Finite Group Databasebeta

← #11714010

Subgroups — #11714010

Order 768.

635 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic, complemented
3#73148C212yesmaximal
4C4⋊#859C211yesmaximal, characteristic, complemented
5#73059C212yesmaximal, complemented
6#32954—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C2×#924—41nomaximal
9C2×C4×SL(2,𝔽3)—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2212yes—
13C2×#859C2212yes—
14#5386—22no—
15C22×C4×Q8D311yescharacteristic
16C2×#475—31no—
17C2×#475—31no—
18#2857—31no—
19#1865—32no—
20#1865—32no—
21#1853—32no—
22#1803—31no—
23#1803—31no—
24#1797—31no—
25#1787—32no—
26C23×C3×C4—41no—
27C22×Dic6—41no—
28C22×SL(2,𝔽3)—21no—
29C22×SL(2,𝔽3)—21no—
30C22×SL(2,𝔽3)Q812yescomplemented
31C22×SL(2,𝔽3)C2311yescharacteristic
32C2×#217—42no—
33C2×C4⋊Dic3—41no—
34#924—48no—
35C2×Dic3⋊C4—42no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)D412yes—
38#859—22no—
39#859—22no—
40C23×Q8C2×D311yescharacteristic
41C2×C4⋊Q8—31no—
42C2×C22⋊Q8—61no—
43C2×C4×Q8—22no—
44C2×C4×Q8—31no—
45C2×C4×Q8—31no—
46C2×C4×Q8—62no—
47C2×C4×Q8C2×D311yescharacteristic
48C2×C4×Q8C2×D311yescharacteristic
49C22×C4⋊C4—31no—
50C22×C42—31no—
51#475—62no—
52#475—64no—
53#475—62no—
54C2×#106—32no—
55C2×#106—32no—
56C2×C4⋊C8—31no—
57C2×C4⋊C8—31no—
58C2×#103—32no—
59C2×#103—32no—
60C2×#103—32no—
61C2×#103—32no—
62C2×C22⋊C8—32no—
63#391—61no—
64#389—32no—
65#389—61no—
66#384—31no—
67#380—32no—
68#380—61no—
69#340—32no—
70#340—32no—
71#336—32no—
72C24×C3—41no—
73C22×C3×C4—44no—
74C22×C3×C4—41no—
75C22×C3×C4—41no—
76C22×C3×C4—82no—
77C22×C3×C4—82no—
78C22×Dic3—42no—
79C22×Dic3—42no—
80C2×Dic6—44no—
81C2×Dic6—82no—
82C2×Dic6—82no—
83C2×SL(2,𝔽3)—22no—
84C2×SL(2,𝔽3)C2×D411yescharacteristic
85C2×SL(2,𝔽3)C2×Q811yescharacteristic
86C2×SL(2,𝔽3)#4111yescharacteristic
87#217—48no—
88C4⋊Dic3—44no—
89Dic3⋊C4—48no—
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—31no—
93C22×Q8—32no—
94C22×Q8—61no—
95C22×Q8—61no—
96C22×Q8—61no—
97C22×Q8Dic612yescomplemented
98C22×Q8C22×D311yescharacteristic
99C23×C4—31no—
100C23×C4—31no—
101C23×C4S411yescharacteristic
102C22×C8—32no—
103C22×C8—32no—
104C4⋊Q8—62no—
105C4⋊Q8—62no—
106C22⋊Q8—124no—
107C4×Q8—21no—
108C4×Q8—32no—
109C4×Q8—34no—
110C4×Q8—62no—
111C4×Q8—62no—
112C4×Q8—61no—
113C4×Q8—62no—
114C4×Q8#6712yes—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C42—31no—
128C2×C42—31no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—62no—
132C2×C42—62no—
133#106—64no—
134#106—64no—
135C4⋊C8—62no—
136C4⋊C8—62no—
137#103—62no—
138#103—64no—
139#103—62no—
140#103—64no—
141#103—62no—
142#103—62no—
143C22⋊C8—64no—
144#95—62no—
145#95—62no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3—41no—
150C23×C3—81no—
151C23×C3—81no—
152C23×C3—81no—
153C23×C3—81no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—88no—
159C2×C3×C4—81no—
160C2×C3×C4—81no—
161C2×Dic3—44no—
162C2×Dic3—44no—
163C2×Dic3—82no—
164C2×Dic3—82no—
165C2×Dic3—82no—
166C2×Dic3—82no—
167Dic6—88no—
168SL(2,𝔽3)C22⋊Q811yescharacteristic, complemented
169C24C2×S411yescharacteristic
