Finite Group Databasebeta

← #11923056

Subgroups — #11923056

Order 1152.

878 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#64264C211yesmaximal, characteristic, complemented
3#10593655C212yesmaximal, complemented
4#10593654C211yesmaximal, characteristic, complemented
5#10593653C212yesmaximal, complemented
6#85507—33nomaximal
7#84059—31nomaximal
8C22×C3×SL(2,𝔽3)—22no—
9C22×C3×SL(2,𝔽3)C2212yes—
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#64264—22no—
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3⋊#226C2211yescharacteristic, complemented
14#64162—41nomaximal
15C3⋊#859C2212yescomplemented
16C23×C3×Q8D311yescharacteristic, complemented
17C23×SL(2,𝔽3)D313yescomplemented
18C2×#983—33no—
19C2×#942—31no—
20C2×#942—31no—
21#5791—31no—
22#5421—36no—
23#5420—33no—
24#5419—36no—
25#5228—32no—
26#5228—32no—
27#5227—32no—
28#5227—32no—
29#5225—31no—
30#5225—31no—
31#5207—31no—
32C24×C32—41no—
33C2×#638—42no—
34C2×#638—41no—
35C2×C3×SL(2,𝔽3)—22no—
36C2×C3×SL(2,𝔽3)D412yescomplemented
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38C3⋊GL(2,𝔽3)—21no—
39C3⋊#226—21no—
40#3757—42no—
41#3757—41no—
42#3004—91noSylow
43C22×C3×Q8—22no—
44C22×C3×Q8—31no—
45C22×C3×Q8—32no—
46C22×C3×Q8—62no—
47C22×C3×Q8C2×D312yes—
48C22×C3×Q8C2×D311yescharacteristic
49C23×C3×C4—31no—
50C2×#235—31no—
51C22×SL(2,𝔽3)—26no—
52C22×SL(2,𝔽3)C2×D316yes—
53C22×SL(2,𝔽3)C2×D313yes—
54#983—66no—
55C2×GL(2,𝔽3)—33no—
56C2×#226—33no—
57#953—61no—
58#953—123no—
59C2×#216—31no—
60C2×#216—32no—
61C2×#216—31no—
62#942—62no—
63#942—64no—
64#942—62no—
65C2×#215—31no—
66C2×#215—32no—
67C2×#215—31no—
68#930—32no—
69#929—61no—
70C4×#67—62no—
71#926—31no—
72#924—32no—
73C2×#208—31no—
74C2×#208—31no—
75#859—36no—
76#835—32no—
77#835—32no—
78#835—32no—
79#835—32no—
80#830—32no—
81#830—32no—
82C23×C32—42no—
83C23×C32—41no—
84C23×C32—84no—
85C23×C32—82no—
86C22×C3⋊D3—41no—
87#638—82no—
88#638—84no—
89C2×C3⋊Dic3—41no—
90C2×C3⋊Dic3—42no—
91C3×SL(2,𝔽3)C2×D411yescharacteristic
92C23×Q8C3⋊D311yescharacteristic, complemented
93C2×#137—91no—
94C2×#137—91no—
95#525—91no—
96#451—92no—
97#451—92no—
98#448—92no—
99#448—92no—
100#419—91no—
101#419—91no—
102#407—91no—
103C24×C3—43no—
104C24×C3S411yescharacteristic
105C2×C3×Q8—22no—
106C2×C3×Q8—31no—
107C2×C3×Q8—32no—
108C2×C3×Q8—32no—
109C2×C3×Q8—34no—
110C2×C3×Q8—61no—
111C2×C3×Q8—62no—
112C2×C3×Q8—64no—
113C2×C3×Q8—62no—
114C2×C3×Q8—64no—
115C2×C3×Q8#6712yescomplemented
116C2×C3×Q8C22×D311yescharacteristic
117C22×C3×C4—31no—
118C22×C3×C4—32no—
119C22×C3×C4—32no—
120C22×C3×C4—31no—
121C22×C3×C4—62no—
122C22×C3×C4—62no—
