Finite Group Databasebeta

← #11922258

Subgroups — #11922258

Order 1152.

519 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 71 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×#5923C211yesmaximal, characteristic
2#5819⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593221C212yesmaximal, complemented
5#10593219C212yesmaximal, complemented
6#85453—31nomaximal
7#84000—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#986C411yescharacteristic, complemented
10C3×#985C2212yes—
11C3×#984C411yescharacteristic, complemented
12C3×#983C412yescomplemented
13#63815—41nomaximal
14C3⋊(#131⋊C3)C2212yescomplemented
15C3⋊GU(2,𝔽3)C2211yescharacteristic, complemented
16C3⋊GU(2,𝔽3)C2211yescharacteristic, complemented
17#5923D311yescharacteristic
18C3×#575—31no—
19#5819—31no—
20#5819D311yescharacteristic, complemented
21#568⋊C3—31no—
22C2×GU(2,𝔽3)—31no—
23C2×GU(2,𝔽3)—61no—
24#5405—32no—
25#5404—32no—
26C2×#837—31no—
27C2×#837—31no—
28#5197—32no—
29#5197—32no—
30#5196—32no—
31#5196—32no—
32#5125—31no—
33Dic3×M16—31no—
34C3×#235—41no—
35C3×#41⋊C3C2×C411yescharacteristic
36C3×#41⋊C3C2311yescharacteristic
37C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
38C3×GL(2,𝔽3)C2×C412yescomplemented
39C3×#226C2×C412yescomplemented
40C2×C3⋊(C3⋊C8)—41no—
41C3×#208—41no—
42#3711—42no—
43D3×C3⋊C8—42no—
44#2956—91noSylow
45C2×C3×#41—31no—
46C2×C3×#41C2×D311yescharacteristic
47C2×#41⋊C3—21no—
48C2×#41⋊C3C2×D311yescharacteristic
49#986Dic311yescharacteristic, complemented
50#985C2×D312yes—
51#984Dic311yescharacteristic, complemented
52#983Dic312yescomplemented
53C3×#137—31no—
54C3×#137—31no—
55C3×#137—32no—
56C3×AGL(1,ℤ/8)—32no—
57C3×AGL(1,ℤ/8)—31no—
58C3×AGL(1,ℤ/8)—31no—
59C3×#135—32no—
60C3×#135—32no—
61C2×C3×M16—31no—
62#948—32no—
63#948—32no—
64#948—32no—
65#948—61no—
66#948—61no—
67#948C2×D312yes—
68C2×C4×Dic3—31no—
69C2×#208—31no—
70C2×#208—31no—
71C2×#208—31no—
72C2×#208—62no—
73#907—121no—
74#131⋊C3—32no—
75GU(2,𝔽3)—31no—
76GU(2,𝔽3)—31no—
77GU(2,𝔽3)—61no—
78GU(2,𝔽3)—61no—
79#837—32no—
80#837—31no—
81#837—32no—
82#837—31no—
83#837—31no—
84#837—31no—
85#819—32no—
86#819—32no—
87#815—32no—
88C8×Dic3—32no—
89C2×C32×C4—41no—
90C3×#67—42no—
91C3×D12—41no—
92C3×C4×D3—42no—
93C3×Dic6—41no—
94C3×SL(2,𝔽3)C22×C411yescharacteristic
95C3⋊(C3⋊C8)—41no—
96C3⋊(C3⋊C8)—41no—
97C3×C3⋊C8—42no—
98#575—31no—
99#568—31no—
100#568—91no—
101#444—92no—
102#444—92no—
103#443—92no—
104#443—92no—
105#430—91no—
106C2×#104—91no—
107C2×#104—91no—
108C4×M16—91no—
109C3×#41—31no—
110C3×#41—31no—
111C3×#41—32no—
112C3×#41—32no—
113C3×#41—61no—
114C3×#41—61no—
115C3×#41C4×D311yescharacteristic
116C3×#41C22×D311yescharacteristic
117C2×C3×Q8—31no—
118C2×C3×Q8C4×D311yescharacteristic
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4—61no—
122C22×C3×C4—31no—
123C22×C3×C4—61no—
124C22×Dic3—61no—
125#235—41no—
126#41⋊C3—21no—
127#41⋊C3—21no—
128#41⋊C3C2×Dic311yescharacteristic
129#41⋊C3C22×D311yescharacteristic
130C2×SL(2,𝔽3)—21no—
131C2×SL(2,𝔽3)C2×Dic311yescharacteristic
132GL(2,𝔽3)C2×Dic312yescomplemented
133#226C2×Dic312yescomplemented
134C3×Q16—32no—
135C3×Q16—32no—
136C3×SD16—32no—
137C3×SD16—32no—
138C3×SD16—32no—
139C3×SD16—32no—
140C3×D8—32no—
141C3×D8—32no—
142C3×M16—32no—
143C3×M16—31no—
144C3×M16—31no—
145C3×M16—121no—
146C2×C3×C8—32no—
147C4×Dic3—31no—
148C4×Dic3—31no—
149C4×Dic3—61no—
150#208—31no—
151#208—32no—
152#208—32no—
153#208—31no—
154#208—31no—
155#208—62no—
156#208—61no—
157#208—62no—
158#208S411yescharacteristic
159C2×C3⋊C8—32no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8—62no—
162C2×C3⋊C8—61no—
163C2×C3⋊C8—62no—
164C2×C3⋊C8—121no—
165C2×C3⋊C8—241no—
166ΓL(1,𝔽25)—122no—
167D3×C8—122no—
168C22×C32—41no—
169C2×C3×D3—42no—
170C32×C4—41no—
171C32×C4—41no—
172C3×Dic3—42no—
173C2×#41—31no—
174C2×#41D3211yescharacteristic
175#137—31no—
176#137—31no—
177#137—32no—
178AGL(1,ℤ/8)—32no—
179AGL(1,ℤ/8)—31no—
180AGL(1,ℤ/8)—31no—
