Finite Group Databasebeta

← #11923002

Subgroups — #11923002

Order 1152.

616 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 85 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
2C3×#568⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593639C212yesmaximal, complemented
5#10593638C212yesmaximal, complemented
6#85453—33nomaximal
7#83811—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267C411yescharacteristic, complemented
10#64266C2212yes—
11#64265C411yescharacteristic, complemented
12#64264C412yescomplemented
13#64116—41nomaximal
14C3×#131⋊C3C2212yescomplemented
15C3⋊GU(2,𝔽3)C2211yescharacteristic, complemented
16C3⋊GU(2,𝔽3)C2211yescharacteristic, complemented
17#5923—33no—
18C3×#568D311yescharacteristic, complemented
19#5822—31no—
20#5745—31no—
21#568⋊C3D313yescomplemented
22C2×GU(2,𝔽3)—33no—
23#5405—36no—
24#5404—36no—
25C2×#837—31no—
26#5145—32no—
27#5145—32no—
28#5144—32no—
29#5144—32no—
30C2×#825—31no—
31#5124—31no—
32Dic3×M16—31no—
33#3828—41no—
34C3×#41⋊C3C2×C411yescharacteristic
35C3×#41⋊C3C2311yescharacteristic
36C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
37C3⋊GL(2,𝔽3)C2×C412yescomplemented
38C3⋊#226C2×C412yescomplemented
39C32×M16—41no—
40C2×C3⋊(C3⋊C8)—41no—
41#3743—42no—
42C8×C3⋊D3—42no—
43#2956—91noSylow
44C2×C3×#41C2×D311yescharacteristic
45C2×#235—31no—
46C2×#41⋊C3C2×D313yes—
47#986—33no—
48#985—36no—
49#984—33no—
50#983—36no—
51C3×#131—32no—
52C3×#131C2×D312yes—
53C2×C3×M16—31no—
54C2×C3×M16—31no—
55#951—31no—
56#951—31no—
57#950—32no—
58#950—32no—
59#949—31no—
60#949—31no—
61#942—32no—
62#932—32no—
63C2×C4×Dic3—31no—
64C2×#208—31no—
65C2×#208—31no—
66#907—61no—
67#907—61no—
68#907—123no—
69D3×M16—62no—
70#901—32no—
71#901—32no—
72#131⋊C3C2×D316yescomplemented
73GU(2,𝔽3)—33no—
74GU(2,𝔽3)—33no—
75#837—31no—
76#837—31no—
77#837—31no—
78#837—31no—
79#825—32no—
80#825—32no—
81#822—32no—
82#822—32no—
83#815—32no—
84C8×Dic3—32no—
85C2×C32×C4—41no—
86#638—42no—
87C4⋊(C3⋊D3)—41no—
88C4×C3⋊D3—42no—
89#634—41no—
90C3×SL(2,𝔽3)C22×C411yescharacteristic
91C32×C8—42no—
92C3⋊(C3⋊C8)—41no—
93C3⋊(C3⋊C8)—41no—
94#575—91no—
95#568—91no—
96#568C3⋊D311yescharacteristic, complemented
97#444—92no—
98#444—92no—
99#443—92no—
100#443—92no—
101#430—91no—
102C2×#104—91no—
103C2×#104—91no—
104C4×M16—91no—
105C3×#41—31no—
106C3×#41—31no—
107C3×#41C4×D311yescharacteristic
108C3×#41C22×D311yescharacteristic
109C2×C3×Q8C4×D311yescharacteristic
110C2×C3×D4—31no—
111C22×C3×C4—31no—
112C2×#67—61no—
113C22×Dic3—61no—
114#235—32no—
115#235—32no—
116#235—61no—
117#235—61no—
118#235—123no—
119C2×D12—31no—
120C2×C4×D3—61no—
121C2×Dic6—31no—
122#41⋊C3C4×D313yes—
123#41⋊C3C22×D313yes—
124C2×SL(2,𝔽3)C4×D313yes—
125GL(2,𝔽3)—36no—
126#226—36no—
127C3×M16—31no—
128C3×M16—32no—
129C3×M16—32no—
130C3×M16—43no—
131C3×M16S411yescharacteristic
132C2×C3×C8—32no—
133C2×C3×C8—61no—
134#216—32no—
135#216—32no—
136#215—32no—
137#215—32no—
138#214—32no—
139#214—32no—
140#213—32no—
141#213—32no—
142C4×Dic3—31no—
143C4×Dic3—31no—
144C4×Dic3—61no—
145#208—32no—
146#208—31no—
147#208—31no—
148#208—31no—
149#208—31no—
150#208—61no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—123no—
154ΓL(1,𝔽25)—62no—
155ΓL(1,𝔽25)—62no—
156ΓL(1,𝔽25)—126no—
157D3×C8—62no—
158D3×C8—62no—
159D3×C8—126no—
160C22×C32—41no—
161C2×C3⋊D3—42no—
162C32×C4—41no—
163C32×C4—41no—
164C3⋊Dic3—42no—
165C2×#41—91no—
166C2×#41C2×C3⋊D311yescharacteristic
167#137—91no—
168#137—91no—
169#137—92no—
170AGL(1,ℤ/8)—92no—
171AGL(1,ℤ/8)—91no—
172AGL(1,ℤ/8)—91no—
173#135—92no—
174#135—92no—
175#131—32no—
176#131—92no—
177#131—92no—
178#131—181no—
179#131—181no—
180#131C2×C3⋊D312yes—
181C2×M16—31no—
182C2×M16—31no—
183C2×M16—91no—
184C2×M16—91no—
185C2×M16—182no—
186C2×C42—91no—
