Finite Group Databasebeta

← #85804⋊C3 #11923190

Subgroups — #85804⋊C3 #11923190

Order 1152.

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

normal subgroups only

This group has 589 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
2SL(2,𝔽3)×#67C212yesmaximal, complemented
3#5572⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
5#5570⋊C3C211yesmaximal, characteristic, complemented
6#85804C311yesmaximal, characteristic, complemented
7#3523⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×D3×SL(2,𝔽3)C2212yes—
10C3×#930—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C2×C3×C4×Q8C2×C311yescharacteristic, complemented
15Q8×#67C2×C312yescomplemented
16#5584—31no—
17#5583—32no—
18Q8×D12C2×C311yescharacteristic, complemented
19#5572C2×C312yescomplemented
20#5570C2×C311yescharacteristic, complemented
21#5567—31no—
22#5513—32no—
23#5479—31no—
24#5470—31no—
25#557⋊C3—31no—
26D4×SL(2,𝔽3)—32no—
27D4×SL(2,𝔽3)—31no—
28#519⋊C3—32no—
29C2×C4×SL(2,𝔽3)—21no—
30C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
31C22×C32×C4—41no—
32C2×C3×#67—42no—
33C2×C3×D12—41no—
34C2×C3×SL(2,𝔽3)—21no—
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36D3×SL(2,𝔽3)—22no—
37C3×(C2×D3)⋊C4—42no—
38C3×C4⋊Dic3—41no—
39#3523—31noSylow
40C22×C3×Q8C22×C311yescharacteristic
41C2×D3×Q8C22×C312yes—
42C22×SL(2,𝔽3)—21no—
43C22×SL(2,𝔽3)—32no—
44C22×SL(2,𝔽3)C2×D311yescharacteristic
45C3×#121—121no—
46C3×C4×Q8—31no—
47C3×C4×Q8—31no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×Q8C22×C311yescharacteristic
50C2×C3×C4⋊C4—31no—
51C2×C3×C42—31no—
52#946—32no—
53Q8×Dic3C22×C312yes—
54#944—32no—
55#930A411yescharacteristic
56#929—32no—
57C4×#67—32no—
58#926—32no—
59#924—31no—
60#924—32no—
61#898—34no—
62#897—32no—
63#892—32no—
64#890—32no—
65#889—34no—
66#887—32no—
67#876—32no—
68#875—31no—
69C4×D12—31no—
70#872—32no—
71#870—32no—
72#870—31no—
73#869—31no—
74#869—31no—
75C4×Dic6—32no—
76C4×Dic6—31no—
77C4×Dic6—32no—
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—
84C22×C3×D3—42no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4—82no—
88C3×#67—84no—
89C2×C3×Dic3—42no—
90C3×D12—82no—
91C3×SL(2,𝔽3)#4111yescharacteristic
92#557—31no—
93#557—91no—
94#556—91no—
95#552—92no—
96D4×Q8—32no—
97D4×Q8—31no—
98#541—92no—
99#533—91no—
100#529—91no—
101#519—32no—
102C2×C4×Q8C3×D311yescharacteristic, complemented
103C2×C3×Q8—21no—
104C2×C3×Q8—31no—
105C2×C3×Q8—61no—
106C2×C3×Q8C23×C311yescharacteristic
107C2×C3×D4—121no—
108C2×C3×D4—122no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—41no—
112C22×C3×C4—81no—
113C22×C3×C4C2×A411yescharacteristic
114C2×#67C2×A412yes—
115D3×Q8—22no—
116D3×Q8—62no—
117C2×D12C2×A411yescharacteristic
118C2×C4×D3—32no—
119C2×Dic6—31no—
120C2×Dic6—32no—
121C2×SL(2,𝔽3)—22no—
122C2×SL(2,𝔽3)—21no—
123C2×SL(2,𝔽3)—21no—
124C2×SL(2,𝔽3)—62no—
125C2×SL(2,𝔽3)C22×D311yescharacteristic
126C3×C4⋊C4—31no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—121no—
130C3×C22⋊C4—122no—
131C3×C42—31no—
132C3×C42—31no—
133C3×C42—31no—
134C3×C42—31no—
135#217—32no—
136(C2×D3)⋊C4—32no—
137(C2×D3)⋊C4—32no—
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4C2×A412yes—
140C4⋊Dic3—32no—
141C4⋊Dic3—32no—
142C4⋊Dic3—31no—
143C4⋊Dic3—32no—
144C4⋊Dic3C2×A411yescharacteristic
145Dic3⋊C4—34no—
146Dic3⋊C4—32no—
147Dic3⋊C4—32no—
148Dic3⋊C4—32no—
149Dic3⋊C4—32no—
150C4×Dic3—34no—
151C4×Dic3—32no—
152C22×C32—42no—
153C22×C32—41no—
154C22×C32—81no—
155C22×C32—81no—
156C2×C3×D3—82no—
157C2×C3×D3—82no—
158C2×C3×D3—82no—
159C32×C4—42no—
160C32×C4—81no—
161C3×Dic3—82no—
162C22×Q8—32no—
163C22×Q8C2×C3×D311yescharacteristic
164C4⋊Q8—91no—
165C4⋊Q8—91no—
166C4⋊Q8—92no—
167#126—92no—
168#126—94no—
169#125—91no—
170#125—92no—
171#125—92no—
172#125—94no—
173#124—91no—
174#123—92no—
175C22⋊Q8—91no—
176C22⋊Q8—92no—
177C22⋊Q8—92no—
178C22⋊Q8—92no—
