Finite Group Databasebeta

← #11922315

Subgroups — #11922315

Order 1152.

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

normal subgroups only

This group has 557 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
2Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4#10593204C212yesmaximal, complemented
5#10593198C212yesmaximal, complemented
6#85341—31nomaximal
7#83897—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#985C414yescomplemented
10C2×C3×GL(2,𝔽3)C2211yescharacteristic
11C2×C3×#226C2211yescharacteristic
12#63855—41nomaximal
13Dic3×SL(2,𝔽3)C2212yes—
14C3⋊#859C2212yes—
15C2×#985D311yescharacteristic
16C2×C3×#135—31no—
17Dic3×#41—31no—
18Dic3×#41D311yescharacteristic, complemented
19C4×#41⋊C3—31no—
20#5424—31no—
21#5424—61no—
22#5394—32no—
23#5388—32no—
24#5233—31no—
25#5233—31no—
26#5184—32no—
27#5164—32no—
28#5164—32no—
29#5152—32no—
30#5106—31no—
31C2×#815—31no—
32C2×C3×C4×D3—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3×GL(2,𝔽3)C2×C412yescomplemented
36C3×#226C2×C412yescomplemented
37C4×C3⋊Dic3—41no—
38C3×C4×Dic3—41no—
39Dic3⋊Dic3—42no—
40(C2×D3)⋊Dic3—42no—
41#2943—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—31no—
63#926—61no—
64C2×C4×Dic3—31no—
65C2×C4×Dic3—61no—
66C22×C3⋊C8—31no—
67#872—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#818—32no—
76#817—32no—
77#815—32no—
78#815—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
86C2×#135—31no—
87C4×#41—31no—
88C4×#41—91no—
89#442—92no—
90#441—92no—
91#440—92no—
92#440—92no—
93#427—91no—
94#416—91no—
95#416—91no—
96C2×#97—91no—
97C3×#41—32no—
98C3×#41—34no—
99C3×#41—62no—
100C3×#41C4×D312yes—
101C2×C3×Q8—31no—
102C2×C3×Q8C22×D311yescharacteristic
103C2×C3×D4—31no—
104C2×C3×D4—31no—
105C2×C3×D4—61no—
106C22×C3×C4—31no—
107C22×C3×C4—61no—
108C22×Dic3—32no—
109C22×Dic3—62no—
110C2×C4×D3—41no—
111#41⋊C3—22no—
112#41⋊C3C2×Dic312yescomplemented
113C2×SL(2,𝔽3)—21no—
114C2×SL(2,𝔽3)C22×D311yescharacteristic
115GL(2,𝔽3)C2×Dic312yescomplemented
116#226C2×Dic312yescomplemented
117C3×Q16—32no—
118C3×Q16—32no—
119C3×SD16—34no—
120C3×SD16—32no—
121C3×SD16—32no—
122C3×D8—34no—
123C2×C3×C8—31no—
124C2×C3×C8—31no—
125C2×C3×C8—34no—
126C3×C42—121no—
127#217—61no—
128#217—61no—
129#217—62no—
130(C2×D3)⋊C4—122no—
131C4⋊Dic3—32no—
132C4⋊Dic3—32no—
133C4⋊Dic3—62no—
134Dic3⋊C4—122no—
135C4×Dic3—31no—
136C4×Dic3—32no—
137C4×Dic3—62no—
138C4×Dic3—61no—
139C4×Dic3—121no—
140C4×Dic3—241no—
141C4×Dic3S411yescharacteristic
142C2×C3⋊C8—32no—
143C2×C3⋊C8—62no—
144C22×C32—41no—
145C2×C3×D3—42no—
146C2×C3×D3—82no—
147C32×C4—42no—
148C3⋊Dic3—82no—
149C3×Dic3—42no—
150C3×Dic3—82no—
151C2×#41—31no—
152C2×#41D3211yescharacteristic
153#135—34no—
154#135—34no—
155C2×Q16—31no—
156C2×SD16—31no—
157C2×SD16—31no—
158C2×D8—31no—
159C22×C8—31no—
160C22×C8—91no—
161C4×Q8—32no—
162C4×Q8—92no—
163C4×D4—92no—
164C4×D4—92no—
165C4×D4—182no—
166#117—91no—
167#117—91no—
168#117—181no—
169C2×C42—91no—
170C2×C42—181no—
171#107—92no—
172#106—92no—
173#103—92no—
174#103—92no—
175#102—92no—
176#102—92no—
177#97—92no—
178#97—92no—
179C23×C3—31no—
180C23×C3—61no—
181C22×D3—41no—
182C3×Q8—31no—
