Finite Group Databasebeta

← #11922918

Subgroups — #11922918

Order 1152.

716 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×#64267C211yesmaximal, characteristic, complemented
2C2×#64266C211yesmaximal, characteristic, complemented
3C3×#520⋊C3C211yesmaximal, characteristic, complemented
4#10593629C212yesmaximal, complemented
5C4⋊(C3⋊GL(2,𝔽3))C212yesmaximal, complemented
6#85368—33nomaximal
7#82565—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267—22no—
10#64266—22no—
11C2×C3⋊GL(2,𝔽3)C2212yes—
12C2×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3⋊#226C2211yescharacteristic, complemented
14#64104—41nomaximal
15C3×C4×SL(2,𝔽3)C2212yescomplemented
16C2×#986—33no—
17C2×#985—33no—
18C3×#520D311yescharacteristic, complemented
19C2×#950—31no—
20C2×#949—31no—
21#5525—31no—
22#520⋊C3D313yescomplemented
23#5395—36no—
24C4⋊GL(2,𝔽3)—36no—
25#5028—32no—
26#5027—32no—
27#5016—32no—
28#5015—32no—
29#4999—31no—
30#4966—31no—
31C2×#809—31no—
32C2×C4⋊(C3⋊D3)—42no—
33C2×C4⋊(C3⋊D3)—41no—
34C2×C4×C3⋊D3—41no—
35C3×#41⋊C3D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C3⋊GL(2,𝔽3)—22no—
38C3⋊GL(2,𝔽3)—21no—
39C3⋊#226—21no—
40C32×C4⋊C4—41no—
41#3752—42no—
42#3045—91noSylow
43C2×C3×#41C2×D311yescharacteristic
44C2×#235—31no—
45C22×D12—31no—
46C2×#41⋊C3C2×D313yes—
47#986—66no—
48#985—66no—
49C2×GL(2,𝔽3)—36no—
50C2×GL(2,𝔽3)—33no—
51C2×#226—33no—
52C3×C4×Q8C2×D312yes—
53C3×C4×D4—32no—
54C3×#117—31no—
55C2×C3×C4⋊C4—31no—
56#950—62no—
57#950—62no—
58#949—62no—
59#949—62no—
60C2×#216—31no—
61C2×#215—32no—
62C2×#215—31no—
63C2×#214—31no—
64C2×#213—31no—
65C2×#213—32no—
66C2×#208—31no—
67C22×C3⋊C8—31no—
68#895—61no—
69#895—61no—
70#895—123no—
71#882—62no—
72#875—32no—
73#874—32no—
74C4×SL(2,𝔽3)C2×D316yescomplemented
75#810—32no—
76#809—32no—
77#809—32no—
78#809—32no—
79C4⋊(C3⋊C8)—32no—
80C4⋊(C3⋊C8)—32no—
81C22×C3⋊D3—41no—
82C22×C3⋊D3—42no—
83C2×C32×C4—41no—
84C2×C32×C4—42no—
85C2×C3⋊Dic3—41no—
86C4⋊(C3⋊D3)—82no—
87C4⋊(C3⋊D3)—84no—
88C4×C3⋊D3—82no—
89C3×SL(2,𝔽3)C2×D411yescharacteristic
90C2×AGL(1,ℤ/8)—91no—
91C2×#135—91no—
92#535—91no—
93#520C3⋊D311yescharacteristic, complemented
94#464—92no—
95#463—92no—
96#460—92no—
97#459—92no—
98#424—91no—
99#417—91no—
100C2×#102—91no—
101C23×D3—61no—
102C3×#41—32no—
103C3×#41D1212yescomplemented
104C2×C3×Q8C22×D311yescharacteristic
105C2×C3×D4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—61no—
108C2×#67—61no—
109#235—62no—
110#235—62no—
111C2×D12—31no—
112C2×D12—32no—
113C2×D12—62no—
114C2×D12—62no—
115C2×D12—61no—
116C2×D12—62no—
117C2×D12—62no—
