Finite Group Databasebeta

← #11713978

Subgroups — #11713978

Order 768.

706 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 61 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×#519⋊C3C211yesmaximal, characteristic, complemented
2C2×#5421C212yesmaximal, complemented
3C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73065C212yesmaximal, complemented
5#73064C211yesmaximal, characteristic, complemented
6#32912—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C22×GL(2,𝔽3)C2212yes—
9#519⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12#5421—24no—
13C2×#859C2212yes—
14C4⋊GL(2,𝔽3)—22no—
15#4960—41nomaximal
16C2×#519D311yescharacteristic, complemented
17C2×#460—31no—
18C2×#460—31no—
19C2×#448—32no—
20#2619—31no—
21#1880—32no—
22#1880—32no—
23#1875—32no—
24#1805—31no—
25#1805—31no—
26#1798—31no—
27C22×#67—42no—
28C22×D12—41no—
29C22×SL(2,𝔽3)—21no—
30C22×SL(2,𝔽3)—21no—
31C22×SL(2,𝔽3)D412yescomplemented
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×GL(2,𝔽3)—22no—
34C2×GL(2,𝔽3)—22no—
35C2×GL(2,𝔽3)—22no—
36C2×C3×C22⋊C4—41no—
37C2×(C2×D3)⋊C4—42no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)D412yescomplemented
40#859—22no—
41C3⋊#95—41no—
42C23×Q8C2×D311yescharacteristic
43C22×SD16—32no—
44C22×SD16—32no—
45C2×#127—31no—
46C2×#121—32no—
47#519—22no—
48#519—62no—
49C2×C4×Q8—31no—
50C2×C4×Q8—31no—
51C2×C4×Q8C2×D311yescharacteristic
52C2×C4×Q8C2×D311yescharacteristic
53C2×#117—31no—
54C2×#117—31no—
55#460—62no—
56#460—64no—
57#460—62no—
58#448—64no—
59#448—64no—
60C2×C4⋊C8—31no—
61C2×C4⋊C8—31no—
62C2×#103—32no—
63C2×#103—32no—
64C2×#102—32no—
65C2×#102—32no—
66C2×C22⋊C8—32no—
67#394—61no—
68#394—61no—
69#392—61no—
70#392—61no—
71#390—32no—
72#382—31no—
73#340—32no—
74#340—32no—
75#336—32no—
76C24×C3—41no—
77C23×D3—42no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C2×#67—84no—
81C2×#67—84no—
82C2×#67—84no—
83C22×Dic3—42no—
84C2×D12—82no—
85C2×D12—84no—
86C2×SL(2,𝔽3)—22no—
87C2×SL(2,𝔽3)C2×D411yescharacteristic
88C2×SL(2,𝔽3)C2×D411yescharacteristic
89C2×SL(2,𝔽3)#4111yescharacteristic
90GL(2,𝔽3)—42no—
91C3×C22⋊C4—82no—
92(C2×D3)⋊C4—84no—
93C22×Q8—21no—
94C22×Q8—21no—
95C22×Q8—31no—
96C22×Q8—32no—
97C22×Q8—61no—
98C22×Q8—61no—
99C22×Q8D1212yescomplemented
100C22×Q8C22×D311yescharacteristic
101C22×D4—32no—
102C22×D4—61no—
103C22×D4—62no—
104C22×D4—61no—
105C23×C4—31no—
106C2×SD16—62no—
107C2×SD16—64no—
108C2×SD16—64no—
109C2×SD16—62no—
110C2×SD16—62no—
111C2×SD16—64no—
112C2×SD16—62no—
113C2×SD16—62no—
114C2×SD16—62no—
115C22×C8—32no—
116C22×C8—32no—
117#127—62no—
118#127—62no—
119#121—64no—
120#121—64no—
121C4×Q8—21no—
122C4×Q8—32no—
123C4×Q8—34no—
124C4×Q8—62no—
125C4×Q8—62no—
126C4×Q8—62no—
127C4×Q8—61no—
128C4×Q8#6712yescomplemented
129#117—62no—
130#117—62no—
131#117—62no—
132#117—62no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—32no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4S411yescharacteristic
143C2×C42—31no—
144C2×C42—31no—
145C2×C42—61no—
146C4⋊C8—62no—
147C4⋊C8—62no—
148#103—64no—
149#103—62no—
150#103—62no—
151#102—64no—
152#102—64no—
153C22⋊C8—64no—
154#95—61no—
155#95—61no—
156C23×C3—42no—
157C23×C3—41no—
158C23×C3—81no—
159C23×C3—81no—
160C23×C3—82no—
161C23×C3—82no—
162C22×D3—44no—
163C22×D3—82no—
164C22×D3—82no—
165C22×D3—82no—
166C22×D3—82no—
167C22×D3—82no—
168C22×D3—82no—
169C2×C3×C4—42no—
170C2×C3×C4—82no—
171C2×C3×C4—81no—
172C2×C3×C4—82no—
173#67—816no—
174C2×Dic3—44no—
175C2×Dic3—82no—
176C2×Dic3—82no—
177D12—88no—
178SL(2,𝔽3)#12111yescharacteristic
