Finite Group Databasebeta

← #17507⋊C3 #11709366

Subgroups — #17507⋊C3 #11709366

Order 768.

324 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 49 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
1D4×#41⋊C3C211yesmaximal, characteristic
2C8×#41⋊C3C211yesmaximal, characteristic, complemented
3C4⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#17507C311yesmaximal, characteristic, complemented, Sylow
5C22×#41⋊C3—21no—
6C2×#131⋊C3—21no—
7C4⋊(#41⋊C3)C411yescharacteristic
8D4×SL(2,𝔽3)C411yescharacteristic
9#548⋊C3—21no—
10C4×#41⋊C3C2211yescharacteristic
11C8×SL(2,𝔽3)—21no—
12C3×D4⋊C8—41nomaximal
13D4×#41C2×C311yescharacteristic
14C8×#41C2×C311yescharacteristic, complemented
15#2906C2×C311yescharacteristic, complemented
16#1492—31no—
17#1486—31no—
18#1480—31no—
19C2×D4⋊C8—31no—
20C2×#41⋊C3—21no—
21C2×#41⋊C3—41no—
22C2×#41⋊C3—41no—
23C2×#41⋊C3D411yescharacteristic
24C22×SL(2,𝔽3)—21no—
25C3×C4×D4—41no—
26#131⋊C3—41no—
27C4×SL(2,𝔽3)C2×C411yescharacteristic
28C4×SL(2,𝔽3)D411yescharacteristic
29C3×C4⋊C8—41no—
30C3×C4×C8—41no—
31C22×#41—21no—
32C2×#131—21no—
33#550C3×C411yescharacteristic
34D4×Q8C3×C411yescharacteristic
35#548—21no—
36#547—31no—
37#546—61no—
38D42—31no—
39#532—31no—
40#525—61no—
41C4×#41C22×C311yescharacteristic
42C2×C4×D4—31no—
43C8×Q8—21no—
44C8×D4—61no—
45#431—31no—
46#424—61no—
47#423—31no—
48C2×C4⋊C8—31no—
49C2×C4×C8—31no—
50D4⋊C8—32no—
51D4⋊C8—31no—
52D4⋊C8—31no—
53D4⋊C8—31no—
54D4⋊C8A411yescharacteristic
55C2×C3×D4—41no—
56C22×C3×C4—81no—
57#41⋊C3—41no—
58#41⋊C3D811yescharacteristic
59#41⋊C3SD1611yescharacteristic
60C2×SL(2,𝔽3)—41no—
61C2×SL(2,𝔽3)C22⋊C411yescharacteristic
62C2×C3×C8—81no—
63C2×C3×C8—81no—
64C3×C4⋊C4—41no—
65C3×C22⋊C4—81no—
66C3×C42—41no—
67C2×#41—21no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41—61no—
71C2×#41—61no—
72C2×#41—121no—
73C2×#41—121no—
74C2×#41C3×D411yescharacteristic
75C22×Q8—21no—
76C22×D4—31no—
77C22×D4—61no—
78C23×C4—61no—
79#131—41no—
80#131—121no—
81C2×M16—61no—
82C22×C8—61no—
83C22×C8—61no—
84C4⋊Q8—31no—
85#127—31no—
86#124—61no—
87#123—62no—
88C22⋊Q8—61no—
89C22⋊Q8—61no—
90#121—32no—
91#121—61no—
92#121—61no—
93#121—61no—
94#120—62no—
95C4×Q8C2×C3×C411yescharacteristic
96C4×Q8C3×D411yescharacteristic
97C4×D4—31no—
98C4×D4—31no—
99C4×D4—31no—
100C4×D4—31no—
101C4×D4—31no—
102C4×D4—61no—
103C4×D4—61no—
104C4×D4—61no—
105C4×D4C2×A411yescharacteristic
106#117—31no—
107C2×C4⋊C4—31no—
108C2×C22⋊C4—61no—
109C2×C42—31no—
110C4⋊C8—31no—
111C4⋊C8—61no—
112C4⋊C8—61no—
113C4⋊C8—61no—
114C4⋊C8C2×A411yescharacteristic
115C22⋊C8—61no—
116C4×C8—31no—
117C4×C8—61no—
118C4×C8C2×A411yescharacteristic
119C23×C3—81no—
120C3×D4—42no—
121C3×D4—81no—
122C2×C3×C4—41no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
125C2×C3×C4—81no—
126C2×C3×C4—161no—
127C2×C3×C4—161no—
128SL(2,𝔽3)#10211yescharacteristic
129C3×C8—82no—
130C3×C8—161no—
131C24—61no—
132#41—31no—
133#41—31no—
134#41—41no—
135#41—121no—
136#41—121no—
137#41—121no—
138#41—121no—
139#41—122no—
140#41—121no—
