Finite Group Databasebeta

← #85832⋊C3 #11923206

Subgroups — #85832⋊C3 #11923206

Order 1152.

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

normal subgroups only

This group has 553 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#5831⋊C3C212yesmaximal, complemented
3#5572⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#85832C311yesmaximal, characteristic, complemented
7#3518⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×D3×SL(2,𝔽3)C2212yes—
10C3×#875—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×C4×#41C2×C311yescharacteristic, complemented
15#5831C2×C312yescomplemented
16Q8×D12C2×C311yescharacteristic
17#5572C2×C312yescomplemented
18Q8×Dic6C2×C311yescharacteristic, complemented
19#5563—31no—
20#5561—32no—
21#5542—31no—
22#5533—32no—
23#5479—31no—
24#5470—31no—
25Q8×SL(2,𝔽3)—31no—
26D4×SL(2,𝔽3)—31no—
27#548⋊C3—32no—
28#519⋊C3—32no—
29C4×#41⋊C3—21no—
30C4×#41⋊C3D311yescharacteristic, complemented
31C2×C3×D12—41no—
32C2×C3×Dic6—41no—
33C3×#41⋊C3—21no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35D3×SL(2,𝔽3)—22no—
36C32×C42—41no—
37C3×(C2×D3)⋊C4—42no—
38C3×(C2×D3)⋊C4—42no—
39#3518—31noSylow
40C2×C3×#41C22×C311yescharacteristic
41C2×D3×Q8C22×C312yes—
42C2×#41⋊C3—21no—
43C2×#41⋊C3C2×D311yescharacteristic
44C22×SL(2,𝔽3)—32no—
45C3×#124—121no—
46C3×C4×Q8C22×C311yescharacteristic
47C3×C4×Q8C22×C311yescharacteristic
48C3×C4×D4—31no—
49C3×C4×D4—31no—
50C3×#117—31no—
51C2×C3×C42—31no—
52Q8×Dic3C22×C312yes—
53#938—32no—
54#936—32no—
55#930—31no—
56#929—32no—
57C4×#67—32no—
58#926—32no—
59#924—32no—
60#924—31no—
61#924—32no—
62#897—32no—
63#892—32no—
64#888—32no—
65#887—32no—
66#885—34no—
67#879—34no—
68#876—32no—
69#875A411yescharacteristic
70C4×D12—31no—
71#872—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
83C22×C3×D3—42no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C3×Dic3—42no—
88C3×D12—82no—
89C3×Dic6—82no—
90C3×SL(2,𝔽3)#4111yescharacteristic
91Q82—31no—
92#556—92no—
93#556—91no—
94#555—91no—
95D4×Q8—31no—
96#548—32no—
97#542—92no—
98#533—91no—
99#529—91no—
100#519—32no—
101C4×#41C3×D311yescharacteristic, complemented
102C3×#41—21no—
103C3×#41—61no—
104C2×C3×Q8—121no—
105C2×C3×Q8C23×C311yescharacteristic
106C2×C3×D4—31no—
107C2×C3×D4—121no—
108C22×C3×C4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C2×#67—32no—
112D3×Q8—22no—
113D3×Q8—62no—
114C2×D12C2×A411yescharacteristic
115C2×C4×D3—32no—
116C2×Dic6—32no—
117C2×Dic6—32no—
118C2×Dic6C2×A411yescharacteristic
119#41⋊C3—22no—
120#41⋊C3—21no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—62no—
123C2×SL(2,𝔽3)C22×D311yescharacteristic
124C3×C4⋊C4—31no—
125C3×C4⋊C4—31no—
126C3×C22⋊C4—32no—
127C3×C22⋊C4—122no—
128C3×C22⋊C4—122no—
129C3×C42—31no—
130C3×C42—31no—
131C3×C42—31no—
132C3×C42—41no—
133C3×C42—81no—
134C3×C42C2×A411yescharacteristic
135#217—34no—
136#217—32no—
137(C2×D3)⋊C4—32no—
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4C2×A412yes—
140(C2×D3)⋊C4C2×A412yes—
141C4⋊Dic3—32no—
142C4⋊Dic3—31no—
143C4⋊Dic3—31no—
144C4⋊Dic3—32no—
145Dic3⋊C4—32no—
146Dic3⋊C4—32no—
147Dic3⋊C4—32no—
148Dic3⋊C4—32no—
149C4×Dic3—34no—
150C4×Dic3—32no—
151C22×C32—41no—
152C2×C3×D3—82no—
153C2×C3×D3—82no—
154C2×C3×D3—82no—
155C32×C4—42no—
156C32×C4—81no—
157C32×C4—81no—
158C3×Dic3—82no—
159C2×#41C2×C3×D311yescharacteristic
160C22×Q8—32no—
161C4⋊Q8—91no—
162C4⋊Q8—91no—
163C4⋊Q8—92no—
164#126—92no—
165#126—94no—
166#125—91no—
167#124—31no—
168#124—94no—
169#124—92no—
170#123—92no—
171C22⋊Q8—91no—
172C22⋊Q8—92no—
173C22⋊Q8—92no—
174C22⋊Q8—92no—
175#121—91no—
176#121—92no—
177C4×Q8—32no—
178C4×Q8—91no—
179C4×Q8—92no—
180C4×Q8—92no—