181#135—32no—
182#135—32no—
183#131—32no—
184#131—92no—
185#131—92no—
186#131—92no—
187#131—181no—
188#131—181no—
189C2×M16—31no—
190C2×M16—91no—
191C2×M16—91no—
192C2×M16—91no—
193C2×M16—182no—
194C2×C42—91no—
195#108—92no—
196#108—92no—
197#104—92no—
198#104—91no—
199#104—92no—
200#104—91no—
201#104—91no—
202#104—91no—
203#97—92no—
204C4×C8—92no—
205C23×C3—31no—
206C23×C3—61no—
207C3×Q8—31no—
208C3×Q8—32no—
209C3×Q8—61no—
210C3×Q8C2×C4×D311yescharacteristic
211C3×D4—31no—
212C3×D4—31no—
213C3×D4—32no—
214C3×D4—61no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—61no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—61no—
226C2×C3×C4—62no—
227C2×C3×C4—81no—
228C2×C3×C4C2×S411yescharacteristic
229#67—42no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—122no—
233D12—41no—
234C4×D3—42no—
235Dic6—41no—
236SL(2,𝔽3)—21no—
237SL(2,𝔽3)C22×Dic311yescharacteristic
238C3×C8—32no—
239C3×C8—32no—
240C3×C8—122no—
241C3⋊C8—32no—
242C3⋊C8—61no—
243C3⋊C8—61no—
244C3⋊C8—121no—
245C3⋊C8—121no—
246C3⋊C8—241no—
247C3⋊C8—241no—
248C3⋊C8C2×S412yes—
249C2×C32—41no—
250C2×C32—81no—
251C3×D3—82no—
252#41—31no—
253#41—31no—
254#41—32no—
255#41—32no—
256#41—61no—
257#41—61no—
258#41D3×Dic311yescharacteristic
259#41C2×D3211yescharacteristic
260C2×Q8—31no—
261C2×Q8D3×Dic311yescharacteristic
262C2×D4—31no—
263C2×D4—31no—
264C2×D4—61no—
265C22×C4—31no—
266C22×C4—61no—
267C22×C4—181no—
268Q16—32no—
269Q16—32no—
270SD16—32no—
271SD16—32no—
272SD16—32no—
273SD16—32no—
274D8—32no—
275D8—32no—
276M16—32no—
277M16—31no—
278M16—31no—
279M16—31no—
280M16—92no—
281M16—91no—
282M16—92no—
283M16—91no—
284M16—91no—
285M16—182no—
286M16—182no—
287M16—181no—
288C2×C8—32no—
289C2×C8—92no—
290C2×C8—182no—
291C2×C8—181no—
292C2×C8—182no—
293C2×C8—181no—
294C42—91no—
295C42—91no—
296C42—181no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—41no—
300C22×C3—62no—
301C22×C3—81no—
302C22×C3—121no—
303C22×C3—241no—
304C22×C3C4×S411yescharacteristic
305C2×D3—42no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—62no—
311C3×C4—81no—
312C3×C4—81no—
313C3×C4C4×S411yescharacteristic
314C3×C4C22×S411yescharacteristic
315Dic3—42no—
316Dic3—62no—
317Dic3—121no—
318C32—41noSylow
319C23—31no—
320C23—61no—
321Q8—31no—
322Q8—32no—
323Q8—61no—
324Q8C2×D3×Dic311yescharacteristic
325D4—31no—
326D4—31no—
327D4—32no—
328D4—61no—
329D4—62no—
330D4—62no—
331D4—61no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—62no—
338C2×C4—61no—
339C2×C4—62no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—362no—
343C2×C4D3×S411yescharacteristic
344C8—32no—
345C8—32no—
346C8—32no—
347C8—92no—
348C8—181no—
349C8—181no—
350C2×C3—21no—
351C2×C3—41no—
352C2×C3—61no—
353C2×C3—61no—
354C2×C3—81no—
355C2×C3—82no—
356C2×C3—81no—
357C2×C3—122no—
358C2×C3C2×C4×S411yescharacteristic
359D3—82no—
360C22—31no—
361C22—31no—
362C22—62no—
363C22—121no—
364C22—241no—
365C22Dic3×S411yescharacteristic
366C4—31no—
367C4—31no—
368C4—62no—
369C4—182no—
370C4—361no—
371C4Dic3×S411yescharacteristic
372C4C2×D3×S411yescharacteristic
373C3—41no—
374C3—81no—
375C3#8545311yescharacteristic, complemented
376C2—21no—
377C2—61no—
378C2—61no—
379C2—122no—
380C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC2×C3C2×C4×S4
prime radicalC3×#5923C2
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. #10593221C2semidirect
5. #10593219C2semidirect
9. C3×#986C4semidirect
11. C3×#984C4semidirect
12. C3×#983C4semidirect
14. C3⋊(#131⋊C3)C22semidirect
15. C3⋊GU(2,𝔽3)C22semidirect
16. C3⋊GU(2,𝔽3)C22semidirect
20. #5819D3semidirect
38. C3×GL(2,𝔽3)C2×C4semidirect
39. C3×#226C2×C4semidirect
49. #986Dic3semidirect
51. #984Dic3semidirect
52. #983Dic3semidirect
132. GL(2,𝔽3)C2×Dic3semidirect
133. #226C2×Dic3semidirect
375. C3#85453semidirect

Maximal subgroups

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

subgroupindex
C3×#59232
#5819⋊C32
C2×C3⋊GU(2,𝔽3)2
#105932212
#105932192
#854533
#840003
#638154