Finite Group Databasebeta

← C2×#10593233 #11922332

Subgroups — C2×#10593233 #11922332

Order 1152.

709 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 59 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×C3⋊#994C211yesmaximal, characteristic, complemented
2C2×C3×#986C211yesmaximal, characteristic, complemented
3C2×C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593233C214yesmaximal, direct product factor
5C2×#5429—31nomaximal
6C2×#5198—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8C3⋊#994—21no—
9C3⋊#994C2212yescomplemented
10C2×C3⋊(#41⋊C3)—21no—
11C3×#986—21no—
12C3×#986C2212yescomplemented
13C2×C3×GL(2,𝔽3)—21no—
14C2×#3715—41nomaximal
15C3⋊GU(2,𝔽3)C2212yescomplemented
16C2×#1009D311yescharacteristic, complemented
17C2×#994—31no—
18C2×#986D311yescharacteristic, complemented
19C2×C3×AGL(1,ℤ/8)—31no—
20C2×#940—31no—
21C2×#932—31no—
22#5429—34no—
23C2×GU(2,𝔽3)—31no—
24C2×GU(2,𝔽3)—61no—
25C2×#837—31no—
26#5198—34no—
27#5198—34no—
28C2×#825—31no—
29C2×#823—31no—
30C2×C3×D12—41no—
31C2×C3×Dic6—41no—
32C3×#41⋊C3D411yescharacteristic, complemented
33C3×#41⋊C3C2311yescharacteristic
34C2×C3×SL(2,𝔽3)D411yescharacteristic, complemented
35C3⋊(#41⋊C3)—22no—
36C3×GL(2,𝔽3)—22no—
37C2×C3⋊(C3⋊C8)—41no—
38#3715—44no—
39C2×#453—91noSylow
40C2×C3×#41C2×D311yescharacteristic
41C22×C3×D4—31no—
42C22×#67—62no—
43#1009—21no—
44#1009—32no—
45#1009—61no—
46#1009C2×D312yescomplemented
47C2×#239—21no—
48C2×D3×D4—61no—
49C2×#235—31no—
50C22×D12—31no—
51#994—32no—
52#994—61no—
53C2×#41⋊C3—21no—
54C2×#41⋊C3—61no—
55C2×#41⋊C3C2×D311yescharacteristic
56#986—21no—
57#986C2×D312yescomplemented
58C2×GL(2,𝔽3)—21no—
59C3×AGL(1,ℤ/8)—32no—
60C3×AGL(1,ℤ/8)—32no—
61C3×AGL(1,ℤ/8)—61no—
62C3×AGL(1,ℤ/8)—61no—
63C2×C3×SD16—61no—
64C2×C3×D8—61no—
65C2×C3×M16—31no—
66C2×#215—121no—
67#940—34no—
68#940—62no—
69C2×#214—61no—
70#932—34no—
71#932—62no—
72C2×#213—61no—
73C2×C4×Dic3—31no—
74C2×#208—31no—
75GU(2,𝔽3)—32no—
76GU(2,𝔽3)—62no—
77#837—32no—
78#837—32no—
79#825—34no—
80#823—34no—
81C22×C3×D3—81no—
82C2×C32×C4—41no—
83C2×C3×Dic3—81no—
84C3×D12—42no—
85C3×D12—81no—
86C3×Dic6—42no—
87C3×Dic6—81no—
88C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
89C3⋊(C3⋊C8)—42no—
90C2×H(5,𝔽2)—31no—
91C2×AGL(1,ℤ/8)—31no—
92C2×AGL(1,ℤ/8)—91no—
93C2×#127—91no—
94#453—94no—
95#453—94no—
96C2×#104—91no—
97C2×#104—91no—
98C2×#100—91no—
99C24×C3—61no—
100C23×D3—32no—
101C3×#41—31no—
102C3×#41—31no—
103C3×#41#6711yescharacteristic
104C3×#41C22×D311yescharacteristic
105C2×C3×Q8—121no—
106C2×C3×Q8#6711yescharacteristic
107C2×C3×D4—31no—
108C2×C3×D4—32no—
109C2×C3×D4—61no—
110C2×C3×D4—61no—
111C2×C3×D4—61no—
112C2×C3×D4—61no—
113C2×C3×D4—61no—
114C22×C3×C4—31no—
115C2×#67—61no—
116C2×#67—68no—
117C2×#67—128no—
118C22×Dic3—61no—
119#239—22no—
120#239—62no—
121D3×D4—64no—
122D3×D4—62no—