179#121—31no—
180C4×Q8—31no—
181C4×Q8—31no—
182C4×Q8—32no—
183C4×Q8—91no—
184C4×Q8—92no—
185C4×Q8—92no—
186C4×Q8—92no—
187C4×Q8C2×C3×D311yescharacteristic
188C4×Q8C2×C3×D311yescharacteristic
189C4×D4—91no—
190C4×D4—92no—
191#117—92no—
192#117—92no—
193#117—92no—
194C2×C4⋊C4—31no—
195C2×C42—31no—
196C23×C3—41no—
197C23×C3—81no—
198C23×C3—122no—
199C23×C3C22×A411yescharacteristic
200C22×D3C22×A412yes—
201C3×Q8—31no—
202C3×Q8—61no—
203C3×Q8—61no—
204C3×Q8C3×#4111yescharacteristic
205C3×D4—242no—
206C3×D4—244no—
207C2×C3×C4—22no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—62no—
217C2×C3×C4—82no—
218C2×C3×C4—84no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4—122no—
222C2×C3×C4C22×A411yescharacteristic
223C2×C3×C4C22×A411yescharacteristic
224#67—24no—
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3C22×A412yes—
228D12—22no—
229C4×D3—62no—
230C4×D3—62no—
231Dic6—62no—
232Dic6—62no—
233Dic6—62no—
234SL(2,𝔽3)—21no—
235SL(2,𝔽3)#23511yescharacteristic
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—82no—
240C3×D3—162no—
241C2×Q8—21no—
242C2×Q8—31no—
243C2×Q8—61no—
244C2×Q8—62no—
245C2×Q8—92no—
246C2×Q8—91no—
247C2×Q8—182no—
248C2×Q8C22×C3×D311yescharacteristic
249C2×D4—32no—
250C2×D4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—92no—
254C22×C4D3×A411yescharacteristic
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—32no—
258C4⋊C4—31no—
259C4⋊C4—92no—
260C4⋊C4—92no—
261C4⋊C4—92no—
262C4⋊C4—92no—
263C4⋊C4—92no—
264C4⋊C4—92no—
265C4⋊C4—94no—
266C4⋊C4—92no—
267C4⋊C4—91no—
268C22⋊C4—32no—
269C22⋊C4—92no—
270C22⋊C4—92no—
271C22⋊C4—92no—
272C22⋊C4—92no—
273C42—31no—
274C42—31no—
275C42—31no—
276C42—31no—
277C42—92no—
278C42—94no—
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—242no—
290C22×C3—242no—
291C22×C3—242no—
292C22×C32−1+4⋊C312yes—
293C22×C3C23×A411yescharacteristic
294C2×D3—22no—
295C2×D3—22no—
296C2×D3—22no—
297C3×C4—21no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—42no—
301C3×C4—61no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—81no—
305C3×C4—82no—
306C3×C4—82no—
307C3×C4—242no—
308C3×C42−1+4⋊C312yes—
309Dic3—22no—
310Dic3—62no—
311Dic3—62no—
312C32—41noSylow
313C23—32no—
314C23C2×D3×A411yescharacteristic
315Q8—31no—
316Q8—61no—
317Q8—61no—
318Q8—182no—
319Q8—182no—
320Q8—182no—
321Q8C3×#23511yescharacteristic
322D4—64no—
323D4—62no—
324C2×C4—22no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—62no—
333C2×C4—92no—
334C2×C4—92no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4C2×D3×A411yescharacteristic
338C2×C4C2×D3×A411yescharacteristic
339C2×C3—22no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—82no—
344C2×C3—84no—
345C2×C3—81no—
346C2×C3—81no—
347C2×C3—81no—
348C2×C3—482no—
349C2×C3A4×#4111yescharacteristic
350C2×C3C2×2−1+4⋊C311yescharacteristic
351C2×C3C2×2−1+4⋊C311yescharacteristic
352D3—42no—
353C22—21no—
354C22—21no—
355C22—62no—
356C22—62no—
357C22—62no—
358C22#1010⋊C312yes—
359C22C22×D3×A411yescharacteristic
360C4—21no—
361C4—31no—
362C4—31no—
363C4—61no—
364C4—61no—
365C4—61no—
366C4—62no—
367C4—182no—
368C4—182no—
369C4#1007⋊C312yes—
370C3—41no—
371C3—81no—
372C3#3523⋊C311yescharacteristic, complemented
373C2—22no—
374C2—122no—
375C2A4×#23511yescharacteristic
376C2C2×#1007⋊C311yescharacteristic
377C2C2×#1010⋊C311yescharacteristic
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. SL(2,𝔽3)×#67C2semidirect
3. #5572⋊C3C2semidirect
4. SL(2,𝔽3)×D12C2semidirect
5. #5570⋊C3C2semidirect
6. #85804C3semidirect
14. C2×C3×C4×Q8C2×C3semidirect
15. Q8×#67C2×C3semidirect
18. Q8×D12C2×C3semidirect
19. #5572C2×C3semidirect
20. #5570C2×C3semidirect
30. C2×C4×SL(2,𝔽3)D3semidirect
102. C2×C4×Q8C3×D3semidirect
372. C3#3523⋊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
SL(2,𝔽3)×#672
#5572⋊C32
SL(2,𝔽3)×D122
#5570⋊C32
#858043
#3523⋊C33
C3×#9304