Finite Group Databasebeta

← #10944220

Subgroups — #10944220

Order 768.

1482 conjugacy classes of subgroups; only the 247 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 247 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
1C3×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C2×#5243C212yesmaximal
3#82373C213yesmaximal, complemented
4#71655C216yesmaximal, complemented
5#71619C213yesmaximal, complemented
6#32927—31nomaximal, Sylow
7C2×C3×H(5,𝔽2)C2211yescharacteristic
8C3×#520—23no—
9C3×#518—23no—
10C3×#518C2213yes—
11C3×C4×#41—23no—
12C2×C3×C4×D4—23no—
13C2×C3×C4×D4C2213yes—
14C2×#935C2216yes—
15C22×C4⋊Dic3C2211yescharacteristic, complemented
16#5243—28no—
17C2×#834C2216yes—
18#5212—23no—
19#5212C2213yescomplemented
20#4981—23no—
21#4953—23no—
22#4943C2216yescomplemented
23#4928—23no—
24#4526C2216yescomplemented
25C4×H(5,𝔽2)D311yescharacteristic
26C2×#458—32no—
27#2486—33no—
28#1890—36no—
29#1799—33no—
30C3×H(5,𝔽2)—26no—
31C3×H(5,𝔽2)D414yes—
32C2×C3×#41—23no—
33C22×C3×D4—23no—
34C22×C3×D4C2313yes—
35C23×C3×C4—21no—
36C23×C3×C4D413yes—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3C2312yes—
39C3×C4×Q8—21no—
40C3×C4×Q8—23no—
41C3×C4×D4—23no—
42C3×C4×D4—26no—
43C3×C4×D4—23no—
44C3×C4×D4—26no—
45C3×C4×D4—26no—
46C3×C4×D4—26no—
47C3×C4×D4—26no—
48C3×#117—26no—
49C3×#117—23no—
50C2×C3×C4⋊C4—23no—
51C2×C3×C4⋊C4D413yes—
52C2×C3×C22⋊C4—23no—
53C2×C3×C22⋊C4—23no—
54C2×C3×C22⋊C4D413yes—
55C2×C3×C22⋊C4C2313yes—
56C2×C3×C42—23no—
57C2×C3×C42D413yes—
58C2×#217—26no—
59C2×#217C2316yes—
60#935—212no—
61#935—212no—
62C2×C4⋊Dic3—23no—
63C2×C4⋊Dic3—23no—
64C2×C4⋊Dic3—23no—
65C2×Dic3⋊C4—26no—
66#834—212no—
67C3⋊#95—26no—
68C3⋊#95—26no—
69C3⋊#95—23no—
70C2×H(5,𝔽2)C2×D311yescharacteristic
71C2×#123—36no—
72#520—23no—
73#518—23no—
74#518C2×D313yes—
75C4×#41—23no—
76C2×C4×D4—23no—
77C2×C4×D4C2×D313yes—
78C22×C4⋊C4—31no—
79#458—68no—
80C2×#99—36no—
81#398—63no—
82#397—63no—
83#389—63no—
84#381—36no—
85#380—33no—
86#380—63no—
87#342—36no—
88C24×C3—21no—
89C24×C3C2×D413yes—
90C24×C3C2411yescharacteristic
91C3×#41—212no—
92C3×#41—212no—
93C2×C3×Q8—21no—
94C2×C3×D4—23no—
95C2×C3×D4—212no—
96C2×C3×D4—212no—
97C2×C3×D4—26no—
98C2×C3×D4—26no—
99C2×C3×D4C2×D416yes—
100C2×C3×D4#4116yes—
101C22×C3×C4—23no—
102C22×C3×C4—23no—
103C22×C3×C4—23no—
104C22×C3×C4—23no—
105C22×C3×C4—23no—
106C22×C3×C4—42no—
107C22×C3×C4—46no—
108C22×C3×C4—82no—
