Finite Group Databasebeta

← #11922108

Subgroups — #11922108

Order 1152.

647 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 59 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×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3×#983C211yesmaximal, characteristic, complemented
3#10593654C211yesmaximal, characteristic, complemented
4#10593203C212yesmaximal, complemented
5#10593194C212yesmaximal, complemented
6#85522—31nomaximal
7#83980—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊(#41⋊C3)—22no—
10C2×C3⋊(#41⋊C3)C2212yes—
11C3×#983—22no—
12C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3×#226C2211yescharacteristic, complemented
14#63961—41nomaximal
15C3⋊#859C2212yescomplemented
16C22×#239D311yescharacteristic, complemented
17C22×#41⋊C3—31no—
18C2×#983D311yescharacteristic, complemented
19C2×C3×#137—31no—
20#5828—31no—
21C2×#949—31no—
22#5427—32no—
23#5425—32no—
24#5420—31no—
25#5420—61no—
26#5225—31no—
27#5187—32no—
28#5187—32no—
29#5172—32no—
30#5172—32no—
31#5131—31no—
32#5126—31no—
33C2×C3×#67—41no—
34C22×C3×Dic3—41no—
35C2×C3×SL(2,𝔽3)D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C3⋊(#41⋊C3)—22no—
38C3×GL(2,𝔽3)—21no—
39C3×#226—21no—
40#3757—41no—
41#3724—42no—
42(C2×D3)⋊Dic3—42no—
43#3005—91noSylow
44C2×C3×#41—31no—
45C22×C3×Q8C2×D311yescharacteristic
46C2×#239—22no—
47C2×#239—32no—
48C2×#239—62no—
49C2×#239C2×D312yes—
50C22×D12—31no—
51C22×C4×D3—31no—
52C2×#41⋊C3—32no—
53C2×#41⋊C3—62no—
54C22×SL(2,𝔽3)—21no—
55C22×SL(2,𝔽3)C2×D311yescharacteristic
56#983—22no—
57C2×GL(2,𝔽3)C2×D311yescharacteristic, complemented
58C2×#226C2×D311yescharacteristic, complemented
59C3×#137—62no—
60C3×#137—62no—
61C2×C3×Q16—31no—
62C2×C3×Q16—31no—
63C2×C3×SD16—31no—
64C2×C3×SD16—31no—
65C2×C3×M16—31no—
66#949—64no—
67#946—32no—
68C2×#215—32no—
69#938—32no—
70#937—61no—
71#935—61no—
72C2×#213—32no—
73#929—121no—
74C4×#67—62no—
75#926—31no—
76C2×#208—31no—
77#859—32no—
78#859—62no—
79#835—32no—
80#835—32no—
81#821—32no—
82#821—32no—
83(C2×D3)⋊C8—32no—
84(C2×D3)⋊C8—32no—
85C23×C32—41no—
86C22×C3×D3—41no—
87C2×C3⋊Dic3—42no—
88C3×#67—82no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—42no—
91C2×C3×Dic3—82no—
92C3×SL(2,𝔽3)C2×D411yescharacteristic
93C22×#41—31no—
94C2×#137—31no—
95C2×AGL(1,ℤ/8)—91no—
96#525—91no—
97#452—92no—
98#452—92no—
99#449—92no—
100#449—92no—
101#419—91no—
102#418—91no—
103#407—91no—
104C23×D3—31no—
105C3×#41—62no—
106C3×#41—62no—
107C2×C3×Q8—31no—
108C2×C3×Q8—32no—
109C2×C3×Q8—31no—
110C2×C3×Q8#6712yes—
111C2×C3×Q8C22×D311yescharacteristic
112C2×C3×D4—31no—
113C2×C3×D4—61no—
114C22×C3×C4—31no—
115C22×C3×C4—61no—