179C24—62no—
180C24C2×S411yescharacteristic
181C2×Q8—22no—
182C2×Q8—32no—
183C2×Q8—31no—
184C2×Q8—31no—
185C2×Q8—31no—
186C2×Q8—32no—
187C2×Q8—32no—
188C2×Q8—61no—
189C2×Q8—61no—
190C2×Q8—61no—
191C2×Q8—64no—
192C2×Q8—62no—
193C2×Q8—62no—
194C2×Q8—62no—
195C2×Q8C2×D1211yescharacteristic
196C2×Q8#23511yescharacteristic
197C2×Q8C2×#6711yescharacteristic
198C2×D4—62no—
199C2×D4—62no—
200C2×D4—62no—
201C2×D4—62no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—122no—
205C2×D4—124no—
206C2×D4—122no—
207C2×D4—242no—
208C2×D4—242no—
209C2×D4—242no—
210C2×D4—242no—
211C22×C4—31no—
212C22×C4—32no—
213C22×C4—32no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—62no—
222C22×C4C2×S411yescharacteristic
223C22×C4C2×S411yescharacteristic
224SD16—122no—
225SD16—122no—
226SD16—122no—
227SD16—128no—
228SD16—122no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—62no—
232C2×C8—62no—
233C2×C8—62no—
234C2×C8—62no—
235C4⋊C4—32no—
236C4⋊C4—32no—
237C4⋊C4—62no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—62no—
241C4⋊C4—62no—
242C4⋊C4—62no—
243C22⋊C4—22no—
244C22⋊C4—62no—
245C22⋊C4—124no—
246C22⋊C4—242no—
247C22⋊C4—242no—
248C42—32no—
249C42—32no—
250C42—61no—
251C42—61no—
252C42—64no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—41no—
258C22×C3—84no—
259C22×C3—82no—
260C22×C3—161no—
261C22×C3—161no—
262C22×C3—162no—
263C2×D3—84no—
264C2×D3—84no—
265C2×D3—84no—
266C2×D3—162no—
267C2×D3—162no—
268C2×D3—162no—
269C2×D3—162no—
270C3×C4—82no—
271C3×C4—161no—
272Dic3—84no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—21no—
277C23—122no—
278C23—122no—
279C23—122no—
280C23—122no—
281C23—122no—
282C23—122no—
283C23—122no—
284C23C4⋊S412yes—
285C23C22×S411yescharacteristic
286Q8—31no—
287Q8—31no—
288Q8—31no—
289Q8—34no—
290Q8—61no—
291Q8—62no—
292Q8—62no—
293Q8—62no—
294Q8#93011yescharacteristic
295D4—122no—
296D4—122no—
297D4—122no—
298D4—122no—
299D4—244no—
300D4—244no—
301D4—248no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—32no—
312C2×C4—32no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—61no—
318C2×C4—62no—
319C2×C4—61no—
320C2×C4—62no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4GL(2,ℤ/4)12yes—
326C8—122no—
327C8—122no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—42no—
331C2×C3—42no—
332C2×C3—41no—
333C2×C3—162no—
334D3—164no—
335C22—24no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—242no—
341C22—242no—
342C22—242no—
343C22—242no—
344C22—242no—
345C22—242no—
346C22—242no—
347C22C2×C4⋊S411yescharacteristic
348C22#591211yescharacteristic
349C22#592612yes—
350C22#592612yes—
351C22C2×GL(2,ℤ/4)11yescharacteristic
352C4—22no—
353C4—31no—
354C4—31no—
355C4—31no—
356C4—31no—
357C4—41no—
358C4—62no—
359C4—62no—
360C4—121no—
361C4—242no—
362C3—41noSylow
363C2—42no—
364C2—482no—
365C2#8535512yes—
366C2#8535812yes—
367C2#8548611yescharacteristic
368C2#8553711yescharacteristic
369C2#8554011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#519D3
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#121
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
1. C2×#519⋊C3C2semidirect
2. C2×#5421C2semidirect
3. C2×C4⋊GL(2,𝔽3)C2semidirect
4. #73065C2semidirect
5. #73064C2semidirect
16. C2×#519D3semidirect
31. C22×SL(2,𝔽3)D4semidirect
39. C4×SL(2,𝔽3)D4semidirect
99. C22×Q8D12semidirect
128. C4×Q8#67semidirect

Maximal subgroups

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

subgroupindex
C2×#519⋊C32
C2×#54212
C2×C4⋊GL(2,𝔽3)2
#730652
#730642
#329123
#49604