Finite Group Databasebeta

← #11922261

Subgroups — #11922261

Order 1152.

536 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#10596586C211yesmaximal, characteristic
2#5819⋊C3C211yesmaximal, characteristic, complemented
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593222C212yesmaximal, complemented
5#10593220C212yesmaximal, complemented
6#85453—31nomaximal
7#78980—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267C411yescharacteristic, complemented
10#64266C2212yes—
11#64265C411yescharacteristic, complemented
12#64264C412yescomplemented
13#63817—41nomaximal
14C3⋊(#131⋊C3)C2212yescomplemented
15C3×GU(2,𝔽3)C2211yescharacteristic, complemented
16C3×GU(2,𝔽3)C2211yescharacteristic, complemented
17#5923—31no—
18#5923—61no—
19#5822—31no—
20#5819D311yescharacteristic, complemented
21#5745—31no—
22#568⋊C3—31no—
23C2×GU(2,𝔽3)D311yescharacteristic, complemented
24#5405—32no—
25#5404—32no—
26C2×C3×#104—31no—
27#5124—31no—
28C2×#805—31no—
29C4×#208—31no—
30#4824—32no—
31#4824—32no—
32#4823—32no—
33#4823—32no—
34#3828—41no—
35C3×#41⋊C3C2×C411yescharacteristic
36C3×#41⋊C3C2311yescharacteristic
37C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
38C3⋊GL(2,𝔽3)C2×C412yescomplemented
39C3⋊#226C2×C412yescomplemented
40C3×#208—41no—
41C2×C3×C3⋊C8—41no—
42#3712—42no—
43#3710—42no—
44#2956—91noSylow
45C2×C3×#41C2×D311yescharacteristic
46C2×#235—31no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3C2×D311yescharacteristic
49#986—31no—
50#986—61no—
51#985—32no—
52#985—62no—
53#984—31no—
54#984—61no—
55#983—32no—
56#983—62no—
57C2×C3×M16—31no—
58C2×C3×C42—31no—
59#951—31no—
60#951—31no—
61#950—32no—
62#950—32no—
63#949—31no—
64#949—31no—
65#948—32no—
66#948C2×D312yes—
67#942—32no—
68#932—32no—
69C2×#208—31no—
70C2×#208—31no—
71C2×#208—31no—
72#907—32no—
73#907—32no—
74#907—121no—
75D3×M16—62no—
76#901—61no—
77#901—61no—
78#131⋊C3—32no—
79GU(2,𝔽3)C2×D311yescharacteristic, complemented
80GU(2,𝔽3)C2×D311yescharacteristic, complemented
81C3×#104—31no—
82C3×#104—31no—
83C3×#104—31no—
84C3×#104—31no—
85#822—32no—
86#822—32no—
87#805—32no—
88#805—32no—
89#803—32no—
90C4×C3⋊C8—32no—
91C2×C32×C4—41no—
92#638—42no—
93C4⋊(C3⋊D3)—41no—
94C4×C3⋊D3—42no—
95#634—41no—
96C3×SL(2,𝔽3)C22×C411yescharacteristic
97C3×C3⋊C8—41no—
98C3×C3⋊C8—41no—
99C3×C3⋊C8—42no—
100#575—91no—
101#568—31no—
102#568—91no—
103#444—92no—
104#444—92no—
105#443—92no—
106#443—92no—
107#430—91no—
108C2×#104—31no—
109C2×#104—91no—
110C4×M16—91no—
111C3×#41—31no—
112C3×#41—31no—
113C3×#41C4×D311yescharacteristic
114C3×#41C22×D311yescharacteristic
115C2×C3×Q8C4×D311yescharacteristic
116C2×C3×D4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—61no—
119C2×#67—61no—
120#235—32no—
121#235—32no—
122#235—61no—
123#235—61no—
124#235—121no—
125#235—241no—
126C2×D12—31no—
127C2×C4×D3—61no—
128C2×Dic6—31no—
129#41⋊C3—21no—
130#41⋊C3—21no—
131#41⋊C3C4×D311yescharacteristic
132#41⋊C3C22×D311yescharacteristic
133C2×SL(2,𝔽3)—21no—
134C2×SL(2,𝔽3)C4×D311yescharacteristic
135GL(2,𝔽3)—32no—
136GL(2,𝔽3)—62no—
137#226—32no—
138#226—62no—
139C3×M16—31no—
140C3×M16—31no—
141C3×M16—61no—
142C3×M16—121no—
143C2×C3×C8—61no—
144C3×C42—31no—
145C3×C42—31no—
146C3×C42—61no—
147#216—32no—
148#216—32no—
149#215—32no—
150#215—32no—
151#214—32no—
152#214—32no—
153#213—32no—
154#213—32no—
155#208—32no—
156#208—31no—
157#208—31no—
158#208—31no—
159#208—32no—
160#208—32no—
161#208S411yescharacteristic
162C2×C3⋊C8—32no—
163C2×C3⋊C8—32no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—61no—
166ΓL(1,𝔽25)—62no—
167ΓL(1,𝔽25)—62no—
168ΓL(1,𝔽25)—122no—
169D3×C8—62no—
170D3×C8—62no—
171D3×C8—122no—
172C22×C32—41no—
173C2×C3⋊D3—42no—
174C32×C4—41no—
175C32×C4—41no—
176C3⋊Dic3—42no—
177C2×#41—91no—
178C2×#41D3211yescharacteristic
179#137—91no—
180#137—91no—
181#137—92no—
182AGL(1,ℤ/8)—92no—
183AGL(1,ℤ/8)—91no—
184AGL(1,ℤ/8)—91no—
185#135—92no—
186#135—92no—
187#131—32no—
188#131—92no—
189#131—92no—
190#131—92no—
