Finite Group Databasebeta

← #11922270

Subgroups — #11922270

Order 1152.

557 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 55 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
2C3×#5923C211yesmaximal, characteristic, complemented
3#5819⋊C3C211yesmaximal, characteristic, complemented
4#10593225C212yesmaximal, complemented
5#10593224C212yesmaximal, complemented
6#85454—31nomaximal
7#84002—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267—21no—
10#64267C2212yes—
11C2×C3⋊GL(2,𝔽3)—21no—
12C3×#986C2211yescharacteristic, complemented
13C3×#985—21no—
14C3×#984C2211yescharacteristic, complemented
15C3×#983—21no—
16#63825—41nomaximal
17C3⋊(#131⋊C3)C2212yescomplemented
18#5923D311yescharacteristic, complemented
19C2×#986—31no—
20C2×#986—61no—
21C3×#575—31no—
22C2×#949—31no—
23C2×#949—31no—
24#5819D311yescharacteristic, complemented
25#568⋊C3—31no—
26#5407—32no—
27#5406—32no—
28#5237—31no—
29#5203—32no—
30#5203—32no—
31#5199—32no—
32#5199—32no—
33C2×#823—31no—
34#5124—31no—
35C2×C4⋊(C3⋊D3)—41no—
36C3×#235—41no—
37C3×#41⋊C3D411yescharacteristic
38C3×#41⋊C3C2311yescharacteristic
39C2×C3×SL(2,𝔽3)D411yescharacteristic
40C3⋊GL(2,𝔽3)—22no—
41C3×GL(2,𝔽3)—21no—
42C3×#226—21no—
43C3×#208—41no—
44#3717—42no—
45#3714—42no—
46#3068—91noSylow
47C2×C3×#41—31no—
48C2×C3×#41C2×D311yescharacteristic
49C22×D12—31no—
50C2×#41⋊C3—21no—
51C2×#41⋊C3C2×D311yescharacteristic
52#986—32no—
53#986—62no—
54#986—62no—
55#986—61no—
56#986C2×D311yescharacteristic, complemented
57#985—21no—
58#984C2×D311yescharacteristic, complemented
59#983—21no—
60C2×GL(2,𝔽3)—62no—
61C2×GL(2,𝔽3)—61no—
62C3×#137—31no—
63C3×#137—31no—
64C3×#137—61no—
65C3×AGL(1,ℤ/8)—31no—
66C3×AGL(1,ℤ/8)—31no—
67C3×AGL(1,ℤ/8)—61no—
68C3×#135—61no—
69C3×#135—61no—
70C2×C3×M16—31no—
71#949—32no—
72#949—32no—
73#949—61no—
74#949—62no—
75#949—61no—
76#949—61no—
77#949—61no—
78#948—32no—
79#948C2×D312yescomplemented
80C2×#215—32no—
81C2×#215—61no—
82C2×#213—32no—
83C2×#213—61no—
84C2×#208—31no—
85C2×#208—31no—
86C2×#208—31no—
87#908—121no—
88#131⋊C3—32no—
89#836—32no—
90#833—32no—
91#823—32no—
92#823—32no—
93#822—32no—
94#822—32no—
95C22×C3⋊D3—81no—
96C2×C32×C4—41no—
97C4⋊(C3⋊D3)—42no—
98C4⋊(C3⋊D3)—81no—
99C3×#67—81no—
100C3×D12—41no—
101C3×C4×D3—81no—
102C3×Dic6—41no—
103C3×SL(2,𝔽3)C2×D411yescharacteristic
104C3×C3⋊C8—42no—
105#575—31no—
106C2×AGL(1,ℤ/8)—91no—
107C2×AGL(1,ℤ/8)—91no—
108#568—31no—
109#472—92no—
110#472—92no—
111#471—92no—
112#471—92no—
113#430—91no—
114#413—91no—
115C2×#100—91no—
116C23×D3—61no—
117C3×#41—31no—
118C3×#41—31no—
119C3×#41—61no—
120C3×#41—61no—
121C3×#41—61no—
122C3×#41—61no—
123C3×#41D1211yescharacteristic
124C3×#41C22×D311yescharacteristic
125C2×C3×Q8—31no—
126C2×C3×Q8D1211yescharacteristic
127C2×C3×D4—31no—
128C2×C3×D4—31no—
129C2×C3×D4—61no—
