Finite Group Databasebeta

← #11922233

Subgroups — #11922233

Order 1152.

719 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 63 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×#983C211yesmaximal, characteristic, complemented
3#10593654C211yesmaximal, characteristic, complemented
4#10593217C212yesmaximal, complemented
5#10593210C212yesmaximal, complemented
6#85507—31nomaximal
7#83979—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×D3×SL(2,𝔽3)—22no—
10C2×D3×SL(2,𝔽3)C2212yes—
11C3×#983—22no—
12C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3×#226C2211yescharacteristic, complemented
14#63976—41nomaximal
15C3⋊#859C2212yescomplemented
16C22×D3×Q8D311yescharacteristic, complemented
17C23×SL(2,𝔽3)—31no—
18C2×#983D311yescharacteristic, complemented
19C2×C3×#137—31no—
20#5828—31no—
21C2×#951—31no—
22#5421—32no—
23#5420—31no—
24#5420—61no—
25#5419—32no—
26#5225—31no—
27#5186—32no—
28#5186—32no—
29#5170—32no—
30#5170—32no—
31#5126—31no—
32#5120—31no—
33C23×C3×D3—41no—
34C2×C3×#67—41no—
35C2×C3×SL(2,𝔽3)D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C2×#625—42no—
38D3×SL(2,𝔽3)—22no—
39C3×GL(2,𝔽3)—21no—
40C3×#226—21no—
41#3757—41no—
42(C2×D3)⋊Dic3—42no—
43#3004—91noSylow
44C2×C3×#41—31no—
45C22×C3×Q8C2×D311yescharacteristic
46C2×D3×Q8—22no—
47C2×D3×Q8—32no—
48C2×D3×Q8—62no—
49C2×D3×Q8C2×D312yes—
50C22×C4×D3—31no—
51C22×Dic6—31no—
52C22×SL(2,𝔽3)—21no—
53C22×SL(2,𝔽3)—32no—
54C22×SL(2,𝔽3)—62no—
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#953—121no—
67#951—64no—
68#946—32no—
69C2×#216—32no—
70#938—32no—
71#937—61no—
72#935—61no—
73C2×#214—32no—
74C4×#67—62no—
75#926—31no—
76C2×#208—31no—
77#859—32no—
78#859—62no—
79#835—32no—
80#835—32no—
81(C2×D3)⋊C8—32no—
82(C2×D3)⋊C8—32no—
83#816—32no—
84#816—32no—
85C23×C32—41no—
86C22×C3×D3—41no—
87C22×C3×D3—42no—
88C22×C3×D3—84no—
89C22×C3×D3—82no—
90C2×C3⋊Dic3—42no—
91C3×#67—82no—
92C2×C3×Dic3—41no—
93C3×SL(2,𝔽3)C2×D411yescharacteristic
94#625—84no—
95C23×Q8—31no—
96C2×#137—31no—
97C2×#137—91no—
98#525—91no—
99#451—92no—
100#451—92no—
101#448—92no—
102#448—92no—
103#419—91no—
104#419—91no—
105#407—91no—
106C24×C3—121no—
107C23×D3S411yescharacteristic
108C3×#41—62no—
109C3×#41—62no—
110C2×C3×Q8—31no—
111C2×C3×Q8—32no—
112C2×C3×Q8—31no—
113C2×C3×Q8#6712yescomplemented
114C2×C3×Q8C22×D311yescharacteristic
115C2×C3×D4—31no—
116C2×C3×D4—61no—
117C22×C3×C4—31no—
118C22×C3×C4—61no—
119C2×#67—41no—
120C2×#67—62no—
121C2×#67—122no—
122C22×Dic3—31no—
123D3×Q8—22no—
124D3×Q8—62no—
125D3×Q8—64no—
126D3×Q8—64no—
127C2×C4×D3—32no—
128C2×C4×D3—32no—
129C2×C4×D3—62no—
130C2×C4×D3—62no—
131C2×Dic6—32no—
