Finite Group Databasebeta

← #11713030

Subgroups — #11713030

Order 768.

461 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 63 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×#5405C211yesmaximal, characteristic
2#85365C211yesmaximal, characteristic
3#85364C211yesmaximal, characteristic
4#2913⋊C3C211yesmaximal, characteristic, complemented
5#73125C211yesmaximal, characteristic, complemented
6#73076C211yesmaximal, characteristic, complemented
7#73075C211yesmaximal, characteristic, complemented
8#22693—31nomaximal, Sylow
9C2×#985C2211yescharacteristic
10C2×#131⋊C3C2211yescharacteristic
11#520⋊C3C2211yescharacteristic
12#520⋊C3C2211yescharacteristic
13#5426C2211yescharacteristic
14#5426C2211yescharacteristic
15C2×GU(2,𝔽3)C2211yescharacteristic
16#5405C414yescomplemented
17#5394—21no—
18C4×GL(2,𝔽3)—21no—
19#5388—21no—
20C4×#226—21no—
21#4881—41nomaximal
22C2×#444—31no—
23#2952—31no—
24#2951—31no—
25#2913D311yescharacteristic, complemented
26#2912—31no—
27#1952—31no—
28#1951—31no—
29#1860—31no—
30#1860—31no—
31#1859—31no—
32#1859—31no—
33#1847—31no—
34#1816—31no—
35#1815—31no—
36#1773—31no—
37C2×#41⋊C3C2311yescharacteristic
38#985C2×C412yes—
39#985Q812yes—
40C2×GL(2,𝔽3)D411yescharacteristic
41C2×#226D411yescharacteristic
42C2×ΓL(1,𝔽25)—41no—
43#892—41no—
44D3×C4⋊C4—41no—
45#131⋊C3C2×C412yescomplemented
46C4×SL(2,𝔽3)—21no—
47C4×SL(2,𝔽3)—21no—
48GU(2,𝔽3)C2×C412yescomplemented
49#859—21no—
50#859—21no—
51C3×#107—41no—
52#817—41no—
53#808—41no—
54#807—41no—
55C2×#135—31no—
56C2×#131—31no—
57C2×#131C2×D311yescharacteristic
58#520—31no—
59#520—31no—
60#520C2×D311yescharacteristic
61#520C2×D311yescharacteristic
62#444—34no—
63#444—34no—
64#442—61no—
65#441—61no—
66#440—61no—
67#440—61no—
68C4×Q16—61no—
69C4×SD16—61no—
70C4×SD16—61no—
71C4×D8—61no—
72C2×#108—31no—
73C2×#108—31no—
74#428—31no—
75#428—31no—
76#428—31no—
77#428—31no—
78#428—61no—
79#428—61no—
80#427—31no—
81#427—61no—
82C2×#107—31no—
83C2×#107—31no—
84C2×#106—31no—
85C2×#106—31no—
86C2×#106—61no—
87C2×#104—31no—
88C2×#104—31no—
89#417—31no—
90#417—31no—
91#417—31no—
92#417—31no—
93#405—31no—
94#405—31no—
95C2×#97—31no—
96#384—31no—
97#382—31no—
98#343—31no—
99#343—31no—
100#343—31no—
101#343—31no—
102#339—31no—
103#339—31no—
104C2×C4×D3—41no—
105C2×C4×D3—81no—
106#41⋊C3C22×C411yescharacteristic
107#41⋊C3C2×Q811yescharacteristic
108C2×SL(2,𝔽3)C2×D411yescharacteristic
109GL(2,𝔽3)C4⋊C412yescomplemented
110#226C4⋊C412yescomplemented
111C2×C3×C8—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114(C2×D3)⋊C4—81no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117Dic3⋊C4—81no—
118C4×Dic3—81no—
119C2×C3⋊C8—41no—
120ΓL(1,𝔽25)—44no—
121C2×#41—31no—
122C2×#41C22×D311yescharacteristic
123#135—32no—
124#135—32no—
125#135—32no—
126#135—32no—
127C2×Q16—31no—
128C2×SD16—31no—
129C2×SD16—31no—
130C2×D8—31no—
131#131—32no—
132#131—34no—
133#131—62no—
134#131C4×D312yescomplemented
135C2×M16—31no—
136C2×M16—31no—
137C2×M16—31no—
138C2×M16—31no—
139C2×M16—61no—
140C22×C8—31no—
141C22×C8—31no—
142C22×C8—61no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×D4—61no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4—61no—
151C4×D4—121no—
152C4×D4—121no—
153#117—31no—
154#117—31no—
155#117—61no—
156#117—61no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—121no—
164C2×C42—31no—
165#108—34no—
166#108—62no—
167#107—31no—
168#107—61no—
169#107—61no—
170#107—61no—
171#107—61no—
172#107—61no—
173#107S411yescharacteristic
174#106—31no—