130C22×C3×C4—31no—
131C22×C3×C4—61no—
132#235—41no—
133C2×D12—32no—
134C2×D12—61no—
135C2×D12—61no—
136C2×D12—61no—
137C2×D12—61no—
138C2×D12—61no—
139C2×D12—121no—
140C2×D12—241no—
141#41⋊C3—21no—
142#41⋊C3—21no—
143#41⋊C3#6711yescharacteristic
144#41⋊C3C22×D311yescharacteristic
145C2×SL(2,𝔽3)—21no—
146C2×SL(2,𝔽3)#6711yescharacteristic
147GL(2,𝔽3)—21no—
148GL(2,𝔽3)—64no—
149GL(2,𝔽3)—62no—
150#226—21no—
151C3×Q16—61no—
152C3×Q16—61no—
153C3×SD16—61no—
154C3×SD16—61no—
155C3×SD16—61no—
156C3×SD16—61no—
157C3×D8—61no—
158C3×D8—61no—
159C3×M16—31no—
160C3×M16—31no—
161C3×M16—61no—
162C3×M16—121no—
163C2×C3×C8—61no—
164#215—62no—
165#215—62no—
166#215—62no—
167#215—62no—
168#213—62no—
169#213—62no—
170#213—62no—
171#213—62no—
172#208—31no—
173#208—32no—
174#208—32no—
175#208—32no—
176#208—61no—
177#208S411yescharacteristic
178C2×C3⋊C8—32no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—61no—
181D24—122no—
182#204—122no—
183C22×C32—41no—
184C2×C3⋊D3—82no—
185C2×C3⋊D3—162no—
186C2×C3×D3—81no—
187C32×C4—41no—
188C32×C4—41no—
189C3×Dic3—81no—
190C2×#41—31no—
191C2×#41D3211yescharacteristic
192C22×D4—91no—
193#137—31no—
194#137—31no—
195#137—61no—
196AGL(1,ℤ/8)—31no—
197AGL(1,ℤ/8)—31no—
198AGL(1,ℤ/8)—61no—
199AGL(1,ℤ/8)—92no—
200AGL(1,ℤ/8)—92no—
201AGL(1,ℤ/8)—181no—
202AGL(1,ℤ/8)—182no—
203AGL(1,ℤ/8)—181no—
204AGL(1,ℤ/8)—181no—
205AGL(1,ℤ/8)—181no—
206#135—61no—
207#135—61no—
208C2×SD16—92no—
209C2×SD16—181no—
210C2×D8—92no—
211C2×D8—181no—
212#131—32no—
213#131—92no—
214C2×M16—31no—
215C2×M16—91no—
216C2×M16—91no—
217C2×M16—91no—
218#108—92no—
219#108—92no—
220#101—92no—
221#100—92no—
222#100—92no—
223#100—92no—
224C23×C3—31no—
225C23×C3—61no—
226C22×D3—62no—
227C22×D3—121no—
228C22×D3—121no—
229C22×D3—241no—
230C22×D3—242no—
231C22×D3—242no—
232C3×Q8—31no—
233C3×Q8—61no—
234C3×Q8—61no—
235C3×Q8C2×D1211yescharacteristic
236C3×D4—31no—
237C3×D4—31no—
238C3×D4—61no—
239C3×D4—61no—
240C3×D4—61no—
241C3×D4—121no—
242C3×D4—121no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—41no—
249C2×C3×C4—61no—
250C2×C3×C4—81no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4C2×S411yescharacteristic
254#67—81no—
255D12—41no—
256D12—62no—
257D12—62no—
258D12—122no—
259D12—121no—
260D12—121no—
261D12—242no—
262D12—241no—
263D12—242no—
264C4×D3—81no—
265Dic6—41no—
266SL(2,𝔽3)—21no—
267SL(2,𝔽3)C2×#6711yescharacteristic
268C3×C8—61no—
269C3×C8—61no—
270C3×C8—122no—
271C3⋊C8—32no—
272C3⋊C8—62no—
273C3⋊C8C2×S412yes—
274C2×C32—41no—
275C2×C32—81no—
276C3⋊D3—162no—
277C3×D3—161no—
278C24—181no—
279#41—31no—
280#41—31no—
281#41—61no—
282#41—61no—
283#41—61no—
284#41—61no—
285#41#62611yescharacteristic
286#41C2×D3211yescharacteristic
287C2×Q8—31no—
288C2×Q8#62611yescharacteristic
289C2×D4—31no—
290C2×D4—31no—
291C2×D4—61no—