123C2×#67—62no—
124C2×#67—61no—
125C2×#67—123no—
126C2×#67—126no—
127#235—62no—
128#235—62no—
129C2×D12—31no—
130C2×C4×D3—61no—
131C2×Dic6—31no—
132C2×SL(2,𝔽3)—26no—
133C2×SL(2,𝔽3)#6716yescomplemented
134C2×SL(2,𝔽3)C22×D313yes—
135GL(2,𝔽3)—63no—
136#226—63no—
137#217—62no—
138#217—61no—
139#217—126no—
140#217—123no—
141#216—62no—
142#216—61no—
143#216—64no—
144#216—61no—
145#215—62no—
146#215—61no—
147#215—64no—
148#215—61no—
149(C2×D3)⋊C4—62no—
150C4⋊Dic3—32no—
151Dic3⋊C4—62no—
152C4×Dic3—61no—
153#208—62no—
154#208—62no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—32no—
158C22×C32—44no—
159C22×C32—42no—
160C22×C32—41no—
161C22×C32—82no—
162C22×C32—82no—
163C22×C32—82no—
164C22×C32—88no—
165C2×C3⋊D3—82no—
166C2×C3⋊D3—81no—
167C3⋊Dic3—82no—
168C3⋊Dic3—81no—
169C2×#41—91no—
170C22×Q8—22no—
171C22×Q8—31no—
172C22×Q8—32no—
173C22×Q8—62no—
174C22×Q8C2×C3⋊D312yes—
175C22×Q8C2×C3⋊D311yescharacteristic
176C23×C4—31no—
177#137—182no—
178#137—184no—
179#137—182no—
180C2×Q16—92no—
181C2×Q16—91no—
182C2×Q16—91no—
183C2×SD16—92no—
184C2×SD16—91no—
185C2×SD16—91no—
186C2×M16—91no—
187C2×M16—91no—
188#123—181no—
189C22⋊Q8—92no—
190#121—92no—
191#120—181no—
192C4×D4—182no—
193#117—91no—
194#103—92no—
195#103—92no—
196#103—92no—
197#103—92no—
198C22⋊C8—92no—
199C22⋊C8—92no—
200C23×C3—24no—
201C23×C3—22no—
202C23×C3—46no—
203C23×C3—43no—
204C23×C3—86no—
205C23×C3—812no—
206C23×C3C2×S412yes—
207C23×C3C2×S411yescharacteristic
208C22×D3—61no—
209C22×D3—123no—
210C3×Q8—32no—
211C3×Q8—31no—
212C3×Q8—34no—
213C3×Q8—61no—
214C3×Q8—62no—
215C3×Q8—64no—
216C3×Q8C2×#6711yescharacteristic
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—32no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—64no—
225C2×C3×C4—62no—
226#67—124no—
227#67—122no—
228#67—246no—
229#67—2412no—
230C2×Dic3—62no—
231C2×Dic3—61no—
232C2×Dic3—126no—
233C2×Dic3—123no—
234D12—61no—
235D12—61no—
236C4×D3—122no—
237Dic6—61no—
238Dic6—61no—
239SL(2,𝔽3)C2×#6713yes—
240C3⋊C8—62no—
241C3⋊C8—61no—
242C3⋊C8—61no—
243C2×C32—42no—
244C2×C32—41no—
245C2×C32—84no—
246C2×C32—82no—
247C3⋊D3—161no—
248C24C3⋊S411yescharacteristic
249#41—182no—
250#41—182no—
251C2×Q8—22no—
252C2×Q8—31no—
253C2×Q8—32no—
254C2×Q8—32no—
255C2×Q8—34no—
256C2×Q8—61no—
257C2×Q8—62no—
258C2×Q8—64no—
259C2×Q8—62no—
260C2×Q8—64no—
261C2×Q8—91no—
262C2×Q8#63812yescomplemented
263C2×Q8C22×C3⋊D311yescharacteristic
264C2×D4—91no—
265C2×D4—182no—
266C2×D4—181no—
267C22×C4—31no—
268C22×C4—32no—
269C22×C4—32no—
270C22×C4—31no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—181no—
274Q16—182no—
