Finite Group Databasebeta

← #11922997

Subgroups — #11922997

Order 1152.

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

normal subgroups only

This group has 612 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
2C2×C3×#131⋊C3C211yesmaximal, characteristic, complemented
3#10593657C211yesmaximal, characteristic, complemented
4#10593639C212yesmaximal, complemented
5#10593638C212yesmaximal, complemented
6#85448—33nomaximal
7#83792—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267C411yescharacteristic, complemented
10#64266C2212yes—
11#64265C411yescharacteristic, complemented
12#64264C412yescomplemented
13#64110—41nomaximal
14C3×#131⋊C3C2211yescharacteristic, complemented
15C3×#131⋊C3C2211yescharacteristic, complemented
16C3⋊GU(2,𝔽3)C2212yescomplemented
17#5923—33no—
18C2×C3×#131D311yescharacteristic, complemented
19#5822—31no—
20C2×#901—31no—
21C2×#131⋊C3D313yescomplemented
22#5423—33no—
23#5405—36no—
24#5404—36no—
25#5236—31no—
26#5145—32no—
27#5145—32no—
28#5144—32no—
29#5144—32no—
30#5139—31no—
31#5124—31no—
32#5101—31no—
33#3828—41no—
34C3×#41⋊C3C2×C411yescharacteristic
35C3×#41⋊C3C2311yescharacteristic
36C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
37C3⋊GL(2,𝔽3)C2×C412yes—
38C3⋊#226C2×C412yes—
39C2×C32×C8—41no—
40#3748—41no—
41#3743—42no—
42C8×C3⋊D3—42no—
43#2955—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—31no—
52C3×#131—31no—
53C3×#131C2×D311yescharacteristic
54C3×#131C2×D311yescharacteristic
55C2×C3×M16—31no—
56C22×C3×C8—31no—
57#951—31no—
58#951—31no—
59#950—32no—
60#950—32no—
61#949—31no—
62#949—31no—
63#942—32no—
64#932—32no—
65#926—31no—
66C2×#208—31no—
67C2×#208—31no—
68#901—32no—
69#901—32no—
70#901—61no—
71#901—61no—
72#901—123no—
73C2×ΓL(1,𝔽25)—61no—
74C2×D3×C8—61no—
75#131⋊C3C2×D313yescomplemented
76#131⋊C3C2×D313yescomplemented
77GU(2,𝔽3)—36no—
78#837—32no—
79#837—32no—
80#825—32no—
81#825—32no—
82#822—32no—
83#822—32no—
84#815—32no—
85C8×Dic3—32no—
86C2×C32×C4—41no—
87#638—42no—
88C4⋊(C3⋊D3)—41no—
89C4×C3⋊D3—42no—
90#634—41no—
91C3×SL(2,𝔽3)C22×C411yescharacteristic
92C32×C8—41no—
93C32×C8—41no—
94C3⋊(C3⋊C8)—42no—
95#575—91no—
96C2×#131—91no—
97C2×#131C3⋊D311yescharacteristic, complemented
98#444—92no—
99#444—92no—
100#443—92no—
101#443—92no—
102#430—91no—
103#421—91no—
104#421—91no—
105#405—91no—
106C3×#41—31no—
107C3×#41—31no—
108C3×#41C4×D311yescharacteristic
109C3×#41C22×D311yescharacteristic
110C2×C3×Q8C4×D311yescharacteristic
111C2×C3×D4—31no—
112C22×C3×C4—31no—
113C2×#67—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—31no—
129C3×M16—61no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—43no—
136C2×C3×C8S411yescharacteristic
137#217—61no—
138#216—32no—
139#216—32no—
140#215—32no—
141#215—32no—
142#214—32no—
143#214—32no—
144#213—32no—
145#213—32no—
146C4⋊Dic3—61no—
147C4×Dic3—32no—
148#208—32no—
149#208—31no—
150#208—31no—
151#208—32no—
152#208—61no—
153#208—123no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—61no—
156ΓL(1,𝔽25)—62no—
157ΓL(1,𝔽25)—62no—
158ΓL(1,𝔽25)—126no—
159D3×C8—62no—
160D3×C8—62no—
161D3×C8—126no—
162C22×C32—41no—
163C2×C3⋊D3—42no—
164C32×C4—41no—
165C32×C4—41no—
166C3⋊Dic3—42no—
167C2×#41—91no—
168C2×#41C2×C3⋊D311yescharacteristic
169#137—91no—
170#137—91no—
171#137—92no—
172AGL(1,ℤ/8)—92no—
173AGL(1,ℤ/8)—91no—
174AGL(1,ℤ/8)—91no—
175#135—92no—
176#135—92no—
177#131—31no—
178#131—31no—
179#131—92no—
180#131—92no—
181#131—181no—
182#131—181no—
183#131C2×C3⋊D311yescharacteristic
184#131C2×C3⋊D311yescharacteristic
185C2×M16—31no—
186C2×M16—91no—
187C2×M16—91no—
188C2×M16—181no—
189C22×C8—31no—
190C22×C8—181no—
191#117—91no—
192#108—92no—
193#108—92no—
194#104—92no—