191#131—181no—
192#131—181no—
193C2×M16—31no—
194C2×M16—91no—
195C2×M16—91no—
196C2×M16—91no—
197C2×M16—182no—
198C2×C42—31no—
199#108—92no—
200#108—92no—
201#104—31no—
202#104—31no—
203#104—31no—
204#104—31no—
205#104—92no—
206#104—92no—
207#97—92no—
208C4×C8—92no—
209C23×C3—31no—
210C22×D3—61no—
211C3×Q8—31no—
212C3×Q8C2×C4×D311yescharacteristic
213C3×D4—31no—
214C3×D4—31no—
215C3×D4—61no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—41no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—81no—
225C2×C3×C4—122no—
226C2×C3×C4C2×S411yescharacteristic
227#67—62no—
228#67—62no—
229#67—122no—
230#67—242no—
231C2×Dic3—61no—
232D12—32no—
233D12—61no—
234D12—121no—
235D12—241no—
236C4×D3—62no—
237C4×D3—62no—
238C4×D3—122no—
239C4×D3—242no—
240Dic6—32no—
241Dic6—61no—
242Dic6—121no—
243Dic6—241no—
244SL(2,𝔽3)—21no—
245SL(2,𝔽3)C2×C4×D311yescharacteristic
246C3×C8—61no—
247C3×C8—61no—
248C3×C8—122no—
249C3⋊C8—32no—
250C3⋊C8—32no—
251C3⋊C8—32no—
252C3⋊C8—41no—
253C3⋊C8—41no—
254C3⋊C8C2×S412yes—
255C2×C32—41no—
256C2×C32—81no—
257C3⋊D3—82no—
258#41—31no—
259#41—31no—
260#41—92no—
261#41—92no—
262#41—181no—
263#41—181no—
264#41#62411yescharacteristic
265#41C2×D3211yescharacteristic
266C2×Q8—91no—
267C2×Q8#62411yescharacteristic
268C2×D4—31no—
269C2×D4—91no—
270C2×D4—181no—
271C22×C4—31no—
272C22×C4—61no—
273C22×C4—181no—
274Q16—92no—
275Q16—92no—
276SD16—92no—
277SD16—92no—
278SD16—92no—
279SD16—92no—
280D8—92no—
281D8—92no—
282M16—31no—
283M16—31no—
284M16—31no—
285M16—61no—
286M16—92no—
287M16—92no—
288M16—91no—
289M16—91no—
290M16—91no—
291M16—92no—
292M16—182no—
293M16—182no—
294C2×C8—61no—
295C2×C8—92no—
296C2×C8—92no—
297C2×C8—182no—
298C2×C8—182no—
299C2×C8—181no—
300C42—31no—
301C42—31no—
302C42—61no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—41no—
306C22×C3—81no—
307C22×C3—121no—
308C22×C3C4×S411yescharacteristic
309C2×D3—62no—
310C2×D3—122no—
311C2×D3—242no—
312C2×D3—241no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—62no—
318C3×C4—81no—
319C3×C4—81no—
320C3×C4—121no—
321C3×C4C4×S411yescharacteristic
322C3×C4C22×S411yescharacteristic
323Dic3—62no—
324Dic3—122no—
325Dic3—242no—
326C32—41noSylow
327C23—31no—
328C23—181no—
329Q8—31no—
330Q8—92no—
331Q8—181no—
332Q8C2×#62411yescharacteristic
333D4—31no—
334D4—31no—
335D4—61no—
336D4—92no—
337D4—182no—
338D4—182no—
339D4—181no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—122no—
348C2×C4—182no—
349C2×C4—181no—
350C2×C4—182no—
351C2×C4D3×S411yescharacteristic
352C8—32no—
353C8—61no—
354C8—61no—
355C8—92no—
356C8—92no—
357C8—92no—
358C2×C3—21no—
359C2×C3—41no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—81no—
363C2×C3—81no—
364C2×C3—161no—
365C2×C3C2×C4×S411yescharacteristic
366D3—122no—
367D3—242no—
368D3—482no—
369C22—31no—
370C22—31no—
371C22—121no—
372C22—182no—
373C22—721no—
374C22(C3⋊S4)⋊C411yescharacteristic
375C4—31no—
376C4—31no—
377C4—62no—
378C4—121no—
379C4—182no—
380C4(C3⋊S4)⋊C411yescharacteristic
381C4C2×D3×S411yescharacteristic
382C3—41no—
383C3—81no—
384C3#8545311yescharacteristic, complemented
385C2—21no—
386C2—61no—
387C2—61no—
388C2—362no—
389C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC2×C3C2×C4×S4
prime radical#10596586C2
cosocleC3×#41⋊C3C23
hypercenterC2×C4D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5819⋊C3C2semidirect
3. C2×C3×GU(2,𝔽3)C2semidirect
4. #10593222C2semidirect
5. #10593220C2semidirect
9. #64267C4semidirect
11. #64265C4semidirect
12. #64264C4semidirect
14. C3⋊(#131⋊C3)C22semidirect
15. C3×GU(2,𝔽3)C22semidirect
16. C3×GU(2,𝔽3)C22semidirect
20. #5819D3semidirect
23. C2×GU(2,𝔽3)D3semidirect
38. C3⋊GL(2,𝔽3)C2×C4semidirect
39. C3⋊#226C2×C4semidirect
79. GU(2,𝔽3)C2×D3semidirect
80. GU(2,𝔽3)C2×D3semidirect
384. C3#85453semidirect

Maximal subgroups

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

subgroupindex
#105965862
#5819⋊C32
C2×C3×GU(2,𝔽3)2
#105932222
#105932202
#854533
#789803
#638174