170C2×Q8—22no—
171C2×Q8—31no—
172C2×Q8—31no—
173C2×Q8—32no—
174C2×Q8—31no—
175C2×Q8—32no—
176C2×Q8—32no—
177C2×Q8—61no—
178C2×Q8—61no—
179C2×Q8—61no—
180C2×Q8—64no—
181C2×Q8—62no—
182C2×Q8—62no—
183C2×Q8—62no—
184C2×Q8—122no—
185C2×Q8—122no—
186C2×Q8—122no—
187C2×Q8C2×Dic611yescharacteristic
188C2×Q8#23511yescharacteristic
189C2×Q8C2×#6711yescharacteristic
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—31no—
193C22×C4—32no—
194C22×C4—32no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4—61no—
202C22×C4—62no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—121no—
207C22×C4GL(2,𝔽3)14yes—
208C22×C4C2×S411yescharacteristic
209C22×C4C2×S411yescharacteristic
210C2×C8—62no—
211C2×C8—62no—
212C2×C8—62no—
213C2×C8—62no—
214C2×C8—62no—
215C2×C8—62no—
216C4⋊C4—32no—
217C4⋊C4—32no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—64no—
221C4⋊C4—62no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C4⋊C4—62no—
225C4⋊C4—122no—
226C4⋊C4—122no—
227C4⋊C4—124no—
228C4⋊C4—122no—
229C22⋊C4—124no—
230C42—32no—
231C42—32no—
232C42—61no—
233C42—61no—
234C42—62no—
235C42—62no—
236C22×C3—44no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—82no—
245C22×C3—82no—
246C22×C3—82no—
247C22×C3—82no—
248C22×C3—82no—
249C22×C3—82no—
250C3×C4—44no—
251C3×C4—82no—
252Dic3—84no—
253Dic3—84no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23A4⋊Q812yes—
259C23C2×GL(2,𝔽3)12yes—
260C23#98412yes—
261C23C22×S411yescharacteristic
262Q8—31no—
263Q8—31no—
264Q8—31no—
265Q8—34no—
266Q8—61no—
267Q8—62no—
268Q8—62no—
269Q8—62no—
270Q8—244no—
271Q8#92411yescharacteristic, complemented
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—28no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—124no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4C2×GL(2,𝔽3)12yes—
302C2×C4C2×GL(2,𝔽3)12yes—
303C2×C4#98312yes—
304C2×C4GL(2,ℤ/4)12yes—
305C8—122no—
306C8—122no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—84no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22#538614yescomplemented
322C22C2×A4⋊Q811yescharacteristic
323C22#591211yescharacteristic
324C22C22×GL(2,𝔽3)11yescharacteristic
325C22C2×#98311yescharacteristic
326C22C2×#98411yescharacteristic
327C22#592311yescharacteristic
328C22C2×GL(2,ℤ/4)11yescharacteristic
329C4—22no—
330C4—31no—
331C4—31no—
332C4—31no—
333C4—31no—
334C4—62no—
335C4—62no—
336C4—121no—
337C4—242no—
338C4—242no—
339C4#542114yes—
340C3—41noSylow
341C2—24no—
342C2C2×#538611yescharacteristic
343C2#8531811yescharacteristic
344C2#8531911yescharacteristic
345C2#8532511yescharacteristic
346C2#8548011yescharacteristic
347C2C2×#542111yescharacteristic
348C2#8551711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC22×C4×Q8D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#5386C2semidirect
4. C4⋊#859C2semidirect
5. #73059C2semidirect
30. C22×SL(2,𝔽3)Q8semidirect
97. C22×Q8Dic6semidirect
168. SL(2,𝔽3)C22⋊Q8semidirect
271. Q8#924semidirect
321. C22#5386semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SL(2,𝔽3)2
C2×#53862
#731482
C4⋊#8592
#730592
#329543
C2×#9244