195#104—92no—
196#104—92no—
197#104—92no—
198#97—92no—
199C4×C8—92no—
200C23×C3—31no—
201C22×D3—61no—
202C3×Q8—31no—
203C3×Q8C2×C4×D311yescharacteristic
204C3×D4—31no—
205C3×D4—31no—
206C3×D4—61no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—43no—
213C2×C3×C4C2×S411yescharacteristic
214#67—62no—
215#67—62no—
216#67—126no—
217C2×Dic3—61no—
218C2×Dic3—62no—
219D12—32no—
220D12—61no—
221D12—123no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—126no—
225Dic6—32no—
226Dic6—61no—
227Dic6—123no—
228SL(2,𝔽3)C2×C4×D313yes—
229C3×C8—31no—
230C3×C8—31no—
231C3×C8—43no—
232C3×C8—43no—
233C3×C8C2×S411yescharacteristic
234C3×C8C2×S411yescharacteristic
235C3⋊C8—32no—
236C3⋊C8—32no—
237C3⋊C8—62no—
238C3⋊C8—126no—
239C2×C32—41no—
240C2×C32—81no—
241C3⋊D3—82no—
242#41—31no—
243#41—31no—
244#41—92no—
245#41—92no—
246#41—181no—
247#41—181no—
248#41C4×C3⋊D311yescharacteristic
249#41C22×C3⋊D311yescharacteristic
250C2×Q8—91no—
251C2×Q8C4×C3⋊D311yescharacteristic
252C2×D4—31no—
253C2×D4—91no—
254C2×D4—181no—
255C22×C4—31no—
256C22×C4—181no—
257Q16—92no—
258Q16—92no—
259SD16—92no—
260SD16—92no—
261SD16—92no—
262SD16—92no—
263D8—92no—
264D8—92no—
265M16—31no—
266M16—31no—
267M16—61no—
268M16—92no—
269M16—91no—
270M16—91no—
271M16—92no—
272M16—182no—
273M16—181no—
274M16—182no—
275C2×C8—31no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—92no—
281C2×C8—182no—
282C2×C8—181no—
283C2×C8—182no—
284C2×C8C3⋊S411yescharacteristic
285C4⋊C4—181no—
286C22⋊C4—181no—
287C42—92no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—43no—
291C22×C3—121no—
292C22×C3C4×S411yescharacteristic
293C2×D3—62no—
294C2×D3—126no—
295C2×D3—122no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—43no—
299C3×C4—43no—
300C3×C4C4×S411yescharacteristic
301C3×C4C22×S411yescharacteristic
302Dic3—62no—
303Dic3—126no—
304Dic3—122no—
305C32—41noSylow
306C23—31no—
307C23—181no—
308Q8—31no—
309Q8—92no—
310Q8—181no—
311Q8C2×C4×C3⋊D311yescharacteristic
312D4—31no—
313D4—31no—
314D4—61no—
315D4—92no—
316D4—182no—
317D4—182no—
318D4—181no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—182no—
325C2×C4—181no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4C2×C3⋊S411yescharacteristic
329C8—31no—
330C8—31no—
331C8—92no—
332C8—92no—
333C8—182no—
334C8C2×C3⋊S411yescharacteristic
335C8C2×C3⋊S411yescharacteristic
336C2×C3—21no—
337C2×C3—43no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3—83no—
341C2×C3C2×C4×S411yescharacteristic
342D3—122no—
343D3—246no—
344C22—31no—
345C22—31no—
346C22—121no—
347C22—182no—
348C22—362no—
349C22C4×C3⋊S411yescharacteristic
350C4—31no—
351C4—31no—
352C4—182no—
353C4—362no—
354C4C4×C3⋊S411yescharacteristic
355C4C22×C3⋊S411yescharacteristic
356C3—43no—
357C3#8544811yescharacteristic, complemented
358C2—21no—
359C2—61no—
360C2—61no—
361C2—362no—
362C2C2×C4×C3⋊S411yescharacteristic
subgroupG/H
centerC8C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC4C22×C3⋊S4
FittingC2×C3×#131D3
socleC2×C3C2×C4×S4
prime radical#10596586C2
cosocleC3×#41⋊C3C23
hypercenterC2×C8C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#131⋊C3C2semidirect
3. #10593657C2semidirect
4. #10593639C2semidirect
5. #10593638C2semidirect
9. #64267C4semidirect
11. #64265C4semidirect
12. #64264C4semidirect
14. C3×#131⋊C3C22semidirect
15. C3×#131⋊C3C22semidirect
16. C3⋊GU(2,𝔽3)C22semidirect
18. C2×C3×#131D3semidirect
21. C2×#131⋊C3D3semidirect
75. #131⋊C3C2×D3semidirect
76. #131⋊C3C2×D3semidirect
97. C2×#131C3⋊D3semidirect
357. C3#85448semidirect

Maximal subgroups

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

subgroupindex
#105965862
C2×C3×#131⋊C32
#105936572
#105936392
#105936382
#854483
#837923
#641104