123D3×D4—62no—
124#235—32no—
125#235—32no—
126#235—61no—
127#235—61no—
128C2×D12—31no—
129C2×D12—32no—
130C2×D12—32no—
131C2×D12—41no—
132C2×D12—61no—
133C2×D12—61no—
134C2×D12—61no—
135C2×C4×D3—61no—
136C2×Dic6S411yescharacteristic
137#41⋊C3—21no—
138#41⋊C3—21no—
139#41⋊C3—62no—
140#41⋊C3#6711yescharacteristic, complemented
141#41⋊C3C22×D311yescharacteristic
142C2×SL(2,𝔽3)—21no—
143C2×SL(2,𝔽3)#6711yescharacteristic, complemented
144GL(2,𝔽3)—22no—
145C3×SD16—62no—
146C3×SD16—62no—
147C3×D8—62no—
148C3×D8—62no—
149C3×M16—32no—
150C3×M16—61no—
151C2×C3×C8—61no—
152#215—124no—
153#214—64no—
154#213—64no—
155C4×Dic3—32no—
156C4×Dic3—61no—
157#208—32no—
158#208—61no—
159C2×C3⋊C8—61no—
160C2×C3⋊C8—121no—
161C2×C3⋊C8—241no—
162C22×C32—41no—
163C2×C3×D3—82no—
164C2×C3×D3—162no—
165C32×C4—41no—
166C32×C4—41no—
167C3×Dic3—82no—
168H(5,𝔽2)—32no—
169H(5,𝔽2)—61no—
170H(5,𝔽2)—92no—
171H(5,𝔽2)—181no—
172C2×#41—61no—
173C2×#41—91no—
174C2×#41D3211yescharacteristic
175C22×D4—31no—
176C22×D4—91no—
177C22×D4—181no—
178C22×D4—182no—
179AGL(1,ℤ/8)—32no—
180AGL(1,ℤ/8)—32no—
181AGL(1,ℤ/8)—61no—
182AGL(1,ℤ/8)—61no—
183AGL(1,ℤ/8)—94no—
184AGL(1,ℤ/8)—182no—
185C2×SD16—61no—
186C2×SD16—181no—
187C2×D8—61no—
188C2×D8—181no—
189C2×M16—31no—
190C2×M16—91no—
191#127—94no—
192#127—182no—
193C2×C42—91no—
194#104—94no—
195#104—92no—
196#104—92no—
197#100—94no—
198C23×C3—31no—
199C23×C3—62no—
200C23×C3—121no—
201C23×C3—124no—
202C23×C3—121no—
203C22×D3—34no—
204C22×D3—61no—
205C22×D3—62no—
206C22×D3—81no—
207C22×D3—124no—
208C3×Q8—31no—
209C3×Q8—122no—
210C3×Q8—241no—
211C3×Q8C2×#6711yescharacteristic
212C3×D4—31no—
213C3×D4—31no—
214C3×D4—61no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—121no—
218C3×D4—121no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—41no—
225C2×C3×C4—81no—
226C2×C3×C4—241no—
227C2×C3×C4C2×S411yescharacteristic
228#67—62no—
229#67—62no—
230#67—1216no—
231C2×Dic3—21no—
232C2×Dic3—62no—
233C2×Dic3—64no—
234D12—32no—
235D12—42no—
236D12—62no—
237D12—62no—
238D12—61no—
239D12—81no—
240C4×D3—62no—
241C4×D3—62no—
242Dic6—21no—
243Dic6C2×S412yes—
244SL(2,𝔽3)—21no—
245SL(2,𝔽3)C2×#6711yescharacteristic, complemented
246C3×C8—62no—
247C3⋊C8—62no—
248C3⋊C8—122no—
249C3⋊C8—242no—
250C2×C32—41no—
251C2×C32—42no—
252C3×D3—162no—
253C24—61no—
254C24—92no—
255#41—31no—
256#41—31no—
257#41—62no—
258#41—92no—
259#41—92no—
260#41—181no—
261#41—181no—
262#41—182no—
263#41#62511yescharacteristic
264#41C2×D3211yescharacteristic
265C2×Q8—31no—
266C2×Q8#62511yescharacteristic
267C2×D4—31no—
268C2×D4—32no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—92no—
275C2×D4—92no—
276C2×D4—91no—
277C2×D4—181no—
278C2×D4—181no—
279C2×D4—181no—
280C2×D4—181no—
281C2×D4—182no—
282C2×D4—182no—
283C2×D4—184no—
284C2×D4—188no—
285C2×D4—368no—
286C22×C4—31no—
