549911yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC23×C4⋊C4D3
socleC24C2×#67
prime radicalC23×A4D4
hypercenterC2×C4⋊C4S4
hypocenterA4#483
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #73052C2semidirect
5. #73051C2semidirect
266. A4#483semidirect
368. C22#4971semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C4⋊C42
#854802
#731412
#730522
#730512
#329823
#49714