Finite Group Databasebeta

← C2×#73105 #11712647

Subgroups — C2×#73105 #11712647

Order 768.

585 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 101 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
2C2×#5409C212yesmaximal
3#73105C214yesmaximal, direct product factor
4C2×#1810—31nomaximal, Sylow
5C22×C4×A4C2211yescharacteristic
6C2×C8×A4C412yes—
7C2×C8×A4C412yes—
8C2×C8×A4C2212yes—
9#5409—22no—
10#5409C2214yes—
11C2×A4⋊C8—22no—
12C2×#819—41nomaximal
13C24×C8D311yescharacteristic
14C22×#108—31no—
15C22×#108—31no—
16C2×#407—32no—
17#1810—34no—
18#1810—34no—
19C2×#341—32no—
20C23×A4Q811yescharacteristic
21C2×C4×A4C2×C411yescharacteristic
22C2×C4×A4C2×C411yescharacteristic
23C2×C4×A4D411yescharacteristic
24C2×C4×A4D411yescharacteristic
25C2×C4×A4Q811yescharacteristic
26C2×C4×A4C2311yescharacteristic
27C22×C3×C8—41no—
28C2×#208—42no—
29C8×A4C2×C414yes—
30A4⋊C8—24no—
31#819—44no—
32C24×C4C2×D311yescharacteristic
33C22×M16—32no—
34C22×M16—32no—
35C23×C8—31no—
36C23×C8—32no—
37C23×C8—31no—
38C23×C8—32no—
39C23×C8—32no—
40C23×C8Dic312yes—
41C23×C8Dic312yes—
42C23×C8C2×D312yes—
43C2×#108—34no—
44C2×#108—34no—
45C2×#108—61no—
46C2×#108—62no—
47C2×#108—62no—
48C2×#108—62no—
49C2×#108—61no—
50#407—34no—
51#407—34no—
52#407—62no—
53#407—62no—
54C2×C22⋊C8—62no—
55C2×C22⋊C8—62no—
56#341—38no—
57C22×A4—22no—
58C22×A4C4⋊C411yescharacteristic
59C22×A4C4⋊C411yescharacteristic
60C22×A4C2×Q811yescharacteristic
61C22×C3×C4—41no—
62C4×A4C4⋊C411yescharacteristic
63C4×A4C4⋊C411yescharacteristic
64C4×A4C22×C411yescharacteristic
65C4×A4C2×D411yescharacteristic
66C2×C3×C8—42no—
67C2×C3×C8—42no—
68C2×C3×C8—42no—
69#208—44no—
70#208—82no—
71C2×C3⋊C8—82no—
72C25Dic611yescharacteristic
73C23×C4—31no—
74C23×C4—31no—
75C23×C4—31no—
76C23×C4—31no—
77C23×C4—31no—
78C23×C4—31no—
79C23×C4—31no—
80C23×C4—31no—
81C23×C4Dic611yescharacteristic
82C23×C4D1211yescharacteristic
83C23×C4D1211yescharacteristic
84C23×C4C2×Dic311yescharacteristic
85C23×C4C2×Dic311yescharacteristic
86C23×C4C22×D311yescharacteristic
87C2×M16—34no—
88C2×M16—34no—
89C2×M16—62no—
90C2×M16—62no—
91C2×M16—62no—
92C2×M16—62no—
93C2×M16—62no—
94C2×M16—62no—
95C2×M16—62no—
96C2×M16—62no—
97C2×M16—62no—
98C2×M16—62no—
99C22×C8—31no—
100C22×C8—32no—
101C22×C8—32no—
102C22×C8—32no—
103C22×C8—32no—
104C22×C8—32no—
105C22×C8—32no—
106C22×C8—32no—
107C22×C8—32no—
108C22×C8—34no—
109C22×C8—34no—
110C22×C8—34no—
111C22×C8—32no—
112C22×C8—64no—
113C22×C8—61no—
114C22×C8—61no—
115C22×C8—61no—
116C22×C8—62no—
117C22×C8—62no—
118C22×C8C2×Dic314yes—
119C22×C8S411yescharacteristic
120#108—64no—
121#108—68no—
122#108—64no—
123C22⋊C8—64no—
124C22⋊C8—64no—
125C22⋊C8—64no—
126C22⋊C8—64no—
127C23×C3—41no—
128C2×A4—21no—
129C2×A4—21no—
130C2×A4#10812yescomplemented
131C2×A4C2×C4⋊C411yescharacteristic
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
138C3×C8—44no—
139C3⋊C8—84no—
140C24—22no—
141C24—31no—
142C24—31no—
143C24—31no—
144C24—31no—
145C24—62no—
146C24C4⋊Dic311yescharacteristic
147C24C4⋊Dic311yescharacteristic
