Finite Group Databasebeta

← #85853⋊C3 #11923196

Subgroups — #85853⋊C3 #11923196

Order 1152.

557 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
2#5834⋊C3C212yesmaximal, complemented
3#5570⋊C3C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5#5574⋊C3C212yesmaximal, complemented
6#85853C311yesmaximal, characteristic, complemented
7#3517⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×#870—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×C4×#41C2×C311yescharacteristic
15#5834C2×C312yes—
16#5577C2×C311yescharacteristic
17#5574C2×C312yes—
18#5570C2×C311yescharacteristic
19#5562—32no—
20#5553—31no—
21#5551—31no—
22#5530—32no—
23#5479—31no—
24#5477—31no—
25#557⋊C3—31no—
26C4⋊(#41⋊C3)—32no—
27C4⋊(#41⋊C3)—31no—
28#520⋊C3—32no—
29C4×#41⋊C3—21no—
30C4×#41⋊C3D311yescharacteristic, complemented
31C3×#41⋊C3—21no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊(#41⋊C3)—22no—
34C32×C42—41no—
35C3×C4⋊Dic3—41no—
36C3×C4⋊Dic3—41no—
37C3×Dic3⋊C4—42no—
38C3×Dic3⋊C4—42no—
39#3517—31noSylow
40C2×C3×#41C22×C311yescharacteristic
41C2×#239C22×C312yes—
42C2×#235—32no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3—32no—
45C2×#41⋊C3C2×D311yescharacteristic
46C3×#125—121no—
47C3×C4×Q8C22×C311yescharacteristic
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×D4—31no—
50C3×C4×D4—31no—
51C3×#117—31no—
52C2×C3×C42—31no—
53Q8×Dic3C22×C312yes—
54#940—32no—
55#938—32no—
56#930—32no—
57#930—32no—
58#930—31no—
59#929—32no—
60C4×#67—32no—
61#924—31no—
62#895—32no—
63#893—32no—
64D3×C4⋊C4—32no—
65#890—32no—
66#884—34no—
67#883—34no—
68#876—32no—
69#875—31no—
70#874—31no—
71C4×D12—32no—
72C4×D12—32no—
73C4×D12—31no—
74#872—32no—
75#870A411yescharacteristic
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
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—42no—
88C3×SL(2,𝔽3)#4111yescharacteristic, complemented
89#557—31no—
90#553—92no—
91#550—32no—
92#550—31no—
93#550—91no—
94#547—91no—
95#540—92no—
96#532—91no—
97#529—91no—
98#520—32no—
99C4×#41C3×D311yescharacteristic
100C3×#41—21no—
101C3×#41—61no—
102C2×C3×Q8C23×C311yescharacteristic
103C2×C3×D4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C2×#67—34no—
108C2×#67—32no—
109#239—22no—
110#239—62no—
111#235—124no—
112C2×D12—32no—
113C2×D12—31no—
114C2×D12—32no—
115C2×C4×D3—32no—
116C2×C4×D3—34no—
117C2×Dic6—31no—
118#41⋊C3—22no—
119#41⋊C3—21no—
120#41⋊C3—62no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)C22×D311yescharacteristic
123C3×C4⋊C4—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—122no—
126C3×C4⋊C4—122no—
127C3×C4⋊C4—121no—
128C3×C4⋊C4—121no—
129C3×C22⋊C4—32no—
130C3×C42—31no—
131C3×C42—31no—
132C3×C42—31no—
133C3×C42—41no—
134C3×C42—81no—
135C3×C42C2×A411yescharacteristic
136#217—32no—
137(C2×D3)⋊C4—32no—
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4—32no—
140(C2×D3)⋊C4—32no—
141C4⋊Dic3—32no—
142C4⋊Dic3—32no—
143C4⋊Dic3C2×A411yescharacteristic
144C4⋊Dic3C2×A411yescharacteristic
145Dic3⋊C4—32no—
146Dic3⋊C4—32no—
147Dic3⋊C4C2×A412yes—
148Dic3⋊C4C2×A412yes—
149C4×Dic3—32no—
150C22×C32—41no—
151C32×C4—42no—
152C32×C4—81no—
153C32×C4—81no—
154C3×Dic3—82no—
155C3×Dic3—82no—
156C2×#41—32no—
157C2×#41—92no—
158C2×#41C2×C3×D311yescharacteristic
159C4⋊Q8—91no—
160#127—91no—
161#127—92no—
162#126—92no—
163#125—31no—
164#125—92no—
165#124—91no—
166#123—94no—
167#123—92no—
168C22⋊Q8—91no—
169#121—94no—
170#121—91no—
171#121—92no—
172#121—92no—
173#121—92no—
174#121—92no—
175C4×Q8—32no—
176C4×Q8—91no—
177C4×Q8C2×C3×D311yescharacteristic
