Finite Group Databasebeta

← #11713979

Subgroups — #11713979

Order 768.

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

normal subgroups only

This group has 670 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×#520⋊C3C211yesmaximal, characteristic, complemented
2C2×#5427C212yesmaximal, complemented
3C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73065C212yesmaximal, complemented
5#73064C211yesmaximal, characteristic, complemented
6#32913—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×GL(2,𝔽3)C2212yes—
9#520⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12#5427—24no—
13C2×#859C2212yes—
14C4⋊GL(2,𝔽3)—22no—
15#4991—41nomaximal
16C2×#520D311yescharacteristic, complemented
17C2×#460—31no—
18C2×#459—31no—
19C2×#449—32no—
20#2619—31no—
21#1880—32no—
22#1879—32no—
23#1874—32no—
24#1805—31no—
25#1804—31no—
26#1798—31no—
27C22×D12—41no—
28C2×#41⋊C3—21no—
29C2×#41⋊C3—21no—
30C2×#41⋊C3D412yescomplemented
31C22×SL(2,𝔽3)C2311yescharacteristic
32C2×GL(2,𝔽3)—22no—
33C2×GL(2,𝔽3)—22no—
34C2×GL(2,𝔽3)—22no—
35C2×C3×C4⋊C4—41no—
36C2×(C2×D3)⋊C4—42no—
37C2×(C2×D3)⋊C4—42no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)D412yescomplemented
40#859—22no—
41C3⋊#95—41no—
42C22×#41C2×D311yescharacteristic
43C22×SD16—32no—
44C22×D8—32no—
45C2×#127—31no—
46C2×#121—32no—
47#520—22no—
48#520—62no—
49C2×C4×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×D4—31no—
52C2×C4×D4—31no—
53C2×#117—31no—
54C22×C4⋊C4—31no—
55#460—62no—
56#460—62no—
57#459—64no—
58#449—64no—
59#449—64no—
60C2×C4⋊C8—31no—
61C2×C4⋊C8—31no—
62C2×#103—32no—
63C2×#102—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—
76C23×D3—42no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×Dic3—42no—
81C2×D12—82no—
82C2×D12—84no—
83#41⋊C3—22no—
84C2×SL(2,𝔽3)C2×D411yescharacteristic
85C2×SL(2,𝔽3)C2×D411yescharacteristic
86C2×SL(2,𝔽3)#4111yescharacteristic
87GL(2,𝔽3)—42no—
88C3×C4⋊C4—82no—
89(C2×D3)⋊C4—84no—
90(C2×D3)⋊C4—84no—
91C2×#41—21no—
92C2×#41—21no—
93C2×#41—32no—
94C2×#41—61no—
95C2×#41—61no—
96C2×#41D1212yescomplemented
97C22×Q8C22×D311yescharacteristic
98C22×D4—31no—
99C22×D4—32no—
100C22×D4—61no—
101C22×D4—62no—
102C22×D4—61no—
103C23×C4—31no—
104C23×C4—61no—
105C2×SD16—62no—
106C2×SD16—62no—
107C2×SD16—62no—
108C2×SD16—62no—
109C2×SD16—62no—
110C2×SD16—62no—
111C2×D8—64no—
112C2×D8—64no—
113C2×D8—64no—
114C22×C8—32no—
115C22×C8—32no—
116#127—62no—
117#127—62no—
118#121—64no—
119#121—64no—
120C4×Q8—21no—
121C4×Q8—32no—
122C4×Q8—61no—
123C4×Q8#6712yes—
124C4×D4—34no—
125C4×D4—62no—
126C4×D4—62no—
127C4×D4—62no—
128#117—62no—
129#117—62no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—32no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4S411yescharacteristic
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C42—31no—
143C2×C42—31no—
144C4⋊C8—62no—
145C4⋊C8—62no—
146#103—62no—
147#103—62no—
148#102—64no—
149#102—64no—
150#102—64no—
151C22⋊C8—64no—
152#95—61no—
153#95—61no—
154C23×C3—41no—
155C22×D3—44no—
156C22×D3—82no—
157C22×D3—82no—
158C22×D3—82no—
159C22×D3—82no—
160C22×D3—82no—
161C22×D3—82no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×C3×C4—82no—
167C2×C3×C4—81no—
168C2×C3×C4—82no—
169C2×Dic3—82no—
170C2×Dic3—82no—
171C2×Dic3—82no—
172D12—88no—
173SL(2,𝔽3)#12111yescharacteristic
174C24—31no—
