Finite Group Databasebeta

← #11713999

Subgroups — #11713999

Order 768.

569 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 55 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×#131⋊C3C211yesmaximal, characteristic, complemented
2#85523C212yesmaximal, complemented
3C2×#5407C211yesmaximal, characteristic, complemented
4#73128C212yesmaximal, complemented
5#73126C211yesmaximal, characteristic, complemented
6#32942—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C2×#986C2212yes—
9C2×#131⋊C3—22no—
10C2×#131⋊C3C2211yescharacteristic
11C2×#131⋊C3C2211yescharacteristic
12#5427—22no—
13#5426C2212yes—
14#5421—22no—
15#5407—22no—
16#5115—41nomaximal
17C22×#131D311yescharacteristic, complemented
18C2×#472—31no—
19C2×#472—31no—
20C2×#466—31no—
21#3006—32no—
22#1907—32no—
23#1907—32no—
24#1900—32no—
25#1816—31no—
26#1816—31no—
27#1810—31no—
28C2×#41⋊C3—21no—
29C2×#41⋊C3—21no—
30C2×#41⋊C3D411yescharacteristic
31C2×#41⋊C3C2311yescharacteristic
32C22×SL(2,𝔽3)D411yescharacteristic
33#986—22no—
34#986—22no—
35C2×GL(2,𝔽3)—22no—
36C22×C3×C8—41no—
37#930—42no—
38C2×D24—41no—
39#131⋊C3—21no—
40#131⋊C3D412yescomplemented
41#859—22no—
42#821—42no—
43#818—41no—
44C22×#41C2×D311yescharacteristic
45C2×AGL(1,ℤ/8)—32no—
46C2×AGL(1,ℤ/8)—32no—
47C22×D8—31no—
48C2×#131—22no—
49C2×#131—31no—
50C2×#131—31no—
51C2×#131—62no—
52C2×#131C2×D311yescharacteristic
53C2×#131C2×D311yescharacteristic
54C22×M16—31no—
55C23×C8—31no—
56C2×#121—32no—
57#472—62no—
58#472—64no—
59#472—62no—
60#466—61no—
61#466—61no—
62#466—61no—
63#466—61no—
64#449—62no—
65#449—62no—
66#448—62no—
67#447—62no—
68C2×#108—31no—
69C2×#108—31no—
70C2×#107—31no—
71#417—32no—
72#417—32no—
73C2×#102—32no—
74C2×#100—32no—
75C2×#100—32no—
76#407—32no—
77#343—32no—
78#343—32no—
79#341—32no—
80C22×C3×C4—41no—
81C2×#67—82no—
82C2×D12—42no—
83#41⋊C3—21no—
84#41⋊C3C2×D411yescharacteristic
85#41⋊C3#4111yescharacteristic
86C2×SL(2,𝔽3)—21no—
87C2×SL(2,𝔽3)C2×D411yescharacteristic
88GL(2,𝔽3)—42no—
89C2×C3×C8—41no—
90C2×C3×C8—41no—
91C2×C3×C8—82no—
92(C2×D3)⋊C4—82no—
93C4⋊Dic3—42no—
94D24—82no—
95C2×#41—21no—
96C2×#41—21no—
97C2×#41—31no—
98C2×#41—31no—
99C2×#41—61no—
100C2×#41—61no—
101C2×#41D1211yescharacteristic
102C2×#41C22×D311yescharacteristic
103C22×Q8D1211yescharacteristic
104C22×D4—31no—
105C22×D4—32no—
106C22×D4—62no—
107C23×C4—31no—
108AGL(1,ℤ/8)—62no—
109AGL(1,ℤ/8)—64no—
110AGL(1,ℤ/8)—62no—
111AGL(1,ℤ/8)—64no—
112AGL(1,ℤ/8)—62no—
113AGL(1,ℤ/8)—62no—
114C2×SD16—62no—
115C2×SD16—62no—
116C2×D8—61no—
117C2×D8—62no—
118C2×D8—62no—
119C2×D8—62no—
120C2×D8—62no—
121C2×D8—61no—
122#131—21no—
123#131—32no—
124#131—34no—
125#131—62no—
126#131—62no—
127#131—61no—
128#131—62no—
129#131#6712yescomplemented
130C2×M16—31no—
131C2×M16—31no—
132C2×M16—32no—
133C2×M16—32no—
134C2×M16—61no—
135C2×M16—62no—
136C2×M16—62no—
137C22×C8—31no—
138C22×C8—31no—
139C22×C8—31no—
140C22×C8—31no—
141C22×C8—31no—
142C22×C8—62no—
143C22×C8—62no—
144C22×C8S411yescharacteristic
145#121—62no—
146#121—62no—
147#121—62no—
148#121—62no—
149C2×C4⋊C4—32no—
150C2×C22⋊C4—62no—
151#108—62no—
152#108—62no—
153#107—61no—
154#107—61no—
155#103—62no—
156#103—62no—
157#102—62no—
158#102—62no—
159#102—62no—
160#102—62no—
161#100—64no—
162#100—64no—
163C22⋊C8—62no—
164C22⋊C8—62no—
165C23×C3—41no—
166C22×D3—82no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—81no—
