Finite Group Databasebeta

← #11922427

Subgroups — #11922427

Order 1152.

707 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 61 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×D3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×#986C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4#10593217C212yesmaximal, complemented
5#10593203C212yesmaximal, complemented
6#85523—31nomaximal
7#83946—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9D3×#41⋊C3—22no—
10C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
11C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
12C3×#986—22no—
13C2×C3×GL(2,𝔽3)C2212yescomplemented
14#63908—41nomaximal
15C3⋊#859C2212yescomplemented
16C2×D3×#41D311yescharacteristic, complemented
17C22×#41⋊C3—31no—
18C2×#986D311yescharacteristic, complemented
19C2×C3×AGL(1,ℤ/8)—31no—
20C2×#938—31no—
21C2×#932—31no—
22#5427—32no—
23#5426—31no—
24#5426—61no—
25#5421—32no—
26#5232—31no—
27#5172—32no—
28#5170—32no—
29#5158—32no—
30#5156—32no—
31#5133—31no—
32#5126—31no—
33C2×C3×D12—41no—
34C2×C3×C4×D3—41no—
35C3×#41⋊C3D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37C2×#625—42no—
38D3×SL(2,𝔽3)—21no—
39C3⋊(#41⋊C3)—21no—
40C3×GL(2,𝔽3)—22no—
41C4⋊(C3⋊Dic3)—41no—
42(C2×D3)⋊Dic3—42no—
43#3006—91noSylow
44C2×C3×#41C2×D311yescharacteristic
45C22×C3×D4—31no—
46C22×#67—61no—
47D3×#41—22no—
48D3×#41—62no—
49C2×#239C2×D311yescharacteristic
50C2×D3×Q8C2×D311yescharacteristic
51C2×#237—31no—
52C2×D3×D4—31no—
53C2×#235—31no—
54C22×C4×D3—31no—
55C2×#41⋊C3—21no—
56C2×#41⋊C3—31no—
57C2×#41⋊C3—62no—
58C2×#41⋊C3C2×D311yescharacteristic
59C22×SL(2,𝔽3)—31no—
60#986—22no—
61C2×GL(2,𝔽3)C2×D312yescomplemented
62C3×AGL(1,ℤ/8)—62no—
63C3×AGL(1,ℤ/8)—62no—
64C2×C3×SD16—32no—
65C2×C3×D8—32no—
66C2×C3×M16—31no—
67C2×#217—61no—
68#938—32no—
69#938—32no—
70#938—61no—
71#938—61no—
72C2×#214—32no—
73#932—64no—
74C2×#213—32no—
75#930—121no—
76C2×C4⋊Dic3—31no—
77C2×#208—31no—
78#859—32no—
79#859—62no—
80#835—32no—
81#832—32no—
82#821—32no—
83(C2×D3)⋊C8—32no—
84(C2×D3)⋊C8—32no—
85#816—32no—
86C22×C3×D3—42no—
87C22×C3×D3—41no—
88C2×C32×C4—41no—
89C2×C3⋊Dic3—42no—
90C2×C3×Dic3—41no—
91C3×D12—82no—
92C3×C4×D3—82no—
93C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
94#625—84no—
95C22×#41—31no—
96C2×AGL(1,ℤ/8)—31no—
97C2×AGL(1,ℤ/8)—91no—
98C2×#121—91no—
99#449—92no—
100#449—92no—
101#448—92no—
102#447—92no—
103#417—91no—
104#417—91no—
105#407—91no—
106C24×C3—61no—
107C23×D3—31no—
108C3×#41—32no—
109C3×#41#6712yescomplemented
110C2×C3×Q8C22×D311yescharacteristic
111C2×C3×D4—31no—
112C2×C3×D4—32no—
113C2×C3×D4—62no—
114C2×C3×D4—62no—
115C2×C3×D4—61no—
116C22×C3×C4—31no—
117C22×C3×C4—121no—
118C2×#67—61no—
119C2×#67—62no—
120C2×#67—62no—
121C2×#67—122no—
122C2×#67—124no—
123C22×Dic3—31no—
124C22×Dic3—61no—
125#239—21no—
126#239—61no—
127D3×Q8—21no—
128D3×Q8—61no—
129#237—62no—
130#237—62no—
131D3×D4—62no—
132D3×D4—62no—
133#235—34no—