109C22×C3×C4C2×D413yes—
110C22×C3×C4C2×D413yes—
111C22×C3×C4C2×D413yes—
112C22×C3×C4C2×D413yes—
113C22×C3×C4C2×D413yes—
114C22×Dic3—26no—
115C22×Dic3—26no—
116C22×Dic3—22no—
117C22×Dic3—26no—
118C3×C4⋊C4—23no—
119C3×C4⋊C4—23no—
120C3×C4⋊C4—26no—
121C3×C22⋊C4—26no—
122C3×C22⋊C4—26no—
123C3×C22⋊C4—26no—
124C3×C42—26no—
125C3×C42—23no—
126C3×C42—23no—
127#217—26no—
128#217—26no—
129#217—412no—
130C4⋊Dic3—46no—
131C4⋊Dic3—43no—
132C4⋊Dic3—41no—
133Dic3⋊C4—412no—
134H(5,𝔽2)—26no—
135H(5,𝔽2)D1214yes—
136C2×#41—23no—
137C22×D4—23no—
138C22×D4C22×D313yes—
139C23×C4—21no—
140C23×C4—32no—
141C23×C4#6713yes—
142C23×C4C22×D311yescharacteristic
143#123—612no—
144#123—612no—
145C4×Q8—21no—
146C4×Q8—23no—
147C4×D4—23no—
148C4×D4—26no—
149C4×D4—23no—
150C4×D4—26no—
151C4×D4—26no—
152C4×D4—26no—
153C4×D4—26no—
154#117—26no—
155#117—23no—
156C2×C4⋊C4—23no—
157C2×C4⋊C4—63no—
158C2×C4⋊C4—63no—
159C2×C4⋊C4—66no—
160C2×C4⋊C4—63no—
161C2×C4⋊C4#6713yes—
162C2×C22⋊C4—23no—
163C2×C22⋊C4—23no—
164C2×C22⋊C4—36no—
165C2×C22⋊C4—66no—
166C2×C22⋊C4#6713yes—
167C2×C22⋊C4C22×D313yes—
168C2×C42—23no—
169C2×C42#6713yes—
170#99—612no—
171#95—63no—
172#95—66no—
173#95—66no—
174C23×C3—24no—
175C23×C3—23no—
176C23×C3—23no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—43no—
180C23×C3—43no—
181C23×C3—81no—
182C23×C3—81no—
183C23×C3#12116yes—
184C23×C3C22×D411yescharacteristic
185C23×C3C22×D413yes—
186C23×C3C2×#4113yes—
187C3×Q8—24no—
188C3×D4—212no—
189C3×D4—212no—
190C3×D4—212no—
191C2×C3×C4—212no—
192C2×C3×C4—23no—
193C2×C3×C4—23no—
194C2×C3×C4—26no—
195C2×C3×C4—23no—
196C2×C3×C4—23no—
197C2×C3×C4—43no—
198C2×C3×C4—46no—
199C2×C3×C4—43no—
200C2×C3×C4—43no—
201C2×C3×C4—83no—
202C2×C3×C4—83no—
203C2×C3×C4—83no—
204C2×C3×C4#12013yes—
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12116yes—
207C2×Dic3—46no—
208C2×Dic3—46no—
209C2×Dic3—42no—
210C2×Dic3—46no—
211C2×Dic3—46no—
212C2×Dic3—46no—
213C24—21no—
214C24C2×#6713yes—
215C24C23×D311yescharacteristic
216#41—212no—
217#41—212no—
218C2×Q8—21no—
219C2×D4—23no—
220C2×D4—212no—
221C2×D4—212no—
222C2×D4—26no—
223C2×D4—26no—
224C2×D4C2×D1216yes—
225C2×D4#23516yes—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—42no—
232C22×C4—46no—
233C22×C4—62no—
234C22×C4—66no—
235C22×C4—66no—
236C22×C4—66no—
237C22×C4—82no—
238C22×C4C2×#6713yes—
239C22×C4C2×#6713yes—
240C22×C4C2×#6713yes—
241C22×C4C2×#6713yes—
242C22×C4C2×#6713yes—