148C24C2×Dic611yescharacteristic
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—61no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4—61no—
186C22×C4—61no—
187C22×C4C4⋊Dic311yescharacteristic
188C22×C4C4⋊Dic311yescharacteristic
189C22×C4C2×D1211yescharacteristic
190C22×C4C22×Dic311yescharacteristic
191C22×C4C2×S411yescharacteristic
192M16—64no—
193M16—64no—
194M16—64no—
195M16—64no—
196M16—122no—
197M16—122no—
198M16—122no—
199M16—122no—
200C2×C8—32no—
201C2×C8—32no—
202C2×C8—32no—
203C2×C8—34no—
204C2×C8—34no—
205C2×C8—34no—
206C2×C8—34no—
207C2×C8—34no—
208C2×C8—62no—
209C2×C8—62no—
210C2×C8—62no—
211C2×C8—64no—
212C2×C8—64no—
213C2×C8—122no—
214C2×C8—122no—
215C2×C8—122no—
216C2×C8—122no—
217C2×C8SL(2,ℤ/4)12yes—
218C2×C8SL(2,ℤ/4)12yes—
219C2×C8C2×S412yes—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—82no—
224A4C2×#10811yescharacteristic, complemented
225C3×C4—41no—
226C3×C4—41no—
227C3×C4—41no—
228C3×C4—41no—
229C23—21no—
230C23—21no—
231C23—31no—
232C23—31no—
233C23—31no—
234C23—31no—
235C23—32no—
236C23—34no—
237C23—31no—
238C23—31no—
239C23—31no—
240C23—31no—
241C23—31no—
242C23—62no—
243C23—62no—
244C23—61no—
245C23—61no—
246C23—61no—
247C23—62no—
248C23—62no—
249C23—62no—
250C23—62no—
251C23—62no—
252C23#81912yescomplemented
253C23C2×C4⋊Dic311yescharacteristic
254C23A4⋊Q811yescharacteristic
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4A4⋊Q811yescharacteristic
288C2×C4C4⋊S411yescharacteristic
289C2×C4C4⋊S411yescharacteristic
290C2×C4C2×SL(2,ℤ/4)11yescharacteristic
291C2×C4C2×SL(2,ℤ/4)11yescharacteristic
292C2×C4C22×S411yescharacteristic
293C8—34no—
294C8—124no—
295C8—124no—
296C8C2×SL(2,ℤ/4)14yes—
297C2×C3—42no—
298C2×C3—41no—
299C2×C3—81no—
300C2×C3—81no—
301C22—22no—
302C22—32no—
303C22—31no—
304C22—32no—
305C22—32no—
306C22—31no—
307C22—31no—
308C22—32no—
309C22—32no—
310C22—31no—
311C22—31no—
312C22—61no—
313C22—61no—
314C22—61no—
315C22—62no—
316C22—61no—
317C22—62no—
318C22—61no—
319C22—61no—
320C22—61no—
321C22—64no—
322C22—62no—
323C22C2×#81911yescharacteristic, complemented
324C22C4⋊SL(2,ℤ/4)11yescharacteristic
325C22C4⋊SL(2,ℤ/4)11yescharacteristic
326C22C2×A4⋊Q811yescharacteristic
327C4—31no—
328C4—31no—
329C4—31no—
330C4—31no—
331C4C4⋊SL(2,ℤ/4)11yescharacteristic
332C4C4⋊SL(2,ℤ/4)11yescharacteristic
333C4C2×C4⋊S411yescharacteristic
334C4C22×SL(2,ℤ/4)11yescharacteristic
335C3—41noSylow
336C2—21no—
337C2—21no—
338C2—31no—
339C2—31no—
340C2—32no—
341C2—61no—
342C2—61no—
343C2#7310512yesdirect product factor
344C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C4C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC24×C8D3
socleC24C4⋊Dic3
prime radicalC23×A4Q8
cosocleC2×C4×A4C23
hypercenterC22×C8S4
hypocenterA4C2×#108
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. #73105C2direct (N × H)
130. C2×A4#108semidirect
224. A4C2×#108semidirect
252. C23#819semidirect
323. C22C2×#819semidirect
343. C2#73105direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C8×A42
C2×#54092
#731052
C2×#18103
C2×#8194