292C2×D4—92no—
293C2×D4—181no—
294C2×D4—181no—
295C2×D4—181no—
296C2×D4—181no—
297C2×D4—181no—
298C22×C4—31no—
299C22×C4—61no—
300Q16—61no—
301Q16—61no—
302SD16—61no—
303SD16—61no—
304SD16—61no—
305SD16—61no—
306SD16—182no—
307SD16—182no—
308SD16—182no—
309SD16—182no—
310D8—61no—
311D8—61no—
312D8—182no—
313D8—182no—
314D8—182no—
315D8—182no—
316M16—31no—
317M16—31no—
318M16—31no—
319M16—61no—
320M16—92no—
321M16—91no—
322M16—92no—
323M16—92no—
324M16—181no—
325C2×C8—61no—
326C2×C8—92no—
327C2×C8—181no—
328C2×C8—181no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—41no—
332C22×C3—81no—
333C22×C3—121no—
334C22×C3—121no—
335C22×C3—241no—
336C22×C3C4⋊S411yescharacteristic
337C2×D3—81no—
338C2×D3—122no—
339C2×D3—242no—
340C2×D3—244no—
341C2×D3—242no—
342C2×D3—242no—
343C2×D3—242no—
344C2×D3—482no—
345C2×D3—484no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—81no—
351C3×C4—81no—
352C3×C4—121no—
353C3×C4C4⋊S411yescharacteristic
354C3×C4C22×S411yescharacteristic
355Dic3—81no—
356C32—41noSylow
357C23—31no—
358C23—61no—
359C23—182no—
360C23—361no—
361C23—361no—
362C23—722no—
363Q8—31no—
364Q8—61no—
365Q8—61no—
366Q8C2×#62611yescharacteristic
367D4—31no—
368D4—31no—
369D4—61no—
370D4—61no—
371D4—61no—
372D4—121no—
373D4—121no—
374D4—182no—
375D4—182no—
376D4—361no—
377D4—361no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—61no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4D3×S411yescharacteristic
387C8—32no—
388C8—61no—
389C8—61no—
390C8—92no—
391C8—182no—
392C2×C3—21no—
393C2×C3—41no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—81no—
397C2×C3—82no—
398C2×C3—81no—
399C2×C3—241no—
400C2×C3C2×C4⋊S411yescharacteristic
401D3—161no—
402D3—242no—
403D3—482no—
404D3—484no—
405C22—31no—
406C22—31no—
407C22—121no—
408C22—121no—
409C22—241no—
410C22—362no—
411C22—722no—
412C22—722no—
413C22—722no—
414C22#6420711yescharacteristic
415C4—31no—
416C4—31no—
417C4—121no—
418C4#6420711yescharacteristic
419C4C2×D3×S411yescharacteristic
420C3—41no—
421C3—81no—
422C3#8545411yescharacteristic, complemented
423C2—21no—
424C2—61no—
425C2—61no—
426C2—241no—
427C2—722no—
428C2C2×#6420711yescharacteristic
subgroupG/H
centerC2C2×#64207
commutator, cosocleC3×#41⋊C3C23
FrattiniC4C2×D3×S4
FittingC2×C3×#41C2×D3
socleC2×C3C2×C4⋊S4
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#64267C2semidirect
2. C3×#5923C2semidirect
3. #5819⋊C3C2semidirect
4. #10593225C2semidirect
5. #10593224C2semidirect
12. C3×#986C22semidirect
14. C3×#984C22semidirect
17. C3⋊(#131⋊C3)C22semidirect
18. #5923D3semidirect
24. #5819D3semidirect
56. #986C2×D3semidirect
58. #984C2×D3semidirect
79. #948C2×D3semidirect
422. C3#85454semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
C3×#59232
#5819⋊C32
#105932252
#105932242
#854543
#840023
#638254