Finite Group Databasebeta

← #11922379

Subgroups — #11922379

Order 1152.

567 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 75 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×C3×#985C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4Dic3×GL(2,𝔽3)C212yesmaximal, complemented
5#10593198C212yesmaximal, complemented
6#85364—31nomaximal
7#83899—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#985C414yescomplemented
10C2×C3×GL(2,𝔽3)C2211yescharacteristic
11C2×C3×#226C2211yescharacteristic
12D3×C4⋊Dic3—41nomaximal
13Dic3×SL(2,𝔽3)C2212yes—
14C3⋊#859C2212yes—
15C2×#985D311yescharacteristic
16C2×C3×#135—31no—
17#5827—31no—
18#5827D311yescharacteristic, complemented
19#520⋊C3—31no—
20#5426—31no—
21#5426—61no—
22C4×GL(2,𝔽3)—32no—
23#5388—32no—
24#5232—31no—
25#5232—31no—
26#5184—32no—
27Dic3×SD16—32no—
28Dic3×SD16—32no—
29#5152—32no—
30C2×#817—31no—
31#5101—31no—
32C2×C3×C4×D3—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×D3×Dic3—42no—
36C3×GL(2,𝔽3)C2×C412yescomplemented
37C3×#226C2×C412yescomplemented
38C4⋊(C3⋊Dic3)—41no—
39C3×C4⋊Dic3—41no—
40Dic3⋊Dic3—42no—
41#2952—91noSylow
42C2×C3×#41—31no—
43C2×C3×#41C2×D311yescharacteristic
44C2×#41⋊C3—21no—
45C2×#41⋊C3C2×D311yescharacteristic
46#985Dic314yescomplemented
47C2×GL(2,𝔽3)C2×D311yescharacteristic
48C2×#226C2×D311yescharacteristic
49C3×#135—34no—
50C3×#135—34no—
51C2×C3×Q16—31no—
52C2×C3×SD16—31no—
53C2×C3×SD16—31no—
54C2×C3×D8—31no—
55C22×C3×C8—31no—
56Q8×Dic3—32no—
57Q8×Dic3C2×D312yes—
58D4×Dic3—32no—
59D4×Dic3—32no—
60D4×Dic3—62no—
61#926—31no—
62#926—61no—
63C2×C4⋊Dic3—31no—
64C2×C4⋊Dic3—31no—
65C2×C4⋊Dic3—61no—
66C2×#208—31no—
67D3×C4⋊C4—121no—
68C4×SL(2,𝔽3)—32no—
69#859—32no—
70#859—62no—
71#835—32no—
72#835—32no—
73#832—32no—
74#832—32no—
75#817—32no—
76#817—32no—
77#815—32no—
78C8×Dic3—32no—
79C22×C3×D3—41no—
80C2×C32×C4—41no—
81C2×C3⋊Dic3—42no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—42no—
84C3×C4×D3—44no—
85C3×SL(2,𝔽3)C22×C411yescharacteristic
86D3×Dic3—84no—
87C2×#135—31no—
88#520—31no—
89#520—91no—
90#442—92no—
91#441—92no—
92C4×SD16—92no—
93C4×SD16—92no—
94C2×#106—91no—
95#417—91no—
96#417—91no—
97#405—91no—
98C3×#41—32no—
99C3×#41—34no—
100C3×#41—62no—
101C3×#41C4×D312yescomplemented
102C2×C3×Q8—31no—
103C2×C3×Q8C22×D311yescharacteristic
104C2×C3×D4—31no—
105C2×C3×D4—31no—
106C2×C3×D4—61no—
107C22×C3×C4—31no—
108C22×C3×C4—61no—
109C22×Dic3—61no—
110C22×Dic3—61no—
111C22×Dic3—62no—
112C2×C4×D3—41no—
113C2×C4×D3—122no—
114#41⋊C3—22no—
115#41⋊C3C2×Dic312yescomplemented
116C2×SL(2,𝔽3)—21no—
117C2×SL(2,𝔽3)C22×D311yescharacteristic
118GL(2,𝔽3)C2×Dic312yescomplemented
119#226C2×Dic312yescomplemented
120C3×Q16—32no—
121C3×Q16—32no—
122C3×SD16—34no—
123C3×SD16—32no—
124C3×SD16—32no—
125C3×D8—34no—
126C2×C3×C8—31no—
127C2×C3×C8—31no—
128C2×C3×C8—34no—
129C3×C4⋊C4—121no—
130#217—32no—
131#217—62no—
132C4⋊Dic3—31no—
133C4⋊Dic3—32no—
134C4⋊Dic3—32no—
135C4⋊Dic3—61no—
136C4⋊Dic3—62no—
137C4⋊Dic3—121no—
138C4⋊Dic3—241no—
139C4⋊Dic3S411yescharacteristic
140Dic3⋊C4—122no—
141C4×Dic3—32no—
142C4×Dic3—62no—
143#208—62no—
144C2×C3⋊C8—32no—
145C22×C32—41no—
146C2×C3×D3—42no—
147C2×C3×D3—42no—
148C2×C3×D3—42no—
149C32×C4—42no—
150C3⋊Dic3—82no—
151C3×Dic3—42no—
152C3×Dic3—82no—
153C2×#41—31no—
154C2×#41D3211yescharacteristic
155#135—34no—
156#135—34no—
157C2×Q16—31no—
158C2×SD16—31no—
159C2×SD16—31no—
160C2×D8—31no—
161C2×M16—91no—
162C22×C8—31no—
163C4×Q8—32no—
164C4×Q8—92no—
165C4×D4—92no—
166C4×D4—92no—
167C4×D4—182no—
168#117—91no—
169#117—181no—
170C2×C4⋊C4—91no—
171C2×C4⋊C4—91no—
172C2×C4⋊C4—181no—
