Finite Group Databasebeta

← #85821⋊C3 #11923177

Subgroups — #85821⋊C3 #11923177

Order 1152.

593 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5817⋊C3C212yesmaximal, complemented
3C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
4#5574⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#85821C311yesmaximal, characteristic, complemented
7#3522⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×#924—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C2×C3×C4×Q8C2×C311yescharacteristic
15#5817C2×C312yes—
16#5585—31no—
17#5582—32no—
18#5579—31no—
19#5577C2×C311yescharacteristic, complemented
20#5574C2×C312yescomplemented
21Q8×Dic6C2×C311yescharacteristic, complemented
22#5507—32no—
23#5479—31no—
24#5477—31no—
25Q8×SL(2,𝔽3)—31no—
26C4⋊(#41⋊C3)—31no—
27#548⋊C3—32no—
28#520⋊C3—32no—
29C2×C4×SL(2,𝔽3)—21no—
30C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
31C22×C32×C4—41no—
32C2×C3×Dic6—41no—
33C2×C3×SL(2,𝔽3)—21no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3⋊(#41⋊C3)—22no—
36C3×#217—42no—
37C3×C4⋊Dic3—41no—
38C3×Dic3⋊C4—42no—
39#3522—31noSylow
40C22×C3×Q8C22×C311yescharacteristic
41C2×#239C22×C312yes—
42C2×#235—32no—
43C2×#41⋊C3—32no—
44C22×SL(2,𝔽3)—21no—
45C22×SL(2,𝔽3)C2×D311yescharacteristic
46C3×C22⋊Q8—121no—
47C3×C4×Q8—31no—
48C3×C4×Q8—31no—
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×Q8C22×C311yescharacteristic
51C2×C3×C4⋊C4—31no—
52C2×C3×C42—31no—
53#947—32no—
54#946—32no—
55Q8×Dic3C22×C312yes—
56#930—31no—
57#930—32no—
58#929—32no—
59C4×#67—32no—
60#924A411yescharacteristic
61#896—34no—
62#895—32no—
63#894—34no—
64#893—32no—
65D3×C4⋊C4—32no—
66#888—32no—
67#876—32no—
68#875—32no—
69#875—31no—
70#874—31no—
71C4×D12—32no—
72C4×D12—32no—
73C4×D12—31no—
74#872—32no—
75#870—31no—
76#869—31no—
77C4×Dic6—31no—
78C4×SL(2,𝔽3)—21no—
79C4×SL(2,𝔽3)—21no—
80C4×SL(2,𝔽3)—32no—
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C4×SL(2,𝔽3)C2×D311yescharacteristic
83C23×C32—41no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4—82no—
87C2×C3×Dic3—42no—
88C2×C3×Dic3—42no—
89C3×Dic6—82no—
90C3×SL(2,𝔽3)#4111yescharacteristic, complemented
91#559—91no—
92Q82—31no—
93#554—91no—
94#550—31no—
95#548—32no—
96#547—92no—
97#542—92no—
98#532—91no—
99#529—91no—
100#520—32no—
101C2×C4×Q8C3×D311yescharacteristic
102C2×C3×Q8—21no—
103C2×C3×Q8—31no—
104C2×C3×Q8—61no—
105C2×C3×Q8—121no—
106C2×C3×Q8C23×C311yescharacteristic
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—41no—
110C22×C3×C4—81no—
111C22×C3×C4C2×A411yescharacteristic
112C2×#67—32no—
113#239—22no—
114#239—62no—
115#235—124no—
116C2×D12—32no—
117C2×D12—31no—
118C2×D12—32no—
119C2×C4×D3—32no—
120C2×C4×D3—34no—
121C2×Dic6—32no—
122C2×Dic6C2×A411yescharacteristic
123#41⋊C3—62no—
124C2×SL(2,𝔽3)—22no—
125C2×SL(2,𝔽3)—21no—
126C2×SL(2,𝔽3)—21no—
127C2×SL(2,𝔽3)C22×D311yescharacteristic
128C3×C4⋊C4—31no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—32no—
131C3×C4⋊C4—122no—
132C3×C4⋊C4—121no—
133C3×C22⋊C4—122no—
134C3×C42—31no—
135C3×C42—31no—
136C3×C42—31no—
137C3×C42—31no—
138#217C2×A412yes—
139(C2×D3)⋊C4—34no—
140(C2×D3)⋊C4—32no—
141(C2×D3)⋊C4—32no—
142(C2×D3)⋊C4—32no—
143(C2×D3)⋊C4—32no—
144C4⋊Dic3—31no—
145C4⋊Dic3—32no—
146C4⋊Dic3C2×A411yescharacteristic
147Dic3⋊C4—32no—
148Dic3⋊C4—32no—
149Dic3⋊C4—32no—
150Dic3⋊C4C2×A412yes—
151C4×Dic3—32no—
152C22×C32—42no—
153C22×C32—41no—
154C22×C32—81no—
155C22×C32—81no—
156C32×C4—42no—
157C32×C4—81no—
158C3×Dic3—82no—
159C3×Dic3—82no—
160C2×#41—32no—
161C2×#41—92no—
162C22×Q8C2×C3×D311yescharacteristic
163C4⋊Q8—91no—
164C4⋊Q8—92no—
165#127—91no—
166#126—92no—
167#125—91no—
168#124—91no—
169#124—92no—
170#124—92no—
171#123—92no—
172#123—94no—
173C22⋊Q8—31no—
174C22⋊Q8—94no—
175C22⋊Q8—92no—
176#121—92no—
177#121—91no—
178#121—92no—
179C4×Q8—31no—
180C4×Q8—31no—
181C4×Q8—32no—
182C4×Q8—91no—
