Finite Group Databasebeta

← #11713972

Subgroups — #11713972

Order 768.

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

normal subgroups only

This group has 653 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×#5419C212yesmaximal
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4#73061C212yesmaximal, complemented
5C4⋊#859C211yesmaximal, characteristic, complemented
6#32906—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C22×#226C2212yes—
9C2×C4×SL(2,𝔽3)—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12#5419—24no—
13C2×#859C2212yes—
14C4⋊#226—22no—
15#5212—41nomaximal
16C22×C4×Q8D311yescharacteristic
17C2×#462—31no—
18C2×#462—31no—
19C2×#451—32no—
20#2857—31no—
21#1878—32no—
22#1878—32no—
23#1872—32no—
24#1803—31no—
25#1803—31no—
26#1797—31no—
27C23×C3×C4—41no—
28C22×SL(2,𝔽3)—21no—
29C22×SL(2,𝔽3)—21no—
30C22×SL(2,𝔽3)D412yes—
31C22×SL(2,𝔽3)C2311yescharacteristic
32C2×#226—22no—
33C2×#226—22no—
34C2×#226—22no—
35C2×#217—42no—
36C2×C4⋊Dic3—41no—
37C2×C4⋊Dic3—41no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)D412yes—
40#859—22no—
41C3⋊#95—42no—
42C23×Q8C2×D311yescharacteristic
43C22×Q16—32no—
44C22×Q16—32no—
45C2×C4⋊Q8—31no—
46C2×C22⋊Q8—32no—
47C2×C4×Q8—22no—
48C2×C4×Q8—31no—
49C2×C4×Q8—31no—
50C2×C4×Q8—62no—
51C2×C4×Q8C2×D311yescharacteristic
52C2×C4×Q8C2×D311yescharacteristic
53C22×C4⋊C4—31no—
54C22×C42—31no—
55#462—62no—
56#462—64no—
57#462—62no—
58#451—64no—
59#451—64no—
60C2×C4⋊C8—31no—
61C2×C4⋊C8—31no—
62C2×#103—32no—
63C2×#103—32no—
64C2×#103—32no—
65C2×#103—32no—
66C2×C22⋊C8—32no—
67#391—32no—
68#389—61no—
69#389—61no—
70#384—31no—
71#380—61no—
72#380—61no—
73#340—32no—
74#340—32no—
75#336—32no—
76C24×C3—41no—
77C22×C3×C4—44no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4—82no—
81C22×C3×C4—82no—
82C22×Dic3—42no—
83C22×Dic3—42no—
84C2×SL(2,𝔽3)—22no—
85C2×SL(2,𝔽3)C2×D411yescharacteristic
86C2×SL(2,𝔽3)C2×D411yescharacteristic
87C2×SL(2,𝔽3)#4111yescharacteristic
88#226—42no—
89#217—84no—
90C4⋊Dic3—82no—
91C4⋊Dic3—82no—
92C22×Q8—21no—
93C22×Q8—21no—
94C22×Q8—31no—
95C22×Q8—32no—
96C22×Q8—32no—
97C22×Q8—61no—
98C22×Q8—61no—
99C22×Q8D1212yes—
100C22×Q8C22×D311yescharacteristic
101C23×C4—31no—
102C23×C4—31no—
103C23×C4S411yescharacteristic
104C2×Q16—62no—
105C2×Q16—64no—
106C2×Q16—64no—
107C2×Q16—62no—
108C2×Q16—62no—
109C2×Q16—64no—
110C2×Q16—62no—
111C2×Q16—62no—
112C2×Q16—62no—
113C22×C8—32no—
114C22×C8—32no—
115C4⋊Q8—62no—
116C4⋊Q8—62no—
117C22⋊Q8—64no—
118C22⋊Q8—64no—
119C4×Q8—21no—
120C4×Q8—32no—
121C4×Q8—34no—
122C4×Q8—62no—
123C4×Q8—62no—
124C4×Q8—61no—
125C4×Q8—62no—
126C4×Q8#6712yes—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C22⋊C4—62no—
139C2×C42—31no—
140C2×C42—31no—
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—62no—
144C2×C42—62no—
145C4⋊C8—62no—
146C4⋊C8—62no—
147#103—64no—
148#103—64no—
149#103—64no—
150#103—62no—
151#103—62no—
152C22⋊C8—64no—
153#95—62no—
154#95—62no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3—42no—
158C23×C3—41no—
159C23×C3—81no—
160C23×C3—81no—
161C23×C3—81no—
162C23×C3—81no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—88no—
168C2×C3×C4—81no—
169C2×C3×C4—81no—
170C2×Dic3—82no—
171C2×Dic3—82no—
172C2×Dic3—82no—
173C2×Dic3—82no—
