Finite Group Databasebeta

← C2×#85536 #11714688

Subgroups — C2×#85536 #11714688

Order 768.

1725 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1725 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×2−1+4⋊C3C211yesmaximal, characteristic
2C22×#984C211yesmaximal, characteristic, complemented
3#85536C218yesmaximal, direct product factor
4C2×#5424C211yesmaximal, characteristic, complemented
5C2×#5419C214yesmaximal, complemented
6C2×#3011—31nomaximal, Sylow
7C2×2−1+4⋊C3—24no—
8C2×2−1+4⋊C3C2214yes—
9C22×#41⋊C3C2211yescharacteristic
10C23×SL(2,𝔽3)C2212yes—
11C2×#984—24no—
12C2×#984C2214yescomplemented
13C22×#226C2212yes—
14C2×#936—41nomaximal
15#5424C2214yescomplemented
16#5419C22116yescomplemented
17C2×#859C2212yes—
18C22×2−1+4D311yescharacteristic
19C22×#137—31no—
20C23×Q16—31no—
21C2×#536—31no—
22#3011—38no—
23#3011—38no—
24C2×#452—34no—
25C2×#451—34no—
26C2×#419—31no—
27C2×#416—31no—
28C2×#408—31no—
29C22×C3×D4—41no—
30C22×Dic6—41no—
312−1+4⋊C3—28no—
32C2×#41⋊C3D414yes—
33C2×#41⋊C3C2312yes—
34C22×SL(2,𝔽3)—24no—
35C22×SL(2,𝔽3)C2314yes—
36C22×SL(2,𝔽3)C2311yescharacteristic
37#984—28no—
38C2×#226—24no—
39C2×#226C2314yes—
40C2×#217—44no—
41#936—48no—
42C2×C4×Dic3—41no—
43#859C2314yes—
44C2×2−1+4—24no—
45C2×2−1+4—34no—
46C2×2−1+4—64no—
47C2×2−1+4C2×D314yes—
48C22×#41—31no—
49C22×#41—32no—
50C22×#41C2×D311yescharacteristic
51C23×Q8—31no—
52C23×Q8—31no—
53C23×Q8C2×D312yes—
54C2×#137—34no—
55C2×#137—34no—
56C2×#137—64no—
57C2×#137—64no—
58C22×Q16—34no—
59C22×Q16—38no—
60C22×Q16—32no—
61C22×Q16—68no—
62C22×SD16—32no—
63C22×M16—31no—
64C23×C8—31no—
65#536—38no—
66#536—64no—
67C2×C4⋊Q8—61no—
68C2×#124—61no—
69C2×#123—62no—
70C2×C22⋊Q8—34no—
71C2×#117—31no—
72#452—332no—
73#451—316no—
74#451—316no—
75#419—38no—
76#416—34no—
77#416—34no—
78C2×#103—34no—
79C2×#103—32no—
80C2×#102—32no—
81#408—38no—
82C2×C22⋊C8—34no—
83C24×C3—42no—
84C2×C3×D4—44no—
85C2×C3×D4—84no—
86C22×C3×C4—41no—
87C22×Dic3—42no—
88C22×Dic3—42no—
89C2×Dic6—44no—
90C2×Dic6—84no—
91#41⋊C3C2×D414yes—
92C2×SL(2,𝔽3)—24no—
93C2×SL(2,𝔽3)C2×D412yes—
94C2×SL(2,𝔽3)C2411yescharacteristic
95#226—24no—
96#217—416no—
97C4×Dic3—44no—
982−1+4—28no—
992−1+4—68no—
1002−1+4—616no—
1012−1+4—616no—
102C2×#41—34no—
103C2×#41—34no—
104C2×#41—32no—
105C2×#41—34no—
106C2×#41—38no—
107C2×#41—64no—
108C2×#41—64no—
109C2×#41—64no—
110C2×#41—68no—
111C2×#41—68no—
112C2×#41#6714yes—
113C2×#41C22×D312yes—
114C22×Q8—24no—
115C22×Q8—31no—
116C22×Q8—32no—
117C22×Q8—34no—
118C22×Q8—34no—
119C22×Q8—38no—
120C22×Q8—32no—
121C22×Q8—34no—
122C22×Q8—62no—
123C22×Q8—64no—
124C22×Q8—62no—
125C22×Q8—64no—
126C22×Q8—64no—
127C22×Q8—61no—
128C22×Q8—64no—
129C22×Q8C22×D314yes—
130C22×Q8C22×D311yescharacteristic
131C22×D4—31no—
132C22×D4—32no—
133C22×D4S411yescharacteristic
134C23×C4—31no—
135C23×C4—32no—
136C23×C4—61no—
137C23×C4—61no—
138#137—616no—
139#137—68no—
140#137—68no—
141C2×Q16—316no—
142C2×Q16—34no—
143C2×Q16—34no—
144C2×Q16—616no—
145C2×Q16—632no—
146C2×Q16—64no—
147C2×Q16—64no—
148C2×SD16—38no—
149C2×SD16—68no—
150C2×M16—34no—
151C2×M16—64no—
152C22×C8—32no—
153C22×C8—34no—
154C22×C8—32no—
155C22×C8—64no—
156C4⋊Q8—68no—
157#124—68no—
158#123—616no—
159C22⋊Q8—316no—
160C22⋊Q8—68no—
161#117—34no—
162#117—62no—
163C2×C4⋊C4—32no—
164C2×C4⋊C4—64no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—64no—
167C2×C42—61no—
168#103—34no—
169#103—316no—
170#103—34no—
171#102—38no—