170C2×C3×C4—81no—
171#67—164no—
172C2×Dic3—82no—
173D12—82no—
174D12—82no—
175SL(2,𝔽3)#12111yescharacteristic
176C3×C8—42no—
177C3×C8—81no—
178C24—31no—
179C24—62no—
180#41—21no—
181#41—31no—
182#41—31no—
183#41—32no—
184#41—32no—
185#41—61no—
186#41—61no—
187#41—62no—
188#41—61no—
189#41—61no—
190#41—61no—
191#41#23511yescharacteristic
192#41C2×#6711yescharacteristic
193C2×Q8—21no—
194C2×Q8—31no—
195C2×Q8—61no—
196C2×Q8C2×D1211yescharacteristic
197C2×D4—31no—
198C2×D4—31no—
199C2×D4—61no—
200C2×D4—62no—
201C2×D4—62no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—62no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4—62no—
208C2×D4—62no—
209C2×D4—122no—
210C2×D4—122no—
211C2×D4—244no—
212C22×C4—31no—
213C22×C4—31no—
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
223SD16—122no—
224SD16—124no—
225SD16—122no—
226D8—122no—
227D8—122no—
228D8—122no—
229D8—124no—
230D8—122no—
231M16—32no—
232M16—32no—
233M16—61no—
234M16—64no—
235M16—61no—
236M16—62no—
237M16—62no—
238M16—62no—
239M16—62no—
240C2×C8—22no—
241C2×C8—31no—
242C2×C8—32no—
243C2×C8—32no—
244C2×C8—31no—
245C2×C8—62no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8C2×S411yescharacteristic
253C2×C8C2×S411yescharacteristic
254C4⋊C4—62no—
255C4⋊C4—62no—
256C22⋊C4—122no—
257C22⋊C4—122no—
258C22×C3—41no—
259C22×C3—42no—
260C22×C3—81no—
261C22×C3—81no—
262C2×D3—162no—
263C2×D3—162no—
264C2×D3—162no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—81no—
268Dic3—162no—
269C23—31no—
270C23—31no—
271C23—61no—
272C23—61no—
273C23—62no—
274C23—121no—
275C23—122no—
276C23—122no—
277C23—122no—
278C23—242no—
279C23—242no—
280C23C4⋊S411yescharacteristic
281Q8—31no—
282Q8—32no—
283Q8—61no—
284Q8#93011yescharacteristic
285D4—31no—
286D4—31no—
287D4—32no—
288D4—61no—
289D4—61no—
290D4—62no—
291D4—62no—
292D4—122no—
293D4—122no—
294D4—122no—
295D4—122no—
296D4—244no—
297D4—244no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
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—122no—
315C2×C4—242no—
316C2×C4C4⋊S411yescharacteristic
317C2×C4C22×S411yescharacteristic
318C8—21no—
319C8—32no—
320C8—61no—
321C8—122no—
322C8—122no—
323C8GL(2,ℤ/4)12yes—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—82no—
328D3—322no—
329C22—21no—
330C22—21no—
331C22—31no—
332C22—31no—
333C22—61no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—61no—
338C22—122no—
339C22—122no—
340C22—242no—
341C22—242no—
342C22—242no—
343C22—482no—
344C22—482no—
345C22#540712yescomplemented
346C22C2×C4⋊S411yescharacteristic
347C4—21no—
348C4—31no—
349C4—31no—
350C4—61no—
351C4—242no—
352C4#591211yescharacteristic
353C4C2×GL(2,ℤ/4)11yescharacteristic
354C3—41noSylow
355C2—22no—
356C2—61no—
357C2—61no—
358C2—121no—
359C2—482no—
360C2C2×#540711yescharacteristic
361C2#8545111yescharacteristic
362C2#8548611yescharacteristic
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC22×#131D3
hypercenterC22×C8S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C22×#131⋊C3C2semidirect
2. #85523C2semidirect
3. C2×#5407C2semidirect
4. #73128C2semidirect
5. #73126C2semidirect
17. C22×#131D3semidirect
40. #131⋊C3D4semidirect
129. #131#67semidirect
345. C22#5407semidirect

Maximal subgroups

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

subgroupindex
C22×#131⋊C32
#855232
C2×#54072
#731282
#731262
#329423
#51154