Finite Group Databasebeta

← #11921731

Subgroups — #11921731

Order 1152.

501 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593625C211yesmaximal, characteristic, complemented
3C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593217C212yesmaximal, complemented
5#10593204C212yesmaximal, complemented
6#85332—31nomaximal
7#80362—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×GL(2,𝔽3)C2212yes—
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11#63908—41nomaximal
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13C3⋊#859C2212yes—
14#5716—31no—
15C4×D3×Q8D311yescharacteristic, complemented
16C2×C4×SL(2,𝔽3)—31no—
17#5421—32no—
18#5394—32no—
19C4⋊GL(2,𝔽3)D311yescharacteristic, complemented
20#5391—31no—
21#5391—61no—
22C3×#460—31no—
23#5170—32no—
24#5164—32no—
25#5075—31no—
26#5021—31no—
27#4850—31no—
28#4835—31no—
29#4774—32no—
30#4757—32no—
31C2×C3×D12—41no—
32C2×C3×C4×D3—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C2×#625—42no—
35D3×SL(2,𝔽3)—21no—
36C3×GL(2,𝔽3)—22no—
37C4⋊(C3⋊Dic3)—41no—
38(C2×D3)⋊Dic3—42no—
39#3112—91noSylow
40C2×D3×Q8C2×D311yescharacteristic
41C2×#237—61no—
42C22×SL(2,𝔽3)—31no—
43C2×GL(2,𝔽3)C2×D312yes—
44C2×C3×SD16—32no—
45C3×#127—31no—
46C3×C4×Q8C2×D311yescharacteristic
47Q8×Dic3C2×D311yescharacteristic
48#938—32no—
49#938—61no—
50#936—61no—
51D4×Dic3—32no—
52D4×Dic3—61no—
53C2×#214—32no—
54#930—121no—
55D3×C4⋊C4—31no—
56#887—31no—
57C42×D3—31no—
58#869—31no—
59C4×Dic6—31no—
60C4×SL(2,𝔽3)—21no—
61C4×SL(2,𝔽3)—31no—
62C4×SL(2,𝔽3)C2×D311yescharacteristic
63#859—32no—
64#859—62no—
65C3×C4⋊C8—31no—
66C3×#102—32no—
67#835—32no—
68#832—32no—
69(C2×D3)⋊C8—32no—
70#818—32no—
71#816—32no—
72#815—32no—
73#807—32no—
74C4⋊(C3⋊C8)—31no—
75C22×C3×D3—42no—
76C22×C3×D3—41no—
77C2×C32×C4—41no—
78C2×C3⋊Dic3—42no—
79C2×C3×Dic3—41no—
80C3×D12—82no—
81C3×C4×D3—82no—
82C3×SL(2,𝔽3)#4111yescharacteristic, complemented
83#625—84no—
84#532—91no—
85C2×C4×Q8—31no—
86#480—92no—
87#477—91no—
88#477—91no—
89#460—31no—
90#460—91no—
91#448—92no—
92#440—92no—
93#440—92no—
94#432—91no—
95C2×C3×Q8C22×D311yescharacteristic
96C2×C3×D4—32no—
97C2×C3×D4—61no—
98C2×C3×D4—61no—
99C22×C3×C4—121no—
100C2×#67—62no—
101C2×#67—122no—
102C22×Dic3—62no—
103D3×Q8—21no—
104D3×Q8—61no—
105#237—124no—
106C2×D12—41no—
107C2×C4×D3—31no—
108C2×C4×D3—31no—
109C2×C4×D3S411yescharacteristic
110C2×Dic6—31no—
111C2×Dic6—61no—
112C2×SL(2,𝔽3)—21no—
113C2×SL(2,𝔽3)—61no—
114C2×SL(2,𝔽3)C22×D311yescharacteristic
115GL(2,𝔽3)—22no—
116C3×SD16—62no—
117C3×SD16—62no—
118C2×C3×C8—32no—
119C3×C4⋊C4—31no—
120C3×C42—31no—
121#217—62no—
122#217—62no—
123#214—64no—
124(C2×D3)⋊C4—122no—
125C4⋊Dic3—31no—
126C4⋊Dic3—32no—
127C4⋊Dic3—61no—
128C4⋊Dic3—121no—
129C4⋊Dic3—241no—
130Dic3⋊C4—61no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C2×C3⋊C8—32no—
135C22×C32—41no—
136C2×C3×D3—42no—
137C2×C3×D3—81no—
138C2×C3×D3—81no—
139C2×C3×D3—82no—
140C2×C3×D3—82no—
141C2×C3×D3—82no—
142C32×C4—42no—
143C3⋊Dic3—82no—
144C3×Dic3—81no—
145C2×#41—181no—
146C22×Q8—31no—
147C2×SD16—32no—
148C2×SD16—92no—
149C4⋊Q8—91no—
150#127—31no—
151#124—181no—
152#121—92no—
153#121—181no—
154C4×Q8—31no—
155C4×Q8—91no—
156C4×Q8—91no—
157C4×Q8D3211yescharacteristic
158C4×D4—92no—
159C4×D4—181no—
160C2×C4⋊C4—91no—
161C2×C42—91no—
162#107—92no—
163#107—92no—
164C4⋊C8—31no—
165C4⋊C8—91no—
166#103—92no—
167#103—92no—
168#102—32no—