183C4×Q8C2×C3×D311yescharacteristic
184C4×Q8C2×C3×D311yescharacteristic
185C4×D4—91no—
186C4×D4—92no—
187C4×D4—92no—
188C4×D4—92no—
189C4×D4—92no—
190#117—92no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—92no—
193C2×C42—31no—
194C23×C3—41no—
195C23×C3—81no—
196C23×C3C22×A411yescharacteristic
197C22×D3—32no—
198C3×Q8—31no—
199C3×Q8—61no—
200C3×Q8—61no—
201C3×Q8—242no—
202C3×Q8C3×#4111yescharacteristic
203C2×C3×C4—22no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—62no—
213C2×C3×C4—82no—
214C2×C3×C4—84no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4—122no—
218C2×C3×C4—122no—
219C2×C3×C4C22×A411yescharacteristic
220C2×C3×C4C22×A411yescharacteristic
221#67—64no—
222C2×Dic3—32no—
223C2×Dic3C22×A412yes—
224C2×Dic3C22×A412yes—
225D12—62no—
226D12—62no—
227D12—62no—
228D12—122no—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—124no—
232Dic6—22no—
233Dic6—122no—
234SL(2,𝔽3)—21no—
235SL(2,𝔽3)#23511yescharacteristic, complemented
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—82no—
240#41—62no—
241#41—182no—
242#41—364no—
243C2×Q8—21no—
244C2×Q8—31no—
245C2×Q8—31no—
246C2×Q8—61no—
247C2×Q8—92no—
248C2×Q8C22×C3×D311yescharacteristic
249C2×D4—92no—
250C2×D4—92no—
251C2×D4—92no—
252C2×D4—91no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—92no—
256C22×C4—94no—
257C22×C4D3×A411yescharacteristic
258C4⋊C4—31no—
259C4⋊C4—31no—
260C4⋊C4—32no—
261C4⋊C4—32no—
262C4⋊C4—31no—
263C4⋊C4—92no—
264C4⋊C4—92no—
265C4⋊C4—92no—
266C4⋊C4—92no—
267C4⋊C4—91no—
268C22⋊C4—32no—
269C22⋊C4—92no—
270C22⋊C4—92no—
271C22⋊C4—92no—
272C22⋊C4—94no—
273C22⋊C4—92no—
274C42—31no—
275C42—31no—
276C42—31no—
277C42—31no—
278C42—92no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—42no—
282C22×C3—41no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—81no—
289C22×C3#99412yes—
290C22×C3C23×A411yescharacteristic
291C2×D3—62no—
292C2×D3—62no—
293C2×D3—62no—
294C3×C4—21no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—42no—
298C3×C4—61no—
299C3×C4—61no—
300C3×C4—61no—
301C3×C4—81no—
302C3×C4—82no—
303C3×C4—82no—
304C3×C4—242no—
305C3×C4—242no—
306C3×C42−1+4⋊C312yes—
307Dic3—22no—
308Dic3—22no—
309Dic3—62no—
310C32—41noSylow
311C23—92no—
312C23C2×D3×A411yescharacteristic
313Q8—31no—
314Q8—61no—
315Q8—61no—
316Q8—62no—
317Q8—362no—
318Q8C3×#23511yescharacteristic
319D4—182no—
320D4—182no—
321D4—184no—
322D4—182no—
323D4—362no—
324C2×C4—22no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—62no—
334C2×C4—92no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4—364no—
338C2×C4C2×D3×A411yescharacteristic
339C2×C4C2×D3×A411yescharacteristic
340C2×C3—22no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—82no—
345C2×C3—84no—
346C2×C3—81no—
347C2×C3—81no—
348C2×C3—81no—
349C2×C3A4×#4111yescharacteristic
350C2×C3C2×#99411yescharacteristic
351C2×C3C2×2−1+4⋊C311yescharacteristic
352D3—122no—
353C22—21no—
354C22—21no—
355C22—182no—
356C22—182no—
357C22—182no—
358C22C3⋊#99412yes—
359C22C22×D3×A411yescharacteristic
360C4—21no—
361C4—31no—
362C4—31no—
363C4—61no—
364C4—62no—
365C4—62no—
366C4—61no—
367C4—61no—
368C4—182no—
369C4#1007⋊C312yes—
370C3—41no—
371C3—81no—
372C3#3522⋊C311yescharacteristic, complemented
373C2—22no—
374C2—362no—
375C2A4×#23511yescharacteristic
376C2C2×#1007⋊C311yescharacteristic
377C2C2×C3⋊#99411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SL(2,𝔽3)C2semidirect
2. #5817⋊C3C2semidirect
3. C4⋊(C3⋊(#41⋊C3))C2semidirect
4. #5574⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #85821C3semidirect
19. #5577C2×C3semidirect
20. #5574C2×C3semidirect
21. Q8×Dic6C2×C3semidirect
30. C2×C4×SL(2,𝔽3)D3semidirect
90. C3×SL(2,𝔽3)#41semidirect
235. SL(2,𝔽3)#235semidirect
372. C3#3522⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#5817⋊C32
C4⋊(C3⋊(#41⋊C3))2
#5574⋊C32
SL(2,𝔽3)×Dic62
#858213
#3522⋊C33
C3×#9244