287C22×C4—181no—
288C22×C4—181no—
289SD16—62no—
290SD16—62no—
291SD16—184no—
292D8—62no—
293D8—62no—
294D8—184no—
295M16—32no—
296M16—61no—
297M16—92no—
298M16—181no—
299C2×C8—61no—
300C2×C8—181no—
301C42—92no—
302C42—181no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—41no—
306C22×C3—62no—
307C22×C3—81no—
308C22×C3—124no—
309C22×C3—122no—
310C22×C3—242no—
311C22×C3—242no—
312C22×C3GL(2,ℤ/4)11yescharacteristic
313C2×D3—62no—
314C2×D3—82no—
315C2×D3—122no—
316C2×D3—124no—
317C2×D3—162no—
318C2×D3—242no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—81no—
324C3×C4—81no—
325C3×C4—242no—
326C3×C4GL(2,ℤ/4)11yescharacteristic
327C3×C4C22×S411yescharacteristic
328Dic3—22no—
329Dic3—64no—
330C32—41noSylow
331C23—31no—
332C23—62no—
333C23—94no—
334C23—121no—
335C23—124no—
336C23—121no—
337C23—182no—
338C23—181no—
339C23—364no—
340Q8—31no—
341Q8—32no—
342Q8—61no—
343Q8C2×#62511yescharacteristic
344D4—31no—
345D4—31no—
346D4—61no—
347D4—62no—
348D4—62no—
349D4—92no—
350D4—121no—
351D4—121no—
352D4—182no—
353D4—182no—
354D4—182no—
355D4—182no—
356D4—181no—
357D4—3616no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—61no—
364C2×C4—182no—
365C2×C4—182no—
366C2×C4—182no—
367C2×C4—184no—
368C2×C4D3×S411yescharacteristic
369C8—62no—
370C8—182no—
371C2×C3—41no—
372C2×C3—42no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—81no—
376C2×C3—82no—
377C2×C3—242no—
378C2×C3#542912yescomplemented
379C2×C3C2×GL(2,ℤ/4)11yescharacteristic
380D3—122no—
381D3—162no—
382C22—31no—
383C22—31no—
384C22—62no—
385C22—124no—
386C22—122no—
387C22—182no—
388C22—242no—
389C22—242no—
390C22—364no—
391C22—362no—
392C22—722no—
393C22#6420911yescharacteristic
394C4—31no—
395C4—31no—
396C4—62no—
397C4—184no—
398C4#6420911yescharacteristic
399C4C2×D3×S411yescharacteristic
400C3—41no—
401C3—81no—
402C3C2×#542911yescharacteristic, complemented
403C2—61no—
404C2—61no—
405C2—242no—
406C2—362no—
407C2#1059323312yesdirect product factor
408C2C2×#6420911yescharacteristic
subgroupG/H
centerC22#64209
commutator, cosocleC3×#41⋊C3C23
FrattiniC4C2×D3×S4
FittingC2×C3×#41C2×D3
socleC22×C3GL(2,ℤ/4)
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊#994C2semidirect
2. C2×C3×#986C2semidirect
3. C2×C3⋊GU(2,𝔽3)C2semidirect
4. #10593233C2direct (N × H)
9. C3⋊#994C22semidirect
12. C3×#986C22semidirect
15. C3⋊GU(2,𝔽3)C22semidirect
16. C2×#1009D3semidirect
18. C2×#986D3semidirect
32. C3×#41⋊C3D4semidirect
34. C2×C3×SL(2,𝔽3)D4semidirect
46. #1009C2×D3semidirect
57. #986C2×D3semidirect
88. C3×SL(2,𝔽3)C2×D4semidirect
140. #41⋊C3#67semidirect
143. C2×SL(2,𝔽3)#67semidirect
245. SL(2,𝔽3)C2×#67semidirect
378. C2×C3#5429semidirect
402. C3C2×#5429semidirect
407. C2#10593233direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
C2×C3×#9862
C2×C3⋊GU(2,𝔽3)2
#105932332
C2×#54293
C2×#51983
C2×#37154