169#102—92no—
170C22⋊C8—92no—
171#97—92no—
172C23×C3—62no—
173C23×C3—121no—
174C22×D3—42no—
175C22×D3C2×S411yescharacteristic
176C3×Q8—31no—
177C3×Q8#23511yescharacteristic
178C3×D4—62no—
179C3×D4—62no—
180C3×D4—122no—
181C3×D4—122no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—41no—
185C2×C3×C4—81no—
186C2×C3×C4—121no—
187C2×C3×C4—242no—
188C2×C3×C4C2×S411yescharacteristic
189#67—124no—
190#67—244no—
191C2×Dic3—31no—
192C2×Dic3—31no—
193C2×Dic3—62no—
194C2×Dic3—122no—
195C2×Dic3—124no—
196C2×Dic3—242no—
197C2×Dic3C2×S411yescharacteristic
198D12—82no—
199C4×D3—22no—
200C4×D3—61no—
201C4×D3—61no—
202C4×D3—62no—
203Dic6—32no—
204Dic6—61no—
205Dic6—122no—
206SL(2,𝔽3)—21no—
207SL(2,𝔽3)#23711yescharacteristic, complemented
208C3×C8—62no—
209C3⋊C8—62no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C3×D3—82no—
214C3×D3—162no—
215#41—364no—
216C2×Q8—61no—
217C2×Q8—91no—
218C2×Q8—181no—
219C2×Q8—181no—
220C2×Q8C2×D3211yescharacteristic
221C2×D4—32no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—182no—
225C22×C4—31no—
226C22×C4—91no—
227C22×C4—91no—
228C22×C4—182no—
229SD16—62no—
230SD16—62no—
231SD16—184no—
232C2×C8—32no—
233C2×C8—92no—
234C4⋊C4—31no—
235C4⋊C4—91no—
236C4⋊C4—92no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C22⋊C4—182no—
240C22⋊C4—182no—
241C42—31no—
242C42—91no—
243C42—91no—
244C42—91no—
245C22×C3—41no—
246C22×C3—81no—
247C22×C3—122no—
248C22×C3—122no—
249C22×C3—122no—
250C22×C3—122no—
251C22×C3—241no—
252C22×C3—241no—
253C22×C3C22×S411yescharacteristic
254C2×D3—21no—
255C2×D3—21no—
256C2×D3—82no—
257C2×D3—82no—
258C2×D3—82no—
259C2×D3#98612yes—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—42no—
263C3×C4—61no—
264C3×C4—82no—
265C3×C4—241no—
266C3×C4#98312yes—
267Dic3—21no—
268Dic3—61no—
269Dic3—61no—
270Dic3—122no—
271Dic3—242no—
272Dic3—482no—
273C32—41noSylow
274C23—31no—
275C23—62no—
276Q8—31no—
277Q8—92no—
278Q8—181no—
279Q8—362no—
280Q8#379511yescharacteristic
281D4—62no—
282D4—62no—
283D4—122no—
284D4—122no—
285D4—364no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—62no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—364no—
297C2×C4D3×S411yescharacteristic
298C8—62no—
299C8—182no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—81no—
304C2×C3—81no—
305C2×C3—81no—
306C2×C3—242no—
307C2×C3—242no—
308C2×C3#591211yescharacteristic
309C2×C3C2×#98311yescharacteristic
310C2×C3C2×#98611yescharacteristic
311D3—22no—
312D3—162no—
313C22—32no—
314C22—61no—
315C22—61no—
316C22—122no—
317C22—122no—
318C22—122no—
319C22C2×D3×S411yescharacteristic
320C4—31no—
321C4—31no—
322C4—61no—
323C4—61no—
324C4—181no—
325C4—181no—
326C4—362no—
327C4#6420112yes—
328C3—41no—
329C3—81no—
330C3#8533211yescharacteristic, complemented
331C2—62no—
332C2—242no—
333C2#1059642711yescharacteristic
334C2C2×#6420111yescharacteristic
335C2D3×#98611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4×D3×SL(2,𝔽3)C2semidirect
2. #10593625C2semidirect
3. C3×C4⋊GL(2,𝔽3)C2semidirect
4. #10593217C2semidirect
5. #10593204C2semidirect
15. C4×D3×Q8D3semidirect
19. C4⋊GL(2,𝔽3)D3semidirect
82. C3×SL(2,𝔽3)#41semidirect
207. SL(2,𝔽3)#237semidirect
330. C3#85332semidirect

Maximal subgroups

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

subgroupindex
C4×D3×SL(2,𝔽3)2
#105936252
C3×C4⋊GL(2,𝔽3)2
#105932172
#105932042
#853323
#803623
#639084