Finite Group Databasebeta

← #11713925

Subgroups — #11713925

Order 768.

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

normal subgroups only

This group has 401 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
1#2351⋊C3C211yesmaximal, characteristic, complemented
2#73065C211yesmaximal, characteristic
3#73065C211yesmaximal, characteristic
4#73065C211yesmaximal, characteristic
5#73064C211yesmaximal, characteristic, complemented
6#73059C211yesmaximal, characteristic, complemented
7C4×#859C211yesmaximal, characteristic, complemented
8#27564—31nomaximal, Sylow
9C22×GL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#4669—41nomaximal
17#2513—31no—
18#2351D311yescharacteristic, complemented
19#2101—31no—
20#2101—31no—
21#2073—31no—
22#2065—31no—
23#1934—31no—
24#1919—31no—
25#1916—31no—
26#1880—31no—
27#1880—31no—
28#1880—31no—
29#1865—31no—
30#1805—31no—
31C4×#103—31no—
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×GL(2,𝔽3)—21no—
34C2×GL(2,𝔽3)—21no—
35C2×GL(2,𝔽3)—21no—
36C2×(C2×D3)⋊C4—41no—
37C2×(C2×D3)⋊C4—41no—
38C2×(C2×D3)⋊C4—41no—
39C2×Dic3⋊C4—41no—
40C2×C4×Dic3—41no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)—21no—
44#859—21no—
45#859—21no—
46#859—21no—
47C3×#95—41no—
48C3⋊#95—41no—
49C22×SD16—31no—
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×Q8C2×D311yescharacteristic
52C2×C4×Q8C2×D311yescharacteristic
53C2×#106—31no—
54C2×C4⋊C8—31no—
55C2×#103—31no—
56C2×#103—31no—
57C2×#103—31no—
58C2×#102—31no—
59C2×#102—31no—
60C2×#102—31no—
61C2×C4×C8—31no—
62#397—61no—
63#397—61no—
64#390—31no—
65#390—31no—
66#390—31no—
67#389—31no—
68#389—31no—
69#389—31no—
70#388—61no—
71#388—61no—
72#382—31no—
73#382—31no—
74C4×C4⋊C4—31no—
75C4×C4⋊C4—31no—
76#340—31no—
77#340—31no—
78#340—31no—
79#340—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#336—31no—
85#336—31no—
86#336—31no—
87#336—31no—
88C23×D3—41no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C2×SL(2,𝔽3)#4111yescharacteristic
96C2×SL(2,𝔽3)#4111yescharacteristic
97C2×SL(2,𝔽3)#4111yescharacteristic
98GL(2,𝔽3)—41no—
99(C2×D3)⋊C4—44no—
100(C2×D3)⋊C4—82no—
101(C2×D3)⋊C4—82no—
102Dic3⋊C4—82no—
103C4×Dic3—82no—
104C22×Q8C22×D311yescharacteristic
105C22×D4—31no—
106C2×SD16—61no—
107C2×SD16—61no—
108C2×SD16—61no—
109C2×SD16—61no—
110C2×SD16—61no—
111C2×SD16—61no—
112C22×C8—31no—
113C22×C8—31no—
114C22×C8—31no—
115C22×C8—31no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—61no—
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—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—61no—
131C2×C22⋊C4—61no—
132C2×C22⋊C4—61no—
133C2×C22⋊C4—61no—
134C2×C22⋊C4—61no—
135C2×C22⋊C4—61no—
136C2×C22⋊C4—61no—
137C2×C42—31no—
138C2×C42—31no—
139C2×C42—31no—
140C2×C42—61no—
141#106—62no—
142C4⋊C8—121no—
143#103—61no—
144#103—61no—
145#103—61no—
146#103—61no—
147#103—61no—
148#103—61no—
149#102—62no—
150#102—121no—
151#102—121no—
152C4×C8—121no—
153#95—31no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95S411yescharacteristic
160C23×C3—41no—
161C22×D3—42no—
162C22×D3—81no—
163C22×D3—81no—
164C22×D3—81no—
165C22×D3—81no—
166C22×D3—81no—
167C22×D3—81no—
168C2×C3×C4—42no—
169C2×C3×C4—81no—
170C2×C3×C4—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—42no—
179C2×Dic3—42no—
180C2×Dic3—81no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183C2×Dic3—81no—
184C2×Dic3—81no—
185C2×Dic3—81no—
186SL(2,𝔽3)#12611yescharacteristic
187C24—61no—
188C2×Q8—31no—
189C2×Q8—31no—
190C2×Q8—31no—
191C2×Q8#23511yescharacteristic
192C2×Q8#23711yescharacteristic
193C2×Q8#23911yescharacteristic
194C2×D4—61no—
195C2×D4—61no—
196C2×D4—61no—
197C2×D4—61no—
198C2×D4—61no—
199C2×D4—61no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4C2×S411yescharacteristic
208C22×C4C2×S411yescharacteristic
209C22×C4C2×S411yescharacteristic
210SD16—121no—
211SD16—121no—
212SD16—121no—
213SD16—121no—
214C2×C8—61no—
215C2×C8—61no—
216C2×C8—61no—
217C2×C8—61no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—61no—
226C4⋊C4—32no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—62no—
239C4⋊C4—121no—
240C4⋊C4—241no—
241C22⋊C4—122no—
242C22⋊C4—122no—
243C22⋊C4—241no—
244C22⋊C4—241no—
245C22⋊C4—241no—
246C22⋊C4—241no—
247C42—61no—
248C42—61no—
249C42—61no—
250C42—61no—
251C42—61no—
252C42—61no—
253C42—241no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C2×D3—82no—
262C2×D3—82no—
263C2×D3—82no—
264C2×D3—161no—
265C2×D3—161no—
266C2×D3—161no—
267C2×D3—161no—
268C3×C4—82no—
269C3×C4—161no—
270C3×C4—161no—
271Dic3—82no—
272Dic3—82no—
273Dic3—82no—
274C23—121no—
275C23—121no—
276C23—121no—
277C23—121no—
278C23—121no—
279C23—121no—
280C23—121no—
281C23C22×S411yescharacteristic
282Q8—31no—
283Q8—31no—
284Q8—31no—
285Q8—61no—
286Q8#89811yescharacteristic
287D4—121no—
288D4—121no—
289D4—121no—
290D4—121no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4#98512yes—
324C8—121no—
325C8—121no—
326C8—121no—
327C8—121no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335D3—162no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22—241no—
343C22#591211yescharacteristic
344C22#591411yescharacteristic
345C22#591911yescharacteristic
346C22C2×#98511yescharacteristic
347C22#592311yescharacteristic
348C22#592611yescharacteristic
349C22#592611yescharacteristic
350C4—22no—
351C4—31no—
352C4—31no—
353C4—31no—
354C4—31no—
355C4—41no—
356C4—41no—
357C4—121no—
358C4—121no—
359C4—121no—
360C4—241no—
361C4—241no—
362C4—241no—
363C3—41noSylow
364C2—481no—
365C2#8531311yescharacteristic
366C2#8534311yescharacteristic
367C2#8535811yescharacteristic
368C2#8538111yescharacteristic
369C2#8538811yescharacteristic
370C2#8542011yescharacteristic
371C2#8542111yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
Fitting#2351D3
prime radicalC22×GL(2,𝔽3)C22
hypercenter#95S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
1. #2351⋊C3C2semidirect
5. #73064C2semidirect
6. #73059C2semidirect
7. C4×#859C2semidirect
18. #2351D3semidirect

Maximal subgroups

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

subgroupindex
#2351⋊C32
#730652
#730642
#730592
C4×#8592
#275643
#46694