174C2×Dic3—82no—
175C2×Dic3—82no—
176SL(2,𝔽3)#12111yescharacteristic
177C24C2×S411yescharacteristic
178C2×Q8—22no—
179C2×Q8—32no—
180C2×Q8—31no—
181C2×Q8—31no—
182C2×Q8—31no—
183C2×Q8—32no—
184C2×Q8—32no—
185C2×Q8—61no—
186C2×Q8—61no—
187C2×Q8—61no—
188C2×Q8—62no—
189C2×Q8—62no—
190C2×Q8—62no—
191C2×Q8—62no—
192C2×Q8—62no—
193C2×Q8—62no—
194C2×Q8—64no—
195C2×Q8—62no—
196C2×Q8—62no—
197C2×Q8—62no—
198C2×Q8C2×D1211yescharacteristic
199C2×Q8#23511yescharacteristic
200C2×Q8C2×#6711yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—31no—
204C22×C4—32no—
205C22×C4—32no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—62no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4—62no—
217C22×C4—121no—
218C22×C4#22614yescomplemented
219C22×C4C2×S411yescharacteristic
220C22×C4C2×S411yescharacteristic
221Q16—122no—
222Q16—122no—
223Q16—122no—
224Q16—128no—
225Q16—122no—
226C2×C8—62no—
227C2×C8—62no—
228C2×C8—62no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—62no—
232C4⋊C4—32no—
233C4⋊C4—32no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—64no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—124no—
240C4⋊C4—122no—
241C4⋊C4—122no—
242C4⋊C4—122no—
243C4⋊C4—122no—
244C22⋊C4—124no—
245C42—32no—
246C42—32no—
247C42—61no—
248C42—61no—
249C42—62no—
250C42—62no—
251C22×C3—44no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—82no—
260C22×C3—82no—
261C22×C3—82no—
262C22×C3—82no—
263C22×C3—82no—
264C22×C3—82no—
265C3×C4—44no—
266C3×C4—82no—
267Dic3—162no—
268Dic3—162no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23C4⋊S412yes—
274C23C2×#22612yes—
275C23#98412yes—
276C23C22×S411yescharacteristic
277Q8—31no—
278Q8—31no—
279Q8—31no—
280Q8—34no—
281Q8—61no—
282Q8—62no—
283Q8—62no—
284Q8—62no—
285Q8—122no—
286Q8—122no—
287Q8—122no—
288Q8—122no—
289Q8#93011yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—28no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—124no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4C2×#22612yes—
320C2×C4C2×#22612yes—
321C2×C4#98312yes—
322C2×C4GL(2,ℤ/4)12yes—
323C8—122no—
324C8—122no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—84no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22C4⋊#22614yescomplemented
340C22C2×C4⋊S411yescharacteristic
341C22#591211yescharacteristic
342C22C22×#22611yescharacteristic
343C22C2×#98311yescharacteristic
344C22C2×#98411yescharacteristic
345C22#592311yescharacteristic
346C22C2×GL(2,ℤ/4)11yescharacteristic
347C4—22no—
348C4—31no—
349C4—31no—
350C4—31no—
351C4—31no—
352C4—62no—
353C4—62no—
354C4—121no—
355C4—242no—
356C4—242no—
357C4#541914yescomplemented
358C3—41noSylow
359C2—24no—
360C2#8531911yescharacteristic
361C2C2×C4⋊#22611yescharacteristic
362C2#8532211yescharacteristic
363C2#8533111yescharacteristic
364C2#8548611yescharacteristic
365C2C2×#541911yescharacteristic
366C2#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)#121
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
3. C2×C4⋊#226C2semidirect
4. #73061C2semidirect
5. C4⋊#859C2semidirect
218. C22×C4#226semidirect
339. C22C4⋊#226semidirect
357. C4#5419semidirect

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×#54192
C2×C4⋊#2262
#730612
C4⋊#8592
#329063
#52124