275Q16—181no—
276Q16—184no—
277Q16—181no—
278SD16—182no—
279SD16—181no—
280SD16—184no—
281SD16—181no—
282M16—182no—
283M16—182no—
284C2×C8—91no—
285C2×C8—91no—
286C2×C8—92no—
287C4⋊C4—92no—
288C4⋊C4—182no—
289C22⋊C4—182no—
290C22⋊C4—181no—
291C22⋊C4—182no—
292C42—181no—
293C22×C3—28no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—412no—
298C22×C3—46no—
299C22×C3—43no—
300C22×C3—86no—
301C22×C3—86no—
302C22×C3—86no—
303C22×C3—824no—
304C22×C3#98314yescomplemented
305C22×C3GL(2,ℤ/4)12yes—
306C22×C3C22×S411yescharacteristic
307C2×D3—122no—
308C2×D3—121no—
309C2×D3—246no—
310C2×D3—243no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—32no—
314C3×C4—62no—
315Dic3—122no—
316Dic3—121no—
317Dic3—246no—
318Dic3—243no—
319C32—41noSylow
320C23—24no—
321C23—22no—
322C23—181no—
323C23C2×C3⋊S412yes—
324C23C2×C3⋊S411yescharacteristic
325Q8—32no—
326Q8—31no—
327Q8—34no—
328Q8—61no—
329Q8—62no—
330Q8—64no—
331Q8—181no—
332Q8—181no—
333Q8C2×#63811yescharacteristic
334D4—181no—
335D4—181no—
336D4—364no—
337D4—362no—
338C2×C4—32no—
339C2×C4—32no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—32no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—64no—
346C2×C4—62no—
347C2×C4—182no—
348C2×C4—181no—
349C2×C4—362no—
350C8—182no—
351C8—181no—
352C8—181no—
353C2×C3—24no—
354C2×C3—22no—
355C2×C3—46no—
356C2×C3—43no—
357C2×C3—812no—
358C2×C3—86no—
359C2×C3C2×#98312yes—
360C2×C3C2×GL(2,ℤ/4)11yescharacteristic
361D3—241no—
362D3—483no—
363C22—28no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—362no—
368C22—361no—
369C22#6426414yescomplemented
370C22#6426912yes—
371C22C22×C3⋊S411yescharacteristic
372C4—31no—
373C4—31no—
374C4—32no—
375C4—62no—
376C4—362no—
377C4—361no—
378C3—43no—
379C3#8550711yescharacteristic, complemented
380C2—24no—
381C2—22no—
382C2—721no—
383C2C2×#6426412yes—
384C2C2×#6426911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC23×C3×Q8D3
socleC22×C3C22×S4
hypercenterC24C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C2×#64264C2semidirect
3. #10593655C2semidirect
4. #10593654C2semidirect
5. #10593653C2semidirect
12. C2×C3⋊GL(2,𝔽3)C22semidirect
13. C2×C3⋊#226C22semidirect
15. C3⋊#859C22semidirect
16. C23×C3×Q8D3semidirect
17. C23×SL(2,𝔽3)D3semidirect
36. C2×C3×SL(2,𝔽3)D4semidirect
92. C23×Q8C3⋊D3semidirect
115. C2×C3×Q8#67semidirect
133. C2×SL(2,𝔽3)#67semidirect
262. C2×Q8#638semidirect
304. C22×C3#983semidirect
369. C22#64264semidirect
379. C3#85507semidirect

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×#642642
#105936552
#105936542
#105936532
#855073
#840593
#641624