116C22×C3×C4—121no—
117C2×#67—41no—
118C2×#67—62no—
119C22×Dic3S411yescharacteristic
120#239—22no—
121#239—64no—
122#239—62no—
123#239—64no—
124C2×D12—32no—
125C2×D12—34no—
126C2×D12—61no—
127C2×D12—62no—
128C2×C4×D3—32no—
129C2×C4×D3—32no—
130C2×C4×D3—62no—
131C2×C4×D3—62no—
132#41⋊C3—62no—
133C2×SL(2,𝔽3)—22no—
134C2×SL(2,𝔽3)—21no—
135C2×SL(2,𝔽3)#6712yescomplemented
136C2×SL(2,𝔽3)C22×D311yescharacteristic
137GL(2,𝔽3)—21no—
138#226—21no—
139C3×Q16—62no—
140C3×Q16—61no—
141C3×Q16—61no—
142C3×SD16—62no—
143C3×SD16—61no—
144C3×SD16—61no—
145C3×M16—62no—
146C2×C3×C8—31no—
147C2×C3×C8—31no—
148#217—61no—
149#217—62no—
150#217—121no—
151#217—241no—
152#215—64no—
153#213—64no—
154(C2×D3)⋊C4—62no—
155(C2×D3)⋊C4—122no—
156C4⋊Dic3—32no—
157Dic3⋊C4—62no—
158Dic3⋊C4—122no—
159C4×Dic3—61no—
160#208—62no—
161C2×C3⋊C8—32no—
162C22×C32—42no—
163C22×C32—41no—
164C22×C32—82no—
165C2×C3×D3—81no—
166C2×C3×D3—82no—
167C3⋊Dic3—82no—
168C3×Dic3—82no—
169C3×Dic3—81no—
170C2×#41—31no—
171C2×#41—32no—
172C2×#41—62no—
173C2×#41—92no—
174C2×#41—182no—
175C22×Q8D3211yescharacteristic
176C22×D4—91no—
177C23×C4—91no—
178#137—62no—
179#137—62no—
180AGL(1,ℤ/8)—184no—
181C2×Q16—31no—
182C2×Q16—31no—
183C2×SD16—31no—
184C2×SD16—31no—
185C2×SD16—92no—
186C2×D8—92no—
187C2×M16—31no—
188C2×M16—91no—
189#123—181no—
190C22⋊Q8—92no—
191#121—92no—
192#120—181no—
193C4×D4—182no—
194#117—91no—
195#103—92no—
196#103—92no—
197#102—92no—
198#102—92no—
199C22⋊C8—92no—
200C22⋊C8—92no—
201C23×C3—41no—
202C23×C3—61no—
203C23×C3—81no—
204C23×C3C2×S411yescharacteristic
205C22×D3—32no—
206C22×D3—41no—
207C22×D3—62no—
208C22×D3—122no—
209C3×Q8—32no—
210C3×Q8—31no—
211C3×Q8—61no—
212C3×Q8—61no—
213C3×Q8—61no—
214C3×Q8C2×#6711yescharacteristic
215C3×D4—61no—
216C3×D4—61no—
217C3×D4—122no—
218C2×C3×C4—32no—
219C2×C3×C4—32no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—122no—
224C2×C3×C4—122no—
225C2×C3×C4—242no—
226#67—82no—
227#67—124no—
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—62no—
231C2×Dic3—122no—
232C2×Dic3—242no—
233C2×Dic3C2×S412yes—
234D12—34no—
235D12—62no—
236D12—64no—
237C4×D3—62no—
238C4×D3—62no—
239C4×D3—64no—
240SL(2,𝔽3)—21no—
241SL(2,𝔽3)C2×#6711yescharacteristic
242C3×C8—61no—
243C3×C8—61no—
244C3⋊C8—62no—
245C2×C32—42no—
246C2×C32—41no—
247C2×C32—82no—
248C3×D3—161no—
249C24—91no—
250#41—62no—
251#41—62no—
252#41—62no—
253#41—184no—
254#41—182no—
255#41—184no—
256C2×Q8—31no—
257C2×Q8—32no—
258C2×Q8—31no—
259C2×Q8#62512yes—
260C2×Q8C2×D3211yescharacteristic
261C2×D4—31no—
262C2×D4—61no—
263C2×D4—92no—
264C2×D4—94no—
265C2×D4—181no—
266C2×D4—182no—
267C2×D4—182no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—61no—