175C24—62no—
176#41—22no—
177#41—64no—
178#41—62no—
179#41—62no—
180#41—62no—
181C2×Q8—31no—
182C2×Q8—31no—
183C2×Q8—31no—
184C2×Q8C2×D1211yescharacteristic
185C2×Q8#23511yescharacteristic
186C2×Q8C2×#6711yescharacteristic
187C2×D4—32no—
188C2×D4—32no—
189C2×D4—32no—
190C2×D4—61no—
191C2×D4—61no—
192C2×D4—61no—
193C2×D4—62no—
194C2×D4—62no—
195C2×D4—62no—
196C2×D4—62no—
197C2×D4—62no—
198C2×D4—62no—
199C2×D4—122no—
200C2×D4—124no—
201C2×D4—122no—
202C2×D4—242no—
203C2×D4—242no—
204C2×D4—242no—
205C2×D4—242no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—32no—
209C22×C4—32no—
210C22×C4—31no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—64no—
216C22×C4—62no—
217C22×C4—242no—
218C22×C4C2×S411yescharacteristic
219C22×C4C2×S411yescharacteristic
220C22×C4C2×S411yescharacteristic
221SD16—122no—
222SD16—122no—
223SD16—122no—
224SD16—122no—
225D8—128no—
226C2×C8—62no—
227C2×C8—62no—
228C2×C8—62no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—62no—
232C4⋊C4—22no—
233C4⋊C4—32no—
234C4⋊C4—32no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C22⋊C4—62no—
241C22⋊C4—62no—
242C22⋊C4—124no—
243C22⋊C4—242no—
244C22⋊C4—242no—
245C42—32no—
246C42—32no—
247C42—61no—
248C42—61no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—41no—
254C2×D3—84no—
255C2×D3—84no—
256C2×D3—84no—
257C2×D3—162no—
258C2×D3—162no—
259C2×D3—162no—
260C2×D3—162no—
261C3×C4—82no—
262C3×C4—82no—
263C3×C4—161no—
264Dic3—162no—
265C23—32no—
266C23—61no—
267C23—62no—
268C23—62no—
269C23—61no—
270C23—122no—
271C23—122no—
272C23—122no—
273C23—122no—
274C23—122no—
275C23—122no—
276C23—122no—
277C23C22×S411yescharacteristic
278Q8—31no—
279Q8—31no—
280Q8—31no—
281Q8—61no—
282Q8#93011yescharacteristic
283D4—34no—
284D4—62no—
285D4—62no—
286D4—62no—
287D4—122no—
288D4—122no—
289D4—122no—
290D4—122no—
291D4—244no—
292D4—244no—
293D4—248no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—32no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—61no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—244no—
317C2×C4C4⋊S412yes—
318C2×C4GL(2,ℤ/4)12yes—
319C8—122no—
320C8—122no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—42no—
324C2×C3—42no—
325C2×C3—41no—
326D3—164no—
327C22—64no—
328C22—62no—
329C22—121no—
330C22—121no—
331C22—122no—
332C22—242no—
333C22—242no—
334C22—242no—
335C22—242no—
336C22—242no—
337C22—242no—
338C22—242no—
339C22C2×C4⋊S411yescharacteristic
340C22#591211yescharacteristic
341C22#592512yes—
342C22#592612yes—
343C22C2×GL(2,ℤ/4)11yescharacteristic
344C4—22no—
345C4—22no—
346C4—31no—
347C4—31no—
348C4—31no—
349C4—31no—
350C4—41no—
351C4—62no—
352C4—121no—
353C4—242no—
354C3—41noSylow
355C2—122no—
356C2—482no—
357C2#8536712yes—
358C2#8536812yes—
359C2#8548611yescharacteristic
360C2#8553011yescharacteristic
361C2#8553711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#520D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#520⋊C3C2semidirect
2. C2×#5427C2semidirect
3. C2×C4⋊GL(2,𝔽3)C2semidirect
4. #73065C2semidirect
5. #73064C2semidirect
16. C2×#520D3semidirect
30. C2×#41⋊C3D4semidirect
39. C4×SL(2,𝔽3)D4semidirect
96. C2×#41D12semidirect

Maximal subgroups

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

subgroupindex
C2×#520⋊C32
C2×#54272
C2×C4⋊GL(2,𝔽3)2
#730652
#730642
#329133
#49914