243C4⋊C4—23no—
244C4⋊C4—23no—
245C4⋊C4—26no—
246C4⋊C4—126no—
247C4⋊C4—1212no—
248C4⋊C4—123no—
249C4⋊C4—121no—
250C22⋊C4—26no—
251C22⋊C4—26no—
252C22⋊C4—26no—
253C22⋊C4—66no—
254C22⋊C4—66no—
255C22⋊C4—1212no—
256C42—26no—
257C42—23no—
258C42—23no—
259C22×C3—23no—
260C22×C3—23no—
261C22×C3—26no—
262C22×C3—41no—
263C22×C3—43no—
264C22×C3—46no—
265C22×C3—43no—
266C22×C3—43no—
267C22×C3—83no—
268C22×C3—84no—
269C22×C3C2×#12011yescharacteristic
270C22×C3C2×#12113yes—
271C22×C3#52513yes—
272C3×C4—26no—
273C3×C4—23no—
274C3×C4—43no—
275C3×C4—43no—
276C3×C4#45812yescomplemented
277Dic3—86no—
278Dic3—82no—
279C23—24no—
280C23—23no—
281C23—23no—
282C23—41no—
283C23—41no—
284C23—43no—
285C23—43no—
286C23—81no—
287C23—81no—
288C23#93016yes—
289C23C22×D1211yescharacteristic
290C23C2×#23513yes—
291C23C22×#6713yes—
292Q8—24no—
293D4—212no—
294D4—212no—
295D4—212no—
296C2×C4—212no—
297C2×C4—23no—
298C2×C4—23no—
299C2×C4—26no—
300C2×C4—23no—
301C2×C4—23no—
302C2×C4—43no—
303C2×C4—46no—
304C2×C4—43no—
305C2×C4—43no—
306C2×C4—83no—
307C2×C4—83no—
308C2×C4—83no—
309C2×C4—126no—
310C2×C4—126no—
311C2×C4—126no—
312C2×C4—126no—
313C2×C4—126no—
314C2×C4—122no—
315C2×C4#93016yes—
316C2×C4#95313yes—
317C2×C4#95311yescharacteristic
318C2×C3—23no—
319C2×C3—23no—
320C2×C3—46no—
321C2×C3#284711yescharacteristic
322C2×C3C2×#45811yescharacteristic
323C2×C3#302811yescharacteristic
324C22—23no—
325C22—23no—
326C22—26no—
327C22—41no—
328C22—43no—
329C22—46no—
330C22—43no—
331C22—43no—
332C22—83no—
333C22—84no—
334C22C2×#93013yes—
335C22#579113yes—
336C22C2×#95311yescharacteristic
337C4—26no—
338C4—23no—
339C4—43no—
340C4—43no—
341C4—246no—
342C4—242no—
343C4#524312yescomplemented
344C3#3292711yescharacteristic, complemented, Sylow
345C2—23no—
346C2—23no—
347C2—46no—
348C2#8403211yescharacteristic
349C2C2×#524311yescharacteristic
350C2#8412411yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×C4×H(5,𝔽2)C2
socleC22×C3C2×#120
prime radicalC2×C3×H(5,𝔽2)C22
hypercenterC4×H(5,𝔽2)D3
hypocenterC3#32927
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
3. #82373C2semidirect
4. #71655C2semidirect
5. #71619C2semidirect
15. C22×C4⋊Dic3C22semidirect
19. #5212C22semidirect
22. #4943C22semidirect
24. #4526C22semidirect
276. C3×C4#458semidirect
343. C4#5243semidirect
344. C3#32927semidirect

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C2×#52432
#823732
#716552
#716192
#329273