172C22⋊C8—316no—
173C23×C3—44no—
174C23×C3—41no—
175C23×C3—84no—
176C23×C3—88no—
177C3×D4—88no—
178C2×C3×C4—44no—
179C2×C3×C4—42no—
180C2×Dic3—44no—
181C2×Dic3—44no—
182C2×Dic3—84no—
183C2×Dic3—84no—
184Dic6—88no—
185SL(2,𝔽3)C22×D411yescharacteristic
186C24—31no—
187C24C2×S412yes—
188#41—38no—
189#41—34no—
190#41—64no—
191#41—68no—
192#41—68no—
193#41—616no—
194#41—616no—
195#41—616no—
196#41—632no—
197#41C2×#6714yes—
198C2×Q8—24no—
199C2×Q8—34no—
200C2×Q8—31no—
201C2×Q8—34no—
202C2×Q8—32no—
203C2×Q8—38no—
204C2×Q8—316no—
205C2×Q8—34no—
206C2×Q8—62no—
207C2×Q8—68no—
208C2×Q8—616no—
209C2×Q8—62no—
210C2×Q8—64no—
211C2×Q8—64no—
212C2×Q8—64no—
213C2×Q8—68no—
214C2×Q8—68no—
215C2×Q8—68no—
216C2×Q8—64no—
217C2×Q8—64no—
218C2×Q8—64no—
219C2×Q8—68no—
220C2×Q8—128no—
221C2×Q8—124no—
222C2×Q8C2×#6712yes—
223C2×Q8C23×D311yescharacteristic
224C2×D4—24no—
225C2×D4—32no—
226C2×D4—34no—
227C2×D4—38no—
228C2×D4—62no—
229C2×D4—61no—
230C2×D4—68no—
231C2×D4C2×S414yes—
232C22×C4—31no—
233C22×C4—34no—
234C22×C4—34no—
235C22×C4—32no—
236C22×C4—32no—
237C22×C4—34no—
238C22×C4—34no—
239C22×C4—32no—
240C22×C4—64no—
241C22×C4—64no—
242C22×C4—64no—
243C22×C4—68no—
244C22×C4—64no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—124no—
248C22×C4C2×S411yescharacteristic
249Q16—632no—
250Q16—68no—
251Q16—64no—
252Q16—64no—
253SD16—616no—
254M16—68no—
255C2×C8—34no—
256C2×C8—34no—
257C2×C8—64no—
258C2×C8—68no—
259C2×C8—64no—
260C4⋊C4—34no—
261C4⋊C4—62no—
262C4⋊C4—616no—
263C22⋊C4—64no—
264C22⋊C4—616no—
265C42—64no—
266C22×C3—44no—
267C22×C3—42no—
268C22×C3—41no—
269C22×C3—84no—
270C22×C3—88no—
271C22×C3—168no—
272C3×C4—44no—
273Dic3—84no—
274Dic3—84no—
275C23—28no—
276C23—24no—
277C23—34no—
278C23—32no—
279C23—64no—
280C23C22×S414yes—
281C23C22×S411yescharacteristic
282Q8—32no—
283Q8—31no—
284Q8—38no—
285Q8—61no—
286Q8—64no—
287Q8—68no—
288Q8—64no—
289Q8—64no—
290Q8—68no—
291Q8—124no—
292Q8—128no—
293Q8C22×#6711yescharacteristic
294D4—28no—
295D4—34no—
296D4—62no—
297D4—64no—
298D4—616no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—34no—
304C2×C4—34no—
305C2×C4—38no—
306C2×C4—34no—
307C2×C4—64no—
308C2×C4—64no—
309C2×C4—68no—
310C2×C4—616no—
311C2×C4—64no—
312C2×C4—64no—
313C2×C4—64no—
314C2×C4—128no—
315C2×C4—124no—
316C2×C4—124no—
317C2×C4GL(2,ℤ/4)14yes—
318C2×C4C22×S412yes—
319C8—64no—
320C8—64no—
321C2×C3—44no—
322C2×C3—42no—
323C2×C3—41no—
324C2×C3—164no—
325C22—28no—
326C22—24no—
327C22—34no—
328C22—48no—
329C22—68no—
330C22—64no—
331C22#592714yes—
332C22C2×GL(2,ℤ/4)12yes—
333C22C23×S411yescharacteristic
334C4—31no—
335C4—31no—
336C4—32no—
337C4—64no—
338C4—124no—
339C4—124no—
340C4C2×GL(2,ℤ/4)14yes—
341C3—41noSylow
342C2—44no—
343C2—64no—
344C2#8553614yesdirect product factor
345C2C2×#592712yes—
346C2C22×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC23C22×S4
commutator, cosocleC2×SL(2,𝔽3)C24
FrattiniC22C23×S4
FittingC22×2−1+4D3
prime radicalC22×2−1+4⋊C3C2
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)C22×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C22×#984C2semidirect
3. #85536C2direct (N × H)
4. C2×#5424C2semidirect
5. C2×#5419C2semidirect
12. C2×#984C22semidirect
15. #5424C22semidirect
16. #5419C22semidirect
344. C2#85536direct (N × H)

Maximal subgroups

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

subgroupindex
C22×2−1+4⋊C32
C22×#9842
#855362
C2×#54242
C2×#54192
C2×#30113
C2×#9364