141#41—121no—
142#41—121no—
143#41C3×D811yescharacteristic
144#41C3×SD1611yescharacteristic
145C2×Q8—41no—
146C2×Q8—61no—
147C2×Q8—121no—
148C2×Q8C3×C22⋊C411yescharacteristic
149C2×D4—31no—
150C2×D4—31no—
151C2×D4—61no—
152C2×D4—61no—
153C2×D4—61no—
154C2×D4—62no—
155C2×D4—62no—
156C2×D4—121no—
157C2×D4—121no—
158C2×D4—121no—
159C2×D4—122no—
160C2×D4C4×A411yescharacteristic
161C22×C4—21no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4—61no—
168C22×C4—61no—
169C22×C4—121no—
170C22×C4—121no—
171C22×C4—121no—
172C22×C4—121no—
173M16—121no—
174M16—121no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—61no—
178C2×C8—61no—
179C2×C8—122no—
180C2×C8—121no—
181C2×C8—121no—
182C4⋊C4—31no—
183C4⋊C4—31no—
184C4⋊C4—31no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4C4×A411yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—32no—
190C22⋊C4—61no—
191C22⋊C4—61no—
192C22⋊C4—62no—
193C22⋊C4—61no—
194C42—31no—
195C42—31no—
196C42—31no—
197C42C22×A411yescharacteristic
198C22×C3—41no—
199C22×C3—82no—
200C22×C3—161no—
201C22×C3—161no—
202C3×C4—41no—
203C3×C4—41no—
204C3×C4—41no—
205C3×C4—41no—
206C3×C4—81no—
207C3×C4—161no—
208C23—21no—
209C23—31no—
210C23—61no—
211C23—122no—
212C23—121no—
213C23—121no—
214C23—241no—
215Q8—31no—
216Q8—121no—
217Q8—121no—
218Q8C3×#10211yescharacteristic
219D4—21no—
220D4—31no—
221D4—31no—
222D4—32no—
223D4—61no—
224D4—61no—
225D4—121no—
226D4—121no—
227D4—122no—
228D4—122no—
229D4—122no—
230D4—241no—
231D4#131⋊C312yescomplemented
232C2×C4—21no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—121no—
246C2×C4—121no—
247C2×C4—121no—
248C2×C4—121no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4C2×C4×A411yescharacteristic
255C2×C4D4×A411yescharacteristic
256C2×C4D4×A411yescharacteristic
257C8—22no—
258C8—41no—
259C8—121no—
260C8—121no—
261C2×C3—41no—
262C2×C3—41no—
263C2×C3—41no—
264C2×C3—162no—
265C22—22no—
266C22—31no—
267C22—31no—
268C22—41no—
269C22—41no—
270C22—61no—
271C22—61no—
272C22—121no—
273C22—121no—
274C22—121no—
275C22—242no—
276C22—242no—
277C22A4×C22⋊C411yescharacteristic
278C4—21no—
279C4—31no—
280C4—31no—
281C4—41no—
282C4—61no—
283C4—61no—
284C4—121no—
285C4C2×#131⋊C311yescharacteristic
286C4#568⋊C311yescharacteristic
287C4A4×D811yescharacteristic
288C4A4×SD1611yescharacteristic
289C3—41noSylow
290C2—42no—
291C2—61no—
292C2—61no—
293C2—241no—
294C2#2880⋊C311yescharacteristic
295C2A4×#10211yescharacteristic
296C2#2901⋊C311yescharacteristic
subgroupG/H
centerC2×C4D4×A4
commutatorC4×Q8C2×C3×C4
FrattiniC42C22×A4
Fitting#17507C3
socleC22A4×C22⋊C4
prime radicalD4×#41⋊C3C2
cosocleC4×#41C22×C3
hypercenterD4⋊C8A4
hypocenterQ8C3×#102
LCSG ▷ C4×Q8 ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C8×#41⋊C3C2semidirect
3. C4⋊(#131⋊C3)C2semidirect
4. #17507C3semidirect
14. C8×#41C2×C3semidirect
15. #2906C2×C3semidirect
231. D4#131⋊C3semidirect

Maximal subgroups

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

subgroupindex
D4×#41⋊C32
C8×#41⋊C32
C4⋊(#131⋊C3)2
#175073
C3×D4⋊C84