134#235—62no—
135C2×D12—31no—
136C2×D12—41no—
137C2×C4×D3—31no—
138C2×C4×D3—31no—
139C2×C4×D3—31no—
140C2×C4×D3—62no—
141C2×C4×D3—62no—
142C2×C4×D3S411yescharacteristic
143C2×Dic6—31no—
144#41⋊C3—22no—
145#41⋊C3—61no—
146#41⋊C3#6712yescomplemented
147C2×SL(2,𝔽3)—21no—
148C2×SL(2,𝔽3)—61no—
149C2×SL(2,𝔽3)C22×D311yescharacteristic
150GL(2,𝔽3)—22no—
151C3×SD16—62no—
152C3×SD16—62no—
153C3×D8—64no—
154C3×M16—62no—
155C2×C3×C8—32no—
156#217—62no—
157#217—62no—
158#214—64no—
159#213—64no—
160(C2×D3)⋊C4—122no—
161C4⋊Dic3—32no—
162C4⋊Dic3—61no—
163C4⋊Dic3—121no—
164C4⋊Dic3—241no—
165#208—62no—
166C2×C3⋊C8—32no—
167C22×C32—41no—
168C2×C3×D3—42no—
169C2×C3×D3—81no—
170C2×C3×D3—81no—
171C2×C3×D3—82no—
172C2×C3×D3—82no—
173C2×C3×D3—82no—
174C32×C4—42no—
175C3⋊Dic3—82no—
176C3×Dic3—81no—
177C2×#41—31no—
178C2×#41—62no—
179C2×#41—91no—
180C2×#41—91no—
181C2×#41—182no—
182C2×#41D3211yescharacteristic
183C22×Q8—31no—
184C22×D4—31no—
185C22×D4—91no—
186C22×D4—181no—
187C23×C4—91no—
188AGL(1,ℤ/8)—62no—
189AGL(1,ℤ/8)—62no—
190AGL(1,ℤ/8)—184no—
191C2×SD16—32no—
192C2×SD16—92no—
193C2×D8—32no—
194C2×D8—92no—
195C2×M16—31no—
196C2×M16—91no—
197#121—92no—
198#121—92no—
199#121—181no—
200#121—181no—
201C2×C4⋊C4—91no—
202C2×C22⋊C4—181no—
203#103—92no—
204#103—92no—
205#102—92no—
206#102—92no—
207C22⋊C8—92no—
208C22⋊C8—92no—
209C23×C3—31no—
210C23×C3—62no—
211C23×C3—121no—
212C23×C3—122no—
213C23×C3—122no—
214C23×C3—122no—
215C22×D3—31no—
216C22×D3—42no—
217C22×D3—62no—
218C22×D3—62no—
219C22×D3—121no—
220C22×D3C2×S411yescharacteristic
221C3×Q8—31no—
222C3×Q8C2×#6711yescharacteristic
223C3×D4—32no—
224C3×D4—61no—
225C3×D4—62no—
226C3×D4—62no—
227C3×D4—122no—
228C2×C3×C4—31no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—41no—
232C2×C3×C4—81no—
233C2×C3×C4—121no—
234C2×C3×C4—242no—
235C2×C3×C4C2×S411yescharacteristic
236#67—64no—
237#67—124no—
238#67—124no—
239#67—244no—
240C2×Dic3—31no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—122no—
244C2×Dic3—122no—
245C2×Dic3—242no—
246C2×Dic3C2×S411yescharacteristic
247D12—32no—
248D12—61no—
249D12—82no—
250C4×D3—22no—
251C4×D3—61no—
252C4×D3—61no—
253C4×D3—61no—
254C4×D3—61no—
255C4×D3—62no—
256Dic6—32no—
257Dic6—61no—
258SL(2,𝔽3)—21no—
259SL(2,𝔽3)C2×#6711yescharacteristic, complemented
260C3×C8—62no—
261C3⋊C8—62no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C3×D3—82no—
266C3×D3—162no—
267C24—61no—
268C24—91no—
269#41—32no—
270#41—61no—
271#41—94no—
272#41—182no—
273#41—182no—
274#41—182no—
275#41—181no—
276#41#62512yescomplemented
277C2×Q8—61no—
278C2×Q8—91no—
279C2×Q8—181no—
280C2×Q8C2×D3211yescharacteristic
281C2×D4—31no—
282C2×D4—32no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—61no—
286C2×D4—91no—
287C2×D4—181no—
288C2×D4—182no—
289C2×D4—182no—
290C2×D4—182no—
291C2×D4—182no—
292C2×D4—364no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—91no—
296C22×C4—91no—
297C22×C4—91no—