118C2×D12—126no—
119C2×D12—123no—
120C2×C4×D3—61no—
121C2×C4×D3—123no—
122C2×Dic6—31no—
123#41⋊C3D1216yescomplemented
124C2×SL(2,𝔽3)C22×D313yes—
125GL(2,𝔽3)—63no—
126GL(2,𝔽3)—66no—
127#226—63no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—43no—
131C3×C4⋊C4S411yescharacteristic
132C3×C22⋊C4—32no—
133C3×C42—32no—
134#216—61no—
135#216—61no—
136#215—62no—
137#215—61no—
138#215—62no—
139#215—61no—
140#214—62no—
141#213—64no—
142#213—62no—
143(C2×D3)⋊C4—62no—
144(C2×D3)⋊C4—62no—
145(C2×D3)⋊C4—126no—
146#208—62no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—62no—
151C22×C32—41no—
152C2×C3⋊D3—42no—
153C2×C3⋊D3—81no—
154C2×C3⋊D3—81no—
155C2×C3⋊D3—82no—
156C2×C3⋊D3—82no—
157C2×C3⋊D3—82no—
158C32×C4—42no—
159C32×C4—82no—
160C3⋊Dic3—81no—
161C2×#41—91no—
162C2×#41C2×C3⋊D311yescharacteristic
163C22×D4—91no—
164AGL(1,ℤ/8)—182no—
165AGL(1,ℤ/8)—182no—
166#135—182no—
167#135—182no—
168C2×Q16—91no—
169C2×SD16—91no—
170C2×SD16—91no—
171C2×SD16—92no—
172C2×D8—91no—
173C2×D8—92no—
174C2×M16—91no—
175C22×C8—91no—
176#127—92no—
177#124—92no—
178#121—181no—
179#121—181no—
180#120—182no—
181C4×Q8C2×C3⋊D312yes—
182C4×D4—32no—
183#117—31no—
184C2×C4⋊C4—31no—
185C4⋊C8—92no—
186C4⋊C8—92no—
187#103—92no—
188#102—92no—
189#102—92no—
190#102—92no—
191C23×C3—31no—
192C22×D3—62no—
193C22×D3—61no—
194C22×D3—123no—
195C22×D3—126no—
196C22×D3—122no—
197C22×D3—241no—
198C22×D3—241no—
199C3×Q8—31no—
200C3×Q8C2×D1211yescharacteristic
201C3×D4—32no—
202C3×D4—61no—
203C2×C3×C4—31no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—32no—
207C2×C3×C4—43no—
208C2×C3×C4—46no—
209C2×C3×C4—241no—
210C2×C3×C4C2×S412yes—
211C2×C3×C4C2×S411yescharacteristic
212#67—122no—
213C2×Dic3—61no—
214C2×Dic3—123no—
215D12—62no—
216D12—61no—
217D12—62no—
218D12—61no—
219D12—122no—
220D12—246no—
221D12—242no—
222D12—242no—
223D12—2412no—
224C4×D3—122no—
225C4×D3—246no—
226Dic6—61no—
227Dic6—61no—
228SL(2,𝔽3)C2×D1213yes—
229C3⋊C8—61no—
230C3⋊C8—61no—
231C3⋊C8—62no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C3⋊D3—82no—
236C3⋊D3—162no—
237C24—181no—
238#41—32no—
239#41—182no—
240#41—182no—
241#41C4⋊(C3⋊D3)12yescomplemented
242C2×Q8—91no—
243C2×Q8C22×C3⋊D311yescharacteristic
244C2×D4—31no—
245C2×D4—92no—
246C2×D4—91no—
247C2×D4—182no—
248C2×D4—182no—
249C2×D4—182no—
250C2×D4—181no—
251C2×D4—181no—
252C2×D4—182no—
253C22×C4—31no—
254C22×C4—61no—
255C22×C4—181no—
256Q16—181no—
257Q16—181no—
258SD16—182no—
259SD16—181no—
260SD16—182no—
261SD16—181no—
262SD16—182no—
263D8—182no—
264D8—184no—
265M16—182no—
266C2×C8—91no—