178C4×Q8C2×C3×D311yescharacteristic
179C4×D4—31no—
180C4×D4—31no—
181C4×D4—91no—
182C4×D4—92no—
183C4×D4—92no—
184C4×D4—92no—
185C4×D4—92no—
186#117—31no—
187#117—92no—
188C2×C4⋊C4—92no—
189C2×C42—31no—
190C23×C3—31no—
191C22×D3—32no—
192C3×Q8—31no—
193C3×Q8C3×#4111yescharacteristic
194C3×D4—61no—
195C3×D4—61no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—62no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—122no—
209C2×C3×C4—122no—
210C2×C3×C4—121no—
211C2×C3×C4C22×A411yescharacteristic
212C2×C3×C4C22×A411yescharacteristic
213C2×C3×C4C22×A411yescharacteristic
214#67—122no—
215#67—124no—
216C2×Dic3—32no—
217C2×Dic3C22×A412yes—
218C2×Dic3C22×A412yes—
219D12—62no—
220D12—62no—
221D12—62no—
222D12—122no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—124no—
226Dic6—62no—
227SL(2,𝔽3)—21no—
228SL(2,𝔽3)#23511yescharacteristic, complemented
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232#41—21no—
233#41—61no—
234#41—62no—
235#41—182no—
236#41—364no—
237C2×Q8—91no—
238C2×Q8C22×C3×D311yescharacteristic
239C2×D4—31no—
240C2×D4—92no—
241C2×D4—92no—
242C2×D4—92no—
243C2×D4—94no—
244C2×D4—91no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—92no—
249C22×C4—94no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—32no—
253C4⋊C4—31no—
254C4⋊C4—32no—
255C4⋊C4—31no—
256C4⋊C4—92no—
257C4⋊C4—92no—
258C4⋊C4—92no—
259C4⋊C4—92no—
260C22⋊C4—32no—
261C22⋊C4—92no—
262C22⋊C4—92no—
263C22⋊C4—92no—
264C22⋊C4—92no—
265C22⋊C4—92no—
266C42—31no—
267C42—31no—
268C42—31no—
269C42—92no—
270C42D3×A411yescharacteristic
271C22×C3—41no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3—61no—
275C22×C3—81no—
276C22×C3C23×A411yescharacteristic
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—42no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—82no—
290C3×C4—82no—
291C3×C4—82no—
292C3×C4—242no—
293C3×C4—242no—
294C3×C4#99412yes—
295Dic3—22no—
296Dic3—22no—
297Dic3—62no—
298C32—41noSylow
299C23—31no—
300C23—92no—
301Q8—31no—
302Q8—182no—
303Q8C3×#23511yescharacteristic
304D4—61no—
305D4—61no—
306D4—182no—
307D4—182no—
308D4—182no—
309D4—362no—
310D4—362no—
311D4—364no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—62no—
320C2×C4—92no—
321C2×C4—121no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4—364no—
325C2×C4C2×D3×A411yescharacteristic
326C2×C4C2×D3×A411yescharacteristic
327C2×C4C2×D3×A411yescharacteristic
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3—81no—
333C2×C3—81no—
334C2×C3—121no—
335C2×C3A4×#4111yescharacteristic
336C2×C3C2×#99411yescharacteristic
337C2×C3#594511yescharacteristic
338D3—122no—
339C22—61no—
340C22—61no—
341C22—61no—
342C22—182no—
343C22—182no—
344C22—182no—
345C22C22×D3×A411yescharacteristic
346C4—21no—
347C4—21no—
348C4—31no—
349C4—31no—
350C4—62no—
351C4—61no—
352C4—61no—
353C4—62no—
354C4—182no—
355C4C3⋊#99412yes—
356C3—41no—
357C3—81no—
358C3#3517⋊C311yescharacteristic, complemented
359C2—121no—
360C2—362no—
361C2A4×#23511yescharacteristic
362C2C2×C3⋊#99411yescharacteristic
363C2#5964⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
hypercenterC42D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#41⋊C3C2semidirect
2. #5834⋊C3C2semidirect
3. #5570⋊C3C2semidirect
4. C4⋊(C3⋊(#41⋊C3))C2semidirect
5. #5574⋊C3C2semidirect
6. #85853C3semidirect
30. C4×#41⋊C3D3semidirect
88. C3×SL(2,𝔽3)#41semidirect
228. SL(2,𝔽3)#235semidirect
358. C3#3517⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#5834⋊C32
#5570⋊C32
C4⋊(C3⋊(#41⋊C3))2
#5574⋊C32
#858533
#3517⋊C33
C3×#8704