Finite Group Databasebeta

← #11713911

Subgroups — #11713911

Order 768.

419 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 53 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#73063C211yesmaximal, characteristic, complemented
5C4⋊#859C211yesmaximal, characteristic, complemented
6#73059C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#27537—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#4948—41nomaximal
17#2497—31no—
18Q8×C22⋊C4D311yescharacteristic
19#2077—31no—
20#2077—31no—
21#2072—31no—
22#2066—31no—
23#1935—31no—
24#1927—31no—
25#1924—31no—
26#1865—31no—
27#1865—31no—
28#1803—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×C4⋊Dic3—41no—
40C2×Dic3⋊C4—41no—
41C2×Dic3⋊C4—41no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)—21no—
44#859—21no—
45#859—21no—
46#859—21no—
47#859—21no—
48C3⋊#95—41no—
49C23×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×Q8C2×D311yescharacteristic
52C2×#106—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#400—61no—
67#396—61no—
68#391—31no—
69#391—31no—
70#389—31no—
71#389—31no—
72#388—61no—
73#387—61no—
74#384—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—
102C4⋊Dic3—82no—
103Dic3⋊C4—82no—
104Dic3⋊C4—82no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—31no—
109C22×Q8—61no—
110C22×Q8—61no—
111C22×Q8—61no—
112C22×Q8C22×D311yescharacteristic
113C23×C4—31no—
114C22×C8—31no—
115C22×C8—31no—
116C22×C8—31no—
117C22×C8—31no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—61no—
121C4×Q8—61no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—61no—
134C2×C22⋊C4—61no—
135C2×C22⋊C4S411yescharacteristic
136C2×C42—31no—
137C2×C42—31no—
138C2×C42—61no—
139#106—121no—
140#106—121no—
141C4⋊C8—121no—
142#103—61no—
143#103—61no—
144#103—61no—
145#103—61no—
146#103—62no—
147#103—62no—
148#103—61no—
149#103—61no—
150#103—61no—
151#103—61no—
152#103—121no—
153#103—121no—
154C22⋊C8—121no—
155C22⋊C8—121no—
156#97—121no—
157#95—31no—
158#95—31no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165C23×C3—42no—
166C23×C3—41no—
167C23×C3—81no—
168C23×C3—81no—
169C23×C3—81no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3—81no—
173C2×C3×C4—42no—
174C2×C3×C4—81no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183C2×Dic3—81no—
184C2×Dic3—81no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188C2×Dic3—81no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191SL(2,𝔽3)#12611yescharacteristic
192C24C2×S411yescharacteristic
193C2×Q8—31no—
194C2×Q8—31no—
195C2×Q8—31no—
196C2×Q8—32no—
197C2×Q8—41no—
198C2×Q8—61no—
199C2×Q8—61no—
200C2×Q8—61no—
201C2×Q8—61no—
202C2×Q8—61no—
203C2×Q8—122no—
204C2×Q8—121no—
205C2×Q8—121no—
206C2×Q8—121no—
207C2×Q8#23511yescharacteristic
208C2×Q8#23711yescharacteristic
209C2×Q8#23711yescharacteristic
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
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×C4—61no—
223C22×C4—61no—
224C22×C4C2×S411yescharacteristic
225C22×C4C2×S411yescharacteristic
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—
236C2×C8—61no—
237C2×C8—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—61no—
249C4⋊C4—61no—
250C4⋊C4—122no—
251C4⋊C4—241no—
252C4⋊C4—241no—
253C4⋊C4—241no—
254C22⋊C4—22no—
255C22⋊C4—62no—
256C22⋊C4—241no—
257C22⋊C4—241no—
258C42—61no—
259C42—61no—
260C42—61no—
261C42—61no—
262C42—241no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—82no—
271C22×C3—82no—
272C22×C3—82no—
273C22×C3—161no—
274C22×C3—161no—
275C22×C3—161no—
276C22×C3—161no—
277C3×C4—82no—
278C3×C4—161no—
279Dic3—82no—
280Dic3—82no—
281Dic3—161no—
282Dic3—161no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#98412yes—
290C23C22×S411yescharacteristic
291Q8—31no—
292Q8—31no—
293Q8—31no—
294Q8—61no—
295Q8—62no—
296Q8—62no—
297Q8—121no—
298Q8—121no—
299Q8#87911yescharacteristic
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
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—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4#98312yes—
337C8—121no—
338C8—121no—
339C8—121no—
340C8—121no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—162no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22#591211yescharacteristic
357C22#591411yescharacteristic
358C22#591411yescharacteristic
359C22C2×#98311yescharacteristic
360C22C2×#98411yescharacteristic
361C22#592611yescharacteristic
362C22#592611yescharacteristic
363C4—22no—
364C4—31no—
365C4—31no—
366C4—31no—
367C4—31no—
368C4—41no—
369C4—121no—
370C4—121no—
371C4—121no—
372C4—241no—
373C4—241no—
374C4—241no—
375C4—241no—
376C3—41noSylow
377C2—42no—
378C2#8529411yescharacteristic
379C2#8531911yescharacteristic
380C2#8535811yescharacteristic
381C2#8537711yescharacteristic
382C2#8537711yescharacteristic
383C2#8540011yescharacteristic
384C2#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. #73063C2semidirect
5. C4⋊#859C2semidirect
6. #73059C2semidirect
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
#730632
C4⋊#8592
#730592
#275373
#49484