298C22×C4—91no—
299C22×C4—182no—
300C22×C4—182no—
301C22×C4—181no—
302SD16—62no—
303SD16—62no—
304SD16—184no—
305D8—64no—
306D8—184no—
307M16—62no—
308M16—182no—
309C2×C8—32no—
310C2×C8—92no—
311C4⋊C4—92no—
312C4⋊C4—181no—
313C22⋊C4—182no—
314C22⋊C4—182no—
315C22×C3—32no—
316C22×C3—41no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—81no—
320C22×C3—122no—
321C22×C3—122no—
322C22×C3—122no—
323C22×C3—122no—
324C22×C3—241no—
325C22×C3—241no—
326C22×C3—244no—
327C22×C3C22×S411yescharacteristic
328C2×D3—21no—
329C2×D3—21no—
330C2×D3—61no—
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—82no—
334C2×D3—82no—
335C2×D3—82no—
336C2×D3—124no—
337C2×D3#98612yescomplemented
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4—241no—
343C3×C4GL(2,ℤ/4)12yes—
344Dic3—21no—
345Dic3—61no—
346Dic3—122no—
347Dic3—242no—
348Dic3—482no—
349C32—41noSylow
350C23—31no—
351C23—31no—
352C23—62no—
353C23—91no—
354C23—122no—
355C23—122no—
356C23—122no—
357C23—182no—
358C23—182no—
359C23—361no—
360Q8—31no—
361Q8—92no—
362Q8—181no—
363Q8C2×#62511yescharacteristic
364D4—32no—
365D4—61no—
366D4—62no—
367D4—62no—
368D4—92no—
369D4—122no—
370D4—181no—
371D4—184no—
372D4—364no—
373D4—364no—
374C2×C4—31no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—31no—
378C2×C4—62no—
379C2×C4—91no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—362no—
388C2×C4D3×S411yescharacteristic
389C8—62no—
390C8—182no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—62no—
395C2×C3—81no—
396C2×C3—81no—
397C2×C3—81no—
398C2×C3—242no—
399C2×C3—242no—
400C2×C3C2×#98611yescharacteristic
401C2×C3#592311yescharacteristic
402C2×C3C2×GL(2,ℤ/4)11yescharacteristic
403D3—22no—
404D3—121no—
405D3—162no—
406C22—32no—
407C22—32no—
408C22—61no—
409C22—61no—
410C22—61no—
411C22—61no—
412C22—122no—
413C22—122no—
414C22—122no—
415C22—181no—
416C22—181no—
417C22—181no—
418C22—244no—
419C22—364no—
420C22C2×D3×S411yescharacteristic
421C4—31no—
422C4—31no—
423C4—61no—
424C4—181no—
425C4—362no—
426C4#6420912yes—
427C3—41no—
428C3—81no—
429C3#8552311yescharacteristic, complemented
430C2—62no—
431C2—62no—
432C2—242no—
433C2—361no—
434C2#1059645511yescharacteristic
435C2D3×#98611yescharacteristic
436C2C2×#6420911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#41⋊C3C2semidirect
2. C2×C3×#986C2semidirect
3. #10593660C2semidirect
4. #10593217C2semidirect
5. #10593203C2semidirect
10. C2×D3×SL(2,𝔽3)C22semidirect
11. C2×C3⋊(#41⋊C3)C22semidirect
13. C2×C3×GL(2,𝔽3)C22semidirect
15. C3⋊#859C22semidirect
16. C2×D3×#41D3semidirect
18. C2×#986D3semidirect
35. C3×#41⋊C3D4semidirect
61. C2×GL(2,𝔽3)C2×D3semidirect
93. C3×SL(2,𝔽3)C2×D4semidirect
109. C3×#41#67semidirect
146. #41⋊C3#67semidirect
259. SL(2,𝔽3)C2×#67semidirect
276. #41#625semidirect
337. C2×D3#986semidirect
429. C3#85523semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#41⋊C32
C2×C3×#9862
#105936602
#105932172
#105932032
#855233
#839463
#639084