132C2×Dic6—34no—
133C2×Dic6—61no—
134C2×Dic6—62no—
135C2×SL(2,𝔽3)—22no—
136C2×SL(2,𝔽3)—21no—
137C2×SL(2,𝔽3)—62no—
138C2×SL(2,𝔽3)#6712yescomplemented
139C2×SL(2,𝔽3)C22×D311yescharacteristic
140GL(2,𝔽3)—21no—
141#226—21no—
142C3×Q16—62no—
143C3×Q16—61no—
144C3×Q16—61no—
145C3×SD16—62no—
146C3×SD16—61no—
147C3×SD16—61no—
148C3×M16—62no—
149C2×C3×C8—31no—
150C2×C3×C8—31no—
151#217—62no—
152#217—61no—
153#217—122no—
154#217—121no—
155#217—241no—
156#216—64no—
157#214—64no—
158(C2×D3)⋊C4—62no—
159C4⋊Dic3—32no—
160Dic3⋊C4—62no—
161C4×Dic3—61no—
162#208—62no—
163C2×C3⋊C8—32no—
164C22×C32—42no—
165C22×C32—41no—
166C22×C32—82no—
167C2×C3×D3—44no—
168C2×C3×D3—82no—
169C2×C3×D3—82no—
170C2×C3×D3—88no—
171C2×C3×D3—82no—
172C2×C3×D3—81no—
173C3⋊Dic3—82no—
174C3×Dic3—81no—
175C2×#41—31no—
176C22×Q8—32no—
177C22×Q8—62no—
178C22×Q8—91no—
179C22×Q8—92no—
180C22×Q8—182no—
181C22×Q8D3211yescharacteristic
182C23×C4—91no—
183#137—62no—
184#137—62no—
185#137—184no—
186C2×Q16—31no—
187C2×Q16—31no—
188C2×Q16—92no—
189C2×SD16—31no—
190C2×SD16—31no—
191C2×SD16—92no—
192C2×M16—31no—
193C2×M16—91no—
194#123—181no—
195C22⋊Q8—92no—
196#121—92no—
197#120—181no—
198C4×D4—182no—
199#117—91no—
200#103—92no—
201#103—92no—
202#103—92no—
203#103—92no—
204C22⋊C8—92no—
205C22⋊C8—92no—
206C23×C3—41no—
207C23×C3—61no—
208C23×C3—81no—
209C23×C3—122no—
210C23×C3—242no—
211C23×C3—244no—
212C23×C3C2×S411yescharacteristic
213C22×D3—22no—
214C22×D3—24no—
215C22×D3—41no—
216C22×D3C2×S412yes—
217C3×Q8—32no—
218C3×Q8—31no—
219C3×Q8—61no—
220C3×Q8—61no—
221C3×Q8—61no—
222C3×Q8C2×#6711yescharacteristic
223C3×D4—61no—
224C3×D4—61no—
225C3×D4—122no—
226C2×C3×C4—32no—
227C2×C3×C4—32no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—61no—
231C2×C3×C4—122no—
232#67—82no—
233#67—124no—
234#67—244no—
235C2×Dic3—32no—
236C2×Dic3—41no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—122no—
240C2×Dic3—242no—
241C4×D3—62no—
242C4×D3—62no—
243C4×D3—64no—
244Dic6—34no—
245Dic6—64no—
246Dic6—62no—
247SL(2,𝔽3)—21no—
248SL(2,𝔽3)C2×#6711yescharacteristic
249C3×C8—61no—
250C3×C8—61no—
251C3⋊C8—62no—
252C2×C32—42no—
253C2×C32—41no—
254C2×C32—82no—
255C3×D3—84no—
256C3×D3—161no—
257C24—31no—
258#41—62no—
259#41—62no—
260C2×Q8—31no—
261C2×Q8—32no—
262C2×Q8—31no—
263C2×Q8—62no—
264C2×Q8—92no—
265C2×Q8—94no—
266C2×Q8—181no—
267C2×Q8—182no—
268C2×Q8—184no—
269C2×Q8—182no—
270C2×Q8—184no—
271C2×Q8#62512yescomplemented
272C2×Q8C2×D3211yescharacteristic
273C2×D4—31no—
274C2×D4—61no—
275C2×D4—182no—
276C22×C4—31no—
277C22×C4—61no—
278C22×C4—92no—
279C22×C4—92no—
280C22×C4—91no—
281C22×C4—182no—
282C22×C4—182no—