175#106—31no—
176#106—61no—
177#106—61no—
178#106—61no—
179#106—61no—
180#106—61no—
181#106—61no—
182#104—34no—
183#104—32no—
184#104—32no—
185#103—61no—
186#103—61no—
187#103—61no—
188#103—61no—
189#102—61no—
190#102—61no—
191#102—61no—
192#102—61no—
193#97—34no—
194#97—61no—
195#97—61no—
196C4×C8—61no—
197C4×C8—61no—
198#95—121no—
199C22×D3—41no—
200C2×C3×C4—41no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×Dic3—41no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C4×D3—42no—
207C4×D3—42no—
208C4×D3—162no—
209SL(2,𝔽3)C2×C4⋊C411yescharacteristic
210C3×C8—42no—
211C3⋊C8—42no—
212#41—31no—
213#41—31no—
214#41—32no—
215#41—32no—
216#41—61no—
217#41—61no—
218#41C2×C4×D311yescharacteristic
219#41D3×Q811yescharacteristic
220C2×Q8—31no—
221C2×Q8D3×D411yescharacteristic
222C2×D4—31no—
223C2×D4—31no—
224C2×D4—61no—
225C22×C4—31no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—121no—
233C22×C4—121no—
234Q16—32no—
235Q16—32no—
236SD16—32no—
237SD16—32no—
238SD16—32no—
239SD16—32no—
240D8—32no—
241D8—32no—
242M16—32no—
243M16—32no—
244M16—61no—
245M16—61no—
246M16—64no—
247M16—61no—
248M16—61no—
249M16—61no—
250M16—61no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—32no—
254C2×C8—32no—
255C2×C8—31no—
256C2×C8—32no—
257C2×C8—32no—
258C2×C8—64no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8C2×S411yescharacteristic
263C4⋊C4—31no—
264C4⋊C4—31no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—242no—
274C4⋊C4C2×S411yescharacteristic
275C4⋊C4C2×S411yescharacteristic
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42—32no—
281C42—61no—
282C42—61no—
283C42—61no—
284C42—121no—
285C42—121no—
286C22×C3—41no—
287C2×D3—42no—
288C2×D3—81no—
289C2×D3—81no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—161no—
293C3×C4—161no—
294Dic3—42no—
295Dic3—161no—
296Dic3—161no—
297C23—31no—
298C23—61no—
299Q8—31no—
300Q8—32no—
301Q8—61no—
302Q8D3×C4⋊C411yescharacteristic
303D4—31no—
304D4—31no—
305D4—32no—
306D4—61no—
307D4—61no—
308D4—62no—
309D4—62no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—61no—
318C2×C4—62no—
319C2×C4—62no—
320C2×C4—62no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4—481no—
332C2×C4—481no—
333C2×C4C22×S411yescharacteristic
334C8—32no—
335C8—32no—
336C8—32no—
337C8—62no—
338C8—121no—
339C8—121no—
340C8C4×S412yes—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344D3—82no—
345C22—31no—
346C22—31no—
347C22—61no—
348C22—61no—
349C22—62no—
350C22—121no—
351C22—121no—
352C22D4×S411yescharacteristic
353C4—31no—
354C4—31no—
355C4—41no—
356C4—41no—
357C4—62no—
358C4—122no—
359C4—121no—
360C4—121no—
361C4—241no—
362C4—241no—
363C4C2×C4×S411yescharacteristic
364C4Q8×S411yescharacteristic
365C3—41noSylow
366C2—61no—
367C2—61no—
368C2—122no—
369C2C4⋊C4×S411yescharacteristic
370C2#8546711yescharacteristic
371C2#8547411yescharacteristic
subgroupG/H
center, socleC22D4×S4
commutator#41⋊C3C22×C4
FrattiniC2×C4C22×S4
Fitting#2913D3
prime radicalC2×#985C22
cosocleC2×#41⋊C3C23
hypercenter#107S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
4. #2913⋊C3C2semidirect
5. #73125C2semidirect
6. #73076C2semidirect
7. #73075C2semidirect
16. #5405C4semidirect
25. #2913D3semidirect
45. #131⋊C3C2×C4semidirect
48. GU(2,𝔽3)C2×C4semidirect
109. GL(2,𝔽3)C4⋊C4semidirect
110. #226C4⋊C4semidirect
134. #131C4×D3semidirect

Maximal subgroups

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

subgroupindex
C2×#54052
#853652
#853642
#2913⋊C32
#731252
#730762
#730752
#226933
#48814