187#108—92no—
188#108—92no—
189#104—92no—
190#104—92no—
191#104—91no—
192#104—91no—
193#104—91no—
194#104—91no—
195#97—92no—
196C4×C8—92no—
197C23×C3—31no—
198C22×D3—61no—
199C3×Q8—31no—
200C3×Q8C2×C4×D311yescharacteristic
201C3×D4—31no—
202C3×D4—31no—
203C3×D4—61no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—43no—
210C2×C3×C4C2×S411yescharacteristic
211#67—62no—
212#67—62no—
213#67—126no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—122no—
218D12—32no—
219D12—61no—
220D12—123no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—126no—
224Dic6—32no—
225Dic6—61no—
226Dic6—123no—
227SL(2,𝔽3)C2×C4×D313yes—
228C3×C8—32no—
229C3×C8—46no—
230C3×C8C2×S412yes—
231C3⋊C8—32no—
232C3⋊C8—32no—
233C3⋊C8—61no—
234C3⋊C8—61no—
235C3⋊C8—123no—
236C3⋊C8—123no—
237C2×C32—41no—
238C2×C32—81no—
239C3⋊D3—82no—
240#41—31no—
241#41—31no—
242#41—92no—
243#41—92no—
244#41—181no—
245#41—181no—
246#41C4×C3⋊D311yescharacteristic
247#41C22×C3⋊D311yescharacteristic
248C2×Q8—91no—
249C2×Q8C4×C3⋊D311yescharacteristic
250C2×D4—31no—
251C2×D4—91no—
252C2×D4—181no—
253C22×C4—31no—
254C22×C4—181no—
255C22×C4—181no—
256Q16—92no—
257Q16—92no—
258SD16—92no—
259SD16—92no—
260SD16—92no—
261SD16—92no—
262D8—92no—
263D8—92no—
264M16—31no—
265M16—32no—
266M16—32no—
267M16—92no—
268M16—91no—
269M16—91no—
270M16—91no—
271M16—91no—
272M16—182no—
273M16—181no—
274M16—182no—
275M16C3⋊S411yescharacteristic
276C2×C8—32no—
277C2×C8—61no—
278C2×C8—92no—
279C2×C8—182no—
280C2×C8—181no—
281C2×C8—182no—
282C42—91no—
283C42—91no—
284C42—181no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—43no—
288C22×C3—121no—
289C22×C3C4×S411yescharacteristic
290C2×D3—62no—
291C2×D3—126no—
292C2×D3—241no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—43no—
296C3×C4—43no—
297C3×C4C4×S411yescharacteristic
298C3×C4C22×S411yescharacteristic
299Dic3—62no—
300Dic3—62no—
301Dic3—126no—
302Dic3—121no—
303C32—41noSylow
304C23—31no—
305C23—181no—
306Q8—31no—
307Q8—92no—
308Q8—181no—
309Q8C2×C4×C3⋊D311yescharacteristic
310D4—31no—
311D4—31no—
312D4—61no—
313D4—92no—
314D4—182no—
315D4—182no—
316D4—181no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—182no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—182no—
326C2×C4—181no—
327C2×C4—362no—
328C2×C4C2×C3⋊S411yescharacteristic
329C8—32no—
330C8—92no—
331C8—92no—
332C8—181no—
333C8—181no—
334C8C2×C3⋊S412yes—
335C2×C3—21no—
336C2×C3—43no—
337C2×C3—61no—
338C2×C3—61no—
339C2×C3—83no—
340C2×C3C2×C4×S411yescharacteristic
341D3—122no—
342D3—246no—
343C22—31no—
344C22—31no—
345C22—121no—
346C22—182no—
347C22—721no—
348C22C4×C3⋊S411yescharacteristic
349C4—31no—
350C4—31no—
351C4—182no—
352C4—182no—
353C4—361no—
354C4C4×C3⋊S411yescharacteristic
355C4C22×C3⋊S411yescharacteristic
356C3—43no—
357C3#8545311yescharacteristic, complemented
358C2—21no—
359C2—61no—
360C2—61no—
361C2—362no—
362C2C2×C4×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC4C22×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC3×#568D3
socleC2×C3C2×C4×S4
prime radical#10596586C2
cosocleC3×#41⋊C3C23
hypercenterM16C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×#568⋊C3C2semidirect
3. C2×C3⋊GU(2,𝔽3)C2semidirect
4. #10593639C2semidirect
5. #10593638C2semidirect
9. #64267C4semidirect
11. #64265C4semidirect
12. #64264C4semidirect
14. C3×#131⋊C3C22semidirect
15. C3⋊GU(2,𝔽3)C22semidirect
16. C3⋊GU(2,𝔽3)C22semidirect
18. C3×#568D3semidirect
21. #568⋊C3D3semidirect
37. C3⋊GL(2,𝔽3)C2×C4semidirect
38. C3⋊#226C2×C4semidirect
72. #131⋊C3C2×D3semidirect
96. #568C3⋊D3semidirect
357. C3#85453semidirect

Maximal subgroups

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

subgroupindex
#105965862
C3×#568⋊C32
C2×C3⋊GU(2,𝔽3)2
#105936392
#105936382
#854533
#838113
#641164