Finite Group Databasebeta

← #11713913

Subgroups — #11713913

Order 768.

407 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 51 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
2#73148C211yesmaximal, characteristic
3#73148C211yesmaximal, characteristic
4#73064C211yesmaximal, characteristic, complemented
5#73063C211yesmaximal, characteristic, complemented
6#73062C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#27540—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#4952—41nomaximal
17#2494—31no—
18Q8×C22⋊C4D311yescharacteristic
19#2077—31no—
20#2076—31no—
21#2074—31no—
22#2068—31no—
23#1935—31no—
24#1927—31no—
25#1924—31no—
26#1865—31no—
27#1863—31no—
28#1805—31no—
29#1787—31no—
30#1787—31no—
31#1763—31no—
32C22×SL(2,𝔽3)—21no—
33C22×SL(2,𝔽3)—21no—
34C22×SL(2,𝔽3)—21no—
35C22×SL(2,𝔽3)C2311yescharacteristic
36C2×C3×C22⋊C4—41no—
37C2×#217—41no—
38C2×#217—41no—
39C2×Dic3⋊C4—41no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)—21no—
42#859—21no—
43#859—21no—
44#859—21no—
45#859—21no—
46C3⋊#95—41no—
47C3⋊#95—41no—
48C3⋊#95—41no—
49C23×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×Q8C2×D311yescharacteristic
52C2×#107—31no—
53C2×#106—31no—
54C2×C4⋊C8—31no—
55C2×#103—31no—
56C2×#103—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#103—31no—
60C2×#103—31no—
61C2×#103—31no—
62C2×#103—31no—
63C2×C22⋊C8—31no—
64C2×C22⋊C8—31no—
65C2×#97—31no—
66#401—61no—
67#397—61no—
68#391—31no—
69#391—31no—
70#389—31no—
71#389—31no—
72#388—61no—
73#387—61no—
74#382—31no—
75#380—31no—
76#380—31no—
77#380—31no—
78C4×C4⋊C4—31no—
79C4×C22⋊C4—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#340—31no—
85#340—31no—
86#336—31no—
87#336—31no—
88C24×C3—41no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×Dic3—41no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C2×SL(2,𝔽3)—41no—
96C2×SL(2,𝔽3)#4111yescharacteristic
97C2×SL(2,𝔽3)#4111yescharacteristic
98C2×SL(2,𝔽3)#4111yescharacteristic
99C3×C22⋊C4—82no—
100#217—82no—
101#217—82no—
102Dic3⋊C4—82no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—31no—
107C22×Q8—61no—
108C22×Q8—61no—
109C22×Q8—61no—
110C22×Q8C22×D311yescharacteristic
111C23×C4—31no—
112C22×C8—31no—
113C22×C8—31no—
114C22×C8—31no—
115C22×C8—31no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—61no—
119C4×Q8—61no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—61no—
130C2×C22⋊C4—61no—
131C2×C22⋊C4S411yescharacteristic
132C2×C42—31no—
133C2×C42—31no—
134C2×C42—61no—
135#107—121no—
136#106—121no—
137C4⋊C8—121no—
138#103—61no—
139#103—61no—
140#103—61no—
141#103—61no—
142#103—62no—
143#103—62no—
144#103—61no—
145#103—61no—
146#103—61no—
147#103—61no—
148#103—121no—
149#103—121no—
150C22⋊C8—121no—
151C22⋊C8—121no—
152#97—121no—
153#95—31no—
154#95—31no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163C23×C3—42no—
164C23×C3—41no—
165C23×C3—81no—
166C23×C3—81no—
167C23×C3—81no—
168C23×C3—81no—
169C23×C3—81no—
170C23×C3—81no—
171C2×C3×C4—81no—
172C2×C3×C4—81no—
173C2×C3×C4—81no—
174C2×C3×C4—81no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×Dic3—42no—
178C2×Dic3—81no—
179C2×Dic3—81no—
180C2×Dic3—81no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183C2×Dic3—81no—
184C2×Dic3—81no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188C2×Dic3—81no—
189SL(2,𝔽3)#12611yescharacteristic
190C24C2×S411yescharacteristic
191C2×Q8—31no—
192C2×Q8—31no—
193C2×Q8—31no—
194C2×Q8—32no—
195C2×Q8—41no—
196C2×Q8—61no—
197C2×Q8—61no—
198C2×Q8—61no—
199C2×Q8—61no—
200C2×Q8—61no—
201C2×Q8—122no—
202C2×Q8—121no—
203C2×Q8—121no—
204C2×Q8—121no—
205C2×Q8#23511yescharacteristic
206C2×Q8#23711yescharacteristic
207C2×Q8#23711yescharacteristic
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C2×S411yescharacteristic
223C22×C4C2×S411yescharacteristic
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—61no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—61no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—122no—
249C4⋊C4—241no—
250C22⋊C4—22no—
251C22⋊C4—121no—
252C22⋊C4—122no—
253C22⋊C4—241no—
254C42—61no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—241no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—82no—
267C22×C3—82no—
268C22×C3—82no—
269C22×C3—161no—
270C22×C3—161no—
271C22×C3—161no—
272C22×C3—161no—
273C3×C4—161no—
274C3×C4—161no—
275Dic3—82no—
276Dic3—161no—
277Dic3—161no—
278Dic3—161no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23#98312yes—
286C23C22×S411yescharacteristic
287Q8—31no—
288Q8—31no—
289Q8—31no—
290Q8—61no—
291Q8—62no—
292Q8—62no—
293Q8—121no—
294Q8—121no—
295Q8#87911yescharacteristic
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C8—121no—
334C8—121no—
335C8—121no—
336C8—121no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—162no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#591211yescharacteristic
353C22#591411yescharacteristic
354C22#591411yescharacteristic
355C22C2×#98311yescharacteristic
356C22#592311yescharacteristic
357C22#592611yescharacteristic
358C22#592711yescharacteristic
359C4—31no—
360C4—31no—
361C4—31no—
362C4—31no—
363C4—41no—
364C4—41no—
365C4—121no—
366C4—121no—
367C4—121no—
368C4—241no—
369C4—241no—
370C4—241no—
371C4—241no—
372C3—41noSylow
373C2—42no—
374C2#8529411yescharacteristic
375C2#8533011yescharacteristic
376C2#8535711yescharacteristic
377C2#8537911yescharacteristic
378C2#8538711yescharacteristic
379C2#8539411yescharacteristic
380C2#8540111yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C22⋊C4D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #73064C2semidirect
5. #73063C2semidirect
6. #73062C2semidirect
7. #73059C2semidirect

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
#731482
#730642
#730632
#730622
#730592
#275403
#49524