181C4×Q8—92no—
182C4×Q8C2×C3×D311yescharacteristic
183C4×Q8C2×C3×D311yescharacteristic
184C4×D4—31no—
185C4×D4—31no—
186C4×D4—91no—
187C4×D4—92no—
188#117—31no—
189#117—92no—
190#117—92no—
191#117—92no—
192C2×C42—31no—
193C23×C3—31no—
194C23×C3—122no—
195C22×D3C22×A412yes—
196C3×Q8—31no—
197C3×Q8—242no—
198C3×Q8C3×#4111yescharacteristic
199C3×D4—61no—
200C3×D4—61no—
201C3×D4—242no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—62no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4—122no—
215C2×C3×C4—121no—
216C2×C3×C4C22×A411yescharacteristic
217C2×C3×C4C22×A411yescharacteristic
218C2×C3×C4C22×A411yescharacteristic
219#67—122no—
220C2×Dic3—32no—
221C2×Dic3—32no—
222C2×Dic3C22×A412yes—
223D12—22no—
224C4×D3—62no—
225C4×D3—62no—
226Dic6—22no—
227Dic6—62no—
228Dic6—62no—
229Dic6—122no—
230SL(2,𝔽3)—21no—
231SL(2,𝔽3)#23511yescharacteristic
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C3×D3—162no—
236#41—21no—
237#41—61no—
238C2×Q8—31no—
239C2×Q8—62no—
240C2×Q8—92no—
241C2×Q8—92no—
242C2×Q8—182no—
243C2×Q8C22×C3×D311yescharacteristic
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—92no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—92no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—92no—
254C4⋊C4—92no—
255C4⋊C4—92no—
256C4⋊C4—92no—
257C4⋊C4—91no—
258C4⋊C4—91no—
259C4⋊C4—92no—
260C4⋊C4—92no—
261C22⋊C4—32no—
262C22⋊C4—32no—
263C22⋊C4—32no—
264C22⋊C4—92no—
265C22⋊C4—92no—
266C22⋊C4—92no—
267C22⋊C4—94no—
268C42—31no—
269C42—31no—
270C42—31no—
271C42—92no—
272C42—94no—
273C42D3×A411yescharacteristic
274C22×C3—41no—
275C22×C3—61no—
276C22×C3—61no—
277C22×C3—61no—
278C22×C3—81no—
279C22×C3—242no—
280C22×C3—242no—
281C22×C3—242no—
282C22×C3C23×A411yescharacteristic
283C2×D3—22no—
284C2×D3—22no—
285C2×D3—22no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—42no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—82no—
296C3×C4—82no—
297C3×C4—82no—
298C3×C4—242no—
299C3×C42−1+4⋊C312yes—
300Dic3—22no—
301Dic3—62no—
302Dic3—62no—
303C32—41noSylow
304C23—31no—
305C23—32no—
306Q8—31no—
307Q8—62no—
308Q8—182no—
309Q8—182no—
310Q8—362no—
311Q8C3×#23511yescharacteristic
312D4—61no—
313D4—61no—
314D4—62no—
315D4—362no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—62no—
323C2×C4—92no—
324C2×C4—92no—
325C2×C4—121no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4C2×D3×A411yescharacteristic
329C2×C4C2×D3×A411yescharacteristic
330C2×C4C2×D3×A411yescharacteristic
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—81no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3—121no—
338C2×C3—482no—
339C2×C3A4×#4111yescharacteristic
340C2×C3C2×2−1+4⋊C311yescharacteristic
341C2×C3#594511yescharacteristic
342D3—42no—
343C22—61no—
344C22—62no—
345C22—61no—
346C22—62no—
347C22—61no—
348C22—62no—
349C22C22×D3×A411yescharacteristic
350C4—21no—
351C4—21no—
352C4—31no—
353C4—31no—
354C4—61no—
355C4—62no—
356C4—61no—
357C4—182no—
358C4—182no—
359C4#1010⋊C312yes—
360C3—41no—
361C3—81no—
362C3#3518⋊C311yescharacteristic, complemented
363C2—122no—
364C2—121no—
365C2A4×#23511yescharacteristic
366C2C2×#1010⋊C311yescharacteristic
367C2#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. #5831⋊C3C2semidirect
3. #5572⋊C3C2semidirect
4. SL(2,𝔽3)×D12C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #85832C3semidirect
14. C3×C4×#41C2×C3semidirect
15. #5831C2×C3semidirect
17. #5572C2×C3semidirect
18. Q8×Dic6C2×C3semidirect
30. C4×#41⋊C3D3semidirect
101. C4×#41C3×D3semidirect
362. C3#3518⋊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
#5831⋊C32
#5572⋊C32
SL(2,𝔽3)×D122
SL(2,𝔽3)×Dic62
#858323
#3518⋊C33
C3×#8754