Finite Group Databasebeta

← #11714020

Subgroups — #11714020

Order 768.

496 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 61 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×#520⋊C3C211yesmaximal, characteristic
2C2×#5391C211yesmaximal, characteristic, complemented
3#73151C212yesmaximal
4#73064C211yesmaximal, characteristic, complemented
5#73062C212yesmaximal, complemented
6#32963—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8#520⋊C3—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859C2212yes—
12C2×#859C2212yes—
13#5391—22no—
14#4986—41nomaximal
15C2×#520D311yescharacteristic
16C2×#477—31no—
17C2×#474—31no—
18#2628—31no—
19#1863—32no—
20#1862—32no—
21#1852—32no—
22#1805—31no—
23#1804—31no—
24#1798—31no—
25#1788—32no—
26C2×#41⋊C3—21no—
27C2×#41⋊C3—21no—
28C2×#41⋊C3Q812yescomplemented
29C22×SL(2,𝔽3)C2311yescharacteristic
30C2×C3×C4⋊C4—41no—
31C2×C4⋊Dic3—41no—
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)D412yes—
34#859—22no—
35#859—22no—
36C3⋊#95—42no—
37C3⋊#95—41no—
38C3⋊#95—42no—
39C22×#41C2×D311yescharacteristic
40C2×C4⋊Q8—31no—
41#520—22no—
42#520—62no—
43C2×C4×Q8C2×D311yescharacteristic
44C2×C4×Q8C2×D311yescharacteristic
45C2×C4×D4—31no—
46C2×C4×D4—31no—
47C2×#117—31no—
48C22×C4⋊C4—31no—
49#477—62no—
50#477—62no—
51#474—64no—
52C2×#107—32no—
53C2×#107—32no—
54C2×C4⋊C8—31no—
55C2×C4⋊C8—31no—
56C2×#103—32no—
57C2×#103—32no—
58C2×#102—32no—
59C2×#102—32no—
60C2×C22⋊C8—32no—
61#400—61no—
62#399—61no—
63#395—61no—
64#393—61no—
65#389—32no—
66#382—31no—
67#380—32no—
68#340—32no—
69#340—32no—
70#336—32no—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×Dic3—42no—
75C22×Dic3—42no—
76#41⋊C3—22no—
77C2×SL(2,𝔽3)C2×D411yescharacteristic
78C2×SL(2,𝔽3)C2×Q811yescharacteristic
79C2×SL(2,𝔽3)#4111yescharacteristic
80C3×C4⋊C4—82no—
81C4⋊Dic3—82no—
82C2×#41—21no—
83C2×#41—21no—
84C2×#41—32no—
85C2×#41—61no—
86C2×#41—61no—
87C2×#41Dic612yes—
88C22×Q8—61no—
89C22×Q8C22×D311yescharacteristic
90C22×D4—31no—
91C23×C4—31no—
92C23×C4—61no—
93C22×C8—32no—
94C22×C8—32no—
95C4⋊Q8—62no—
96C4⋊Q8—62no—
97C4×Q8—21no—
98C4×Q8—32no—
99C4×Q8—61no—
100C4×Q8#6712yes—
101C4×D4—34no—
102C4×D4—62no—
103C4×D4—62no—
104C4×D4—62no—
105#117—62no—
106#117—62no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4—62no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4S411yescharacteristic
117C2×C22⋊C4—31no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—62no—
120C2×C42—31no—
121C2×C42—31no—
122#107—64no—
123#107—64no—
124C4⋊C8—62no—
125C4⋊C8—62no—
126#103—62no—
127#103—62no—
128#103—62no—
129#103—62no—
130#102—64no—
131#102—64no—
132C22⋊C8—64no—
133#95—61no—
134#95—62no—
135#95—61no—
136#95—62no—
137C23×C3—41no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4—81no—
141C2×C3×C4—81no—
142C2×C3×C4—82no—
143C2×C3×C4—81no—
144C2×C3×C4—82no—
145C2×Dic3—82no—
146C2×Dic3—82no—
147C2×Dic3—82no—
148C2×Dic3—82no—
149C2×Dic3—82no—
150C2×Dic3—82no—
151SL(2,𝔽3)C22⋊Q811yescharacteristic
152C24—31no—
153#41—22no—
154#41—64no—
155#41—62no—
156#41—62no—
157#41—62no—
158C2×Q8—31no—
159C2×Q8—31no—
160C2×Q8—31no—
161C2×Q8—122no—
162C2×Q8—242no—
163C2×Q8C2×Dic611yescharacteristic
164C2×Q8#23511yescharacteristic
165C2×Q8C2×#6711yescharacteristic
166C2×D4—32no—
167C2×D4—32no—
168C2×D4—32no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4—61no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—61no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—61no—
181C22×C4—64no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—242no—
185C22×C4C2×S411yescharacteristic
186C22×C4C2×S411yescharacteristic
187C22×C4C2×S411yescharacteristic
188C2×C8—62no—
189C2×C8—62no—
190C2×C8—62no—
191C2×C8—62no—
192C2×C8—62no—
193C2×C8—62no—
194C4⋊C4—22no—
195C4⋊C4—32no—
196C4⋊C4—32no—
197C4⋊C4—62no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4—62no—
201C4⋊C4—62no—
202C4⋊C4—62no—
203C4⋊C4—62no—
204C4⋊C4—122no—
205C4⋊C4—242no—
206C22⋊C4—62no—
207C22⋊C4—62no—
208C22⋊C4—242no—
209C42—32no—
210C42—32no—
211C42—61no—
212C42—61no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—41no—
218C3×C4—82no—
219C3×C4—82no—
220C3×C4—161no—
221Dic3—162no—
222Dic3—162no—
223C23—32no—
224C23—61no—
225C23—62no—
226C23—62no—
227C23—61no—
228C23C22×S411yescharacteristic
229Q8—31no—
230Q8—31no—
231Q8—31no—
232Q8—61no—
233Q8—244no—
234Q8#92411yescharacteristic
235D4—34no—
236D4—62no—
237D4—62no—
238D4—62no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—32no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—244no—
265C2×C4A4⋊Q812yes—
266C2×C4GL(2,ℤ/4)12yes—
267C8—122no—
268C8—122no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—42no—
272C2×C3—42no—
273C2×C3—41no—
274C22—64no—
275C22—62no—
276C22—121no—
277C22—121no—
278C22—122no—
279C22C2×A4⋊Q811yescharacteristic
280C22#591211yescharacteristic
281C22#592512yes—
282C22#592712yes—
283C22C2×GL(2,ℤ/4)11yescharacteristic
284C4—22no—
285C4—22no—
286C4—31no—
287C4—31no—
288C4—31no—
289C4—31no—
290C4—41no—
291C4—62no—
292C4—121no—
293C4—242no—
294C4—242no—
295C3—41noSylow
296C2—122no—
297C2#8536312yes—
298C2#8537112yes—
299C2#8548011yescharacteristic
300C2#8553011yescharacteristic
301C2#8553511yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#5391C2semidirect
4. #73064C2semidirect
5. #73062C2semidirect
28. C2×#41⋊C3Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#53912
#731512
#730642
#730622
#329633
#49864