183C3×Q8—32no—
184C3×Q8—61no—
185C3×Q8C2×C4×D311yescharacteristic
186C3×D4—32no—
187C3×D4—32no—
188C3×D4—61no—
189C3×D4—61no—
190C3×D4—64no—
191C2×C3×C4—31no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—41no—
195C2×C3×C4—61no—
196C2×C3×C4—64no—
197C2×C3×C4—81no—
198C2×C3×C4—122no—
199C2×C3×C4C2×S411yescharacteristic
200C2×Dic3—32no—
201C2×Dic3—41no—
202C2×Dic3—62no—
203C2×Dic3—122no—
204C2×Dic3—122no—
205C2×Dic3—124no—
206C2×Dic3—242no—
207C2×Dic3C2×S412yes—
208C4×D3—44no—
209SL(2,𝔽3)—21no—
210SL(2,𝔽3)C22×Dic311yescharacteristic
211C3×C8—32no—
212C3×C8—32no—
213C3⋊C8—62no—
214C2×C32—41no—
215C2×C32—42no—
216C3×D3—82no—
217#41—32no—
218#41—34no—
219#41—62no—
220#41D3×Dic312yes—
221C2×Q8—31no—
222C2×Q8C2×D3211yescharacteristic
223C2×D4—31no—
224C2×D4—31no—
225C2×D4—61no—
226C22×C4—31no—
227C22×C4—61no—
228C22×C4—92no—
229C22×C4—182no—
230Q16—32no—
231Q16—32no—
232SD16—34no—
233SD16—32no—
234SD16—32no—
235D8—34no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8—34no—
239C2×C8—92no—
240C2×C8—182no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C4⋊C4—182no—
244C22⋊C4—182no—
245C22⋊C4—181no—
246C22⋊C4—181no—
247C42—31no—
248C42—92no—
249C42—91no—
250C42—182no—
251C42—181no—
252C22×C3—32no—
253C22×C3—41no—
254C22×C3—62no—
255C22×C3—62no—
256C22×C3—81no—
257C22×C3—122no—
258C22×C3C22×S411yescharacteristic
259C2×D3—42no—
260C2×D3—82no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—42no—
264C3×C4—62no—
265C3×C4—82no—
266C3×C4—242no—
267C3×C4C4×S412yes—
268Dic3—22no—
269Dic3—42no—
270Dic3—62no—
271Dic3—122no—
272Dic3—242no—
273Dic3—244no—
274C32—41noSylow
275C23—31no—
276C23—61no—
277Q8—31no—
278Q8—32no—
279Q8—61no—
280Q8C2×D3×Dic311yescharacteristic
281D4—32no—
282D4—32no—
283D4—61no—
284D4—61no—
285D4—64no—
286C2×C4—31no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—61no—
291C2×C4—64no—
292C2×C4—92no—
293C2×C4—182no—
294C2×C4—362no—
295C2×C4—364no—
296C2×C4D3×S411yescharacteristic
297C8—32no—
298C8—32no—
299C8—182no—
300C2×C3—42no—
301C2×C3—41no—
302C2×C3—62no—
303C2×C3—82no—
304C2×C3—81no—
305C2×C3—122no—
306C2×C3C2×C4×S411yescharacteristic
307C2×C3#592312yes—
308D3—82no—
309C22—32no—
310C22—62no—
311C22—62no—
312C22—122no—
313C22C2×D3×S411yescharacteristic
314C4—31no—
315C4—31no—
316C4—62no—
317C4—62no—
318C4—182no—
319C4—362no—
320C4Dic3×S412yes—
321C3—41no—
322C3—81no—
323C3#8534111yescharacteristic, complemented
324C2—62no—
325C2—122no—
326C2#1059645412yes—
327C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C2×D3×S4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×C3×#985C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. Dic3×#41⋊C3C2semidirect
3. #10593658C2semidirect
4. #10593204C2semidirect
5. #10593198C2semidirect
9. C3×#985C4semidirect
18. Dic3×#41D3semidirect
33. C3×#41⋊C3C2×C4semidirect
35. C3×GL(2,𝔽3)C2×C4semidirect
36. C3×#226C2×C4semidirect
46. #985Dic3semidirect
112. #41⋊C3C2×Dic3semidirect
115. GL(2,𝔽3)C2×Dic3semidirect
116. #226C2×Dic3semidirect
323. C3#85341semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9852
Dic3×#41⋊C32
#105936582
#105932042
#105931982
#853413
#838973
#638554