267C2×C8—91no—
268C2×C8—92no—
269C2×C8—182no—
270C4⋊C4—32no—
271C4⋊C4—31no—
272C4⋊C4C3⋊S411yescharacteristic
273C22⋊C4—32no—
274C22⋊C4—182no—
275C22⋊C4—182no—
276C42—32no—
277C22×C3—32no—
278C22×C3—43no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3C22×S411yescharacteristic
282C2×D3—126no—
283C2×D3—121no—
284C2×D3—122no—
285C2×D3—121no—
286C2×D3—122no—
287C2×D3—121no—
288C2×D3—122no—
289C2×D3—243no—
290C2×D3—243no—
291C2×D3—246no—
292C2×D3—246no—
293C2×D3—246no—
294C2×D3—244no—
295C3×C4—22no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—46no—
299C3×C4—62no—
300C3×C4—86no—
301C3×C4C4⋊S412yes—
302Dic3—121no—
303Dic3—243no—
304C32—41noSylow
305C23—31no—
306C23—182no—
307C23—181no—
308C23—362no—
309C23—721no—
310C23—721no—
311Q8—31no—
312Q8—181no—
313Q8—181no—
314Q8C2×C4⋊(C3⋊D3)11yescharacteristic
315D4—32no—
316D4—61no—
317D4—182no—
318D4—181no—
319D4—182no—
320D4—181no—
321D4—362no—
322D4—362no—
323D4—722no—
324D4—722no—
325C2×C4—31no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—181no—
330C2×C4—241no—
331C2×C4—362no—
332C2×C4C2×C3⋊S412yes—
333C2×C4C2×C3⋊S411yescharacteristic
334C8—181no—
335C8—181no—
336C8—182no—
337C2×C3—43no—
338C2×C3—43no—
339C2×C3—43no—
340C2×C3—62no—
341C2×C3C2×C4⋊S411yescharacteristic
342C2×C3#592511yescharacteristic
343C2×C3#592611yescharacteristic
344D3—246no—
345D3—242no—
346D3—241no—
347D3—486no—
348C22—32no—
349C22—61no—
350C22—61no—
351C22—361no—
352C22—362no—
353C22—361no—
354C22—362no—
355C22—361no—
356C22—362no—
357C22—724no—
358C22C22×C3⋊S411yescharacteristic
359C4—22no—
360C4—31no—
361C4—31no—
362C4—62no—
363C4—361no—
364C4C4⋊(C3⋊S4)12yes—
365C3—43no—
366C3#8536811yescharacteristic, complemented
367C2—62no—
368C2—722no—
369C2—721no—
370C2C2×C4⋊(C3⋊S4)11yescharacteristic
371C2#1059658811yescharacteristic
372C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×#520D3
socleC22×C3C22×S4
hypercenterC4⋊C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64267C2semidirect
2. C2×#64266C2semidirect
3. C3×#520⋊C3C2semidirect
4. #10593629C2semidirect
5. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
12. C2×C3⋊GL(2,𝔽3)C22semidirect
13. C2×C3⋊#226C22semidirect
15. C3×C4×SL(2,𝔽3)C22semidirect
18. C3×#520D3semidirect
22. #520⋊C3D3semidirect
35. C3×#41⋊C3D4semidirect
74. C4×SL(2,𝔽3)C2×D3semidirect
93. #520C3⋊D3semidirect
103. C3×#41D12semidirect
123. #41⋊C3D12semidirect
241. #41C4⋊(C3⋊D3)semidirect
366. C3#85368semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C2×#642662
C3×#520⋊C32
#105936292
C4⋊(C3⋊GL(2,𝔽3))2
#853683
#825653
#641044