173#106—92no—
174#106—92no—
175#103—92no—
176#103—92no—
177#102—92no—
178#102—92no—
179#97—92no—
180C4×C8—92no—
181C23×C3—31no—
182C23×C3—61no—
183C22×D3—41no—
184C3×Q8—31no—
185C3×Q8—32no—
186C3×Q8—61no—
187C3×Q8C2×C4×D311yescharacteristic
188C3×D4—32no—
189C3×D4—32no—
190C3×D4—61no—
191C3×D4—61no—
192C3×D4—64no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—61no—
198C2×C3×C4—64no—
199C2×C3×C4—81no—
200C2×C3×C4—122no—
201C2×C3×C4C2×S411yescharacteristic
202C2×Dic3—32no—
203C2×Dic3—41no—
204C2×Dic3—62no—
205C2×Dic3—122no—
206C2×Dic3—242no—
207C2×Dic3—242no—
208C2×Dic3—241no—
209C2×Dic3—241no—
210C2×Dic3C2×S412yes—
211C4×D3—44no—
212C4×D3—244no—
213SL(2,𝔽3)—21no—
214SL(2,𝔽3)C22×Dic311yescharacteristic
215C3×C8—32no—
216C3×C8—32no—
217C3⋊C8—62no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C3×D3—44no—
222#41—32no—
223#41—34no—
224#41—62no—
225#41D3×Dic312yescomplemented
226C2×Q8—31no—
227C2×Q8C2×D3211yescharacteristic
228C2×D4—31no—
229C2×D4—31no—
230C2×D4—61no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—182no—
234C22×C4—181no—
235C22×C4—181no—
236Q16—32no—
237Q16—32no—
238SD16—34no—
239SD16—32no—
240SD16—32no—
241D8—34no—
242M16—182no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—34no—
246C2×C8—92no—
247C4⋊C4—31no—
248C4⋊C4—92no—
249C4⋊C4—92no—
250C4⋊C4—91no—
251C4⋊C4—182no—
252C4⋊C4—181no—
253C22⋊C4—92no—
254C22⋊C4—182no—
255C42—92no—
256C42—182no—
257C22×C3—32no—
258C22×C3—41no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—62no—
262C22×C3—81no—
263C22×C3—121no—
264C22×C3—121no—
265C22×C3C22×S411yescharacteristic
266C2×D3—42no—
267C2×D3—42no—
268C2×D3—42no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—42no—
272C3×C4—62no—
273C3×C4—82no—
274C3×C4—242no—
275C3×C4C4×S412yes—
276Dic3—22no—
277Dic3—42no—
278Dic3—62no—
279Dic3—122no—
280Dic3—242no—
281Dic3—482no—
282C32—41noSylow
283C23—31no—
284C23—61no—
285Q8—31no—
286Q8—32no—
287Q8—61no—
288Q8C2×D3×Dic311yescharacteristic
289D4—32no—
290D4—32no—
291D4—61no—
292D4—61no—
293D4—64no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—61no—
299C2×C4—64no—
300C2×C4—92no—
301C2×C4—182no—
302C2×C4—722no—
303C2×C4—721no—
304C2×C4—721no—
305C2×C4D3×S411yescharacteristic
306C8—32no—
307C8—32no—
308C8—182no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—62no—
313C2×C3—81no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3—122no—
317C2×C3C2×C4×S411yescharacteristic
318C2×C3#592411yescharacteristic
319C2×C3#592611yescharacteristic
320D3—44no—
321C22—32no—
322C22—61no—
323C22—61no—
324C22—62no—
325C22—121no—
326C22—121no—
327C22C2×D3×S411yescharacteristic
328C4—31no—
329C4—31no—
330C4—62no—
331C4—62no—
332C4—182no—
333C4—362no—
334C4Dic3×S412yes—
335C3—41no—
336C3—81no—
337C3#8536411yescharacteristic, complemented
338C2—62no—
339C2—122no—
340C2#1059644711yescharacteristic
341C2#1059645911yescharacteristic
342C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×C3×#985C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC2×C4D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. #10593660C2semidirect
4. Dic3×GL(2,𝔽3)C2semidirect
5. #10593198C2semidirect
9. C3×#985C4semidirect
18. #5827D3semidirect
33. C3×#41⋊C3C2×C4semidirect
36. C3×GL(2,𝔽3)C2×C4semidirect
37. C3×#226C2×C4semidirect
46. #985Dic3semidirect
101. C3×#41C4×D3semidirect
115. #41⋊C3C2×Dic3semidirect
118. GL(2,𝔽3)C2×Dic3semidirect
119. #226C2×Dic3semidirect
225. #41D3×Dic3semidirect
337. C3#85364semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9852
#5827⋊C32
#105936602
Dic3×GL(2,𝔽3)2
#105931982
#853643
#838993
D3×C4⋊Dic34