271C22×C4—92no—
272C22×C4—92no—
273C22×C4—182no—
274C22×C4—182no—
275Q16—62no—
276Q16—61no—
277Q16—61no—
278SD16—62no—
279SD16—61no—
280SD16—61no—
281SD16—184no—
282D8—184no—
283M16—62no—
284M16—182no—
285C2×C8—31no—
286C2×C8—31no—
287C2×C8—92no—
288C4⋊C4—92no—
289C4⋊C4—182no—
290C22⋊C4—182no—
291C22⋊C4—182no—
292C22⋊C4—181no—
293C42—181no—
294C22×C3—22no—
295C22×C3—41no—
296C22×C3—42no—
297C22×C3—82no—
298C22×C3—81no—
299C22×C3—82no—
300C22×C3—122no—
301C22×C3—121no—
302C22×C3—162no—
303C22×C3GL(2,ℤ/4)12yes—
304C22×C3C22×S411yescharacteristic
305C2×D3—62no—
306C2×D3—62no—
307C2×D3—62no—
308C2×D3—81no—
309C2×D3—82no—
310C2×D3—124no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—32no—
314C3×C4—121no—
315C3×C4—242no—
316Dic3—22no—
317Dic3—81no—
318Dic3—122no—
319Dic3—242no—
320Dic3—482no—
321C32—41noSylow
322C23—61no—
323C23—92no—
324C23—182no—
325C23—362no—
326C23D3×S411yescharacteristic
327Q8—32no—
328Q8—31no—
329Q8—61no—
330Q8—61no—
331Q8—61no—
332Q8C2×#62511yescharacteristic
333D4—61no—
334D4—61no—
335D4—94no—
336D4—122no—
337D4—182no—
338D4—184no—
339D4—364no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—32no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—62no—
346C2×C4—61no—
347C2×C4—122no—
348C2×C4—182no—
349C2×C4—182no—
350C2×C4—184no—
351C2×C4—182no—
352C8—61no—
353C8—61no—
354C8—182no—
355C2×C3—22no—
356C2×C3—42no—
357C2×C3—41no—
358C2×C3—82no—
359C2×C3—82no—
360C2×C3—81no—
361C2×C3—162no—
362C2×C3—241no—
363C2×C3#592312yes—
364C2×C3C2×GL(2,ℤ/4)11yescharacteristic
365D3—122no—
366D3—161no—
367C22—22no—
368C22—122no—
369C22—121no—
370C22—182no—
371C22—182no—
372C22—182no—
373C22—364no—
374C22#6420912yes—
375C22C2×D3×S411yescharacteristic
376C4—31no—
377C4—31no—
378C4—32no—
379C4—62no—
380C4—121no—
381C4—362no—
382C3—41no—
383C3—81no—
384C3#8552211yescharacteristic, complemented
385C2—22no—
386C2—241no—
387C2—362no—
388C2#1059643512yes—
389C2C2×#6420911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC22×C3×Q8C2×D3
socleC22×C3C22×S4
hypercenterC23D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊(#41⋊C3)C2semidirect
2. C2×C3×#983C2semidirect
3. #10593654C2semidirect
4. #10593203C2semidirect
5. #10593194C2semidirect
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. C3⋊#859C22semidirect
16. C22×#239D3semidirect
18. C2×#983D3semidirect
35. C2×C3×SL(2,𝔽3)D4semidirect
57. C2×GL(2,𝔽3)C2×D3semidirect
58. C2×#226C2×D3semidirect
135. C2×SL(2,𝔽3)#67semidirect
384. C3#85522semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(#41⋊C3)2
C2×C3×#9832
#105936542
#105932032
#105931942
#855223
#839803
#639614