283Q16—62no—
284Q16—61no—
285Q16—61no—
286Q16—184no—
287SD16—62no—
288SD16—61no—
289SD16—61no—
290SD16—184no—
291M16—62no—
292M16—182no—
293C2×C8—31no—
294C2×C8—31no—
295C2×C8—92no—
296C4⋊C4—92no—
297C4⋊C4—182no—
298C22⋊C4—182no—
299C22⋊C4—182no—
300C22⋊C4—181no—
301C42—181no—
302C22×C3—22no—
303C22×C3—41no—
304C22×C3—42no—
305C22×C3—82no—
306C22×C3—81no—
307C22×C3—82no—
308C22×C3—124no—
309C22×C3—122no—
310C22×C3—121no—
311C22×C3—162no—
312C22×C3—242no—
313C22×C3—242no—
314C22×C3—248no—
315C22×C3GL(2,ℤ/4)12yes—
316C22×C3C22×S411yescharacteristic
317C2×D3—28no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—81no—
321C2×D3—82no—
322C2×D3#98314yescomplemented
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—32no—
326C3×C4—121no—
327Dic3—62no—
328Dic3—81no—
329Dic3—122no—
330Dic3—242no—
331Dic3—482no—
332C32—41noSylow
333C23—32no—
334C23—64no—
335C23—62no—
336C23—61no—
337C23D3×S411yescharacteristic
338Q8—32no—
339Q8—31no—
340Q8—61no—
341Q8—61no—
342Q8—61no—
343Q8—94no—
344Q8—182no—
345Q8—184no—
346Q8C2×#62511yescharacteristic
347D4—61no—
348D4—61no—
349D4—122no—
350D4—364no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—61no—
356C2×C4—92no—
357C2×C4—122no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—184no—
361C2×C4—182no—
362C2×C4—182no—
363C8—61no—
364C8—61no—
365C8—182no—
366C2×C3—22no—
367C2×C3—42no—
368C2×C3—41no—
369C2×C3—82no—
370C2×C3—82no—
371C2×C3—81no—
372C2×C3—162no—
373C2×C3—244no—
374C2×C3—241no—
375C2×C3C2×#98312yes—
376C2×C3C2×GL(2,ℤ/4)11yescharacteristic
377D3—24no—
378D3—161no—
379C22—22no—
380C22—34no—
381C22—68no—
382C22—62no—
383C22—62no—
384C22—122no—
385C22—121no—
386C22#6420912yes—
387C22C2×D3×S411yescharacteristic
388C4—31no—
389C4—31no—
390C4—32no—
391C4—121no—
392C4—182no—
393C4—362no—
394C3—41no—
395C3—81no—
396C3#8550711yescharacteristic, complemented
397C2—22no—
398C2—64no—
399C2—241no—
400C2D3×#98312yes—
401C2C2×#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×D3×SL(2,𝔽3)C2semidirect
2. C2×C3×#983C2semidirect
3. #10593654C2semidirect
4. #10593217C2semidirect
5. #10593210C2semidirect
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. C3⋊#859C22semidirect
16. C22×D3×Q8D3semidirect
18. C2×#983D3semidirect
35. C2×C3×SL(2,𝔽3)D4semidirect
57. C2×GL(2,𝔽3)C2×D3semidirect
58. C2×#226C2×D3semidirect
113. C2×C3×Q8#67semidirect
138. C2×SL(2,𝔽3)#67semidirect
271. C2×Q8#625semidirect
322. C2×D3#983semidirect
396. C3#85507semidirect

Maximal subgroups

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

subgroupindex
C22×D3×SL(2,𝔽3)2
C2×C3×#9832
#105936542
#105932172
#105932102
#855073
#839793
#639764