Finite Group Databasebeta

← C2×#11724194 #13180807

Subgroups — C2×#11724194 #13180807

Order 1728.

1025 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1025 conjugacy classes of subgroups — too many to draw. Only its 82 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×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#11724194C214yesmaximal, direct product factor
5C2×#64267—31nomaximal
6C2×#64199—33nomaximal
7C2×#63939—31nomaximal
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C2×#6292—41nomaximal
10C3×C3⋊(#41⋊C3)C2212yescomplemented
11C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
12C3×C3⋊GL(2,𝔽3)C2212yescomplemented
13C2×C3×#239D311yescharacteristic, complemented
14C2×C3×#41⋊C3—31no—
15C2×D3×D12—31no—
16C2×C3⋊(#41⋊C3)D313yescomplemented
17#64267—34no—
18C2×C3⋊GL(2,𝔽3)—31no—
19C2×C3⋊GL(2,𝔽3)—33no—
20C2×C3⋊GL(2,𝔽3)—61no—
21C2×C3⋊GL(2,𝔽3)—63no—
22C2×C3⋊GL(2,𝔽3)D311yescharacteristic, complemented
23C2×C3×GL(2,𝔽3)—33no—
24#64199—312no—
25C2×#3755—31no—
26C2×C3×#215—31no—
27#63939—34no—
28#63939—34no—
29C2×#3714—31no—
30C2×#3713—31no—
31C2×#3711—31no—
32C22×C3⋊(C3⋊D3)—41no—
33C22×C3×C3⋊D3—41no—
34C2×C32×Dic3—41no—
35C32×SL(2,𝔽3)C2311yescharacteristic, complemented
36#6292—44no—
37C2×#986—93no—
38C2×#949—91no—
39C2×#914—91no—
40C22×D32—61no—
41C2×C32×Q8C2×D311yescharacteristic
42C2×C4⋊(C3⋊D3)—31no—
43C2×C4⋊(C3⋊D3)—121no—
44C3×#239—32no—
45C3×#239C2×D312yescomplemented
46C2×C3×D12—31no—
47C2×C3×D12—31no—
48C2×C3×C4×D3—31no—
49C3×#41⋊C3—32no—
50C2×C3×SL(2,𝔽3)—21no—
51C2×C3×SL(2,𝔽3)—23no—
52C2×C3×SL(2,𝔽3)C2×D313yescomplemented
53C2×C3×SL(2,𝔽3)C2×D311yescharacteristic, complemented
54C2×#626—61no—
55C2×#626—123no—
56D3×D12—34no—
57D3×D12—62no—
58C3⋊(#41⋊C3)C2×D316yescomplemented
59C3⋊GL(2,𝔽3)—32no—
60C3⋊GL(2,𝔽3)—36no—
61C3⋊GL(2,𝔽3)—62no—
62C3⋊GL(2,𝔽3)—66no—
63C3⋊GL(2,𝔽3)C2×D312yescomplemented
64C3×GL(2,𝔽3)—36no—
65#3755—32no—
66#3755—32no—
67C2×C3⋊(C3⋊C8)—31no—
68C3×#215—32no—
69C3×#215—32no—
70C2×C3×C3⋊C8—31no—
71#3714—34no—
72#3713—34no—
73#3711—34no—
74C22×C33—41no—
75C2×C3⋊(C3⋊D3)—42no—
76C2×C3⋊(C3⋊D3)—82no—
77C2×C3×C3⋊D3—42no—
78C2×C3×C3⋊D3—82no—
79C32×Dic3—42no—
80C2×C3×#41—31no—
81C2×#239C3⋊D311yescharacteristic, complemented
82C2×D3×D4—91no—
83C22×D12—91no—
84C2×#41⋊C3—33no—
85#986—912no—
86C2×GL(2,𝔽3)—33no—
87C2×GL(2,𝔽3)—93no—
88C2×GL(2,𝔽3)—183no—
89C2×C3×SD16—91no—
90#949—94no—
91#949—94no—
92C2×#215—31no—
93C2×#215—91no—
94C2×#215—91no—
95C2×#215—181no—
96C2×#213—91no—
97C2×#213—91no—
98C2×#213—91no—
99C2×#208—91no—
100#914—94no—
101#914—94no—
102C2×D24—91no—
103C2×ΓL(1,𝔽25)—91no—
104C22×C3⋊D3—41no—
105C22×C3⋊D3—61no—
106C22×C3⋊D3—121no—
107C22×C3⋊D3—123no—
108C22×C3⋊D3—241no—
109C22×C3⋊D3—243no—
110C22×C3×D3—31no—
111C22×C3×D3—61no—
112C22×C3×D3—123no—
113C2×D32—64no—
114C2×D32—124no—
115C32×Q8—31no—
116C32×Q8C22×D311yescharacteristic
117C2×C32×C4—31no—
118C2×C32×C4—121no—
119C4⋊(C3⋊D3)—32no—
120C4⋊(C3⋊D3)—61no—
121C4⋊(C3⋊D3)—124no—
122C2×C3×Dic3—43no—
123C2×C3×Dic3S411yescharacteristic
124C3×D12—32no—
125C3×D12—32no—
126C3×D12—61no—
127C3×D12—61no—
128C3×C4×D3—32no—
129C3×C4×D3—32no—
130C3×SL(2,𝔽3)—21no—
131C3×SL(2,𝔽3)—23no—
132C3×SL(2,𝔽3)C22×D313yescomplemented
133C3×SL(2,𝔽3)C22×D311yescharacteristic, complemented
134#626—64no—
135#626—1212no—
136C3⋊(C3⋊C8)—32no—
137C3×C3⋊C8—32no—
138C2×AGL(1,ℤ/8)—271noSylow
139C2×C33—42no—
140C2×C33—41no—
141C3⋊(C3⋊D3)—82no—
142C3×C3⋊D3—82no—
143C23×D3—181no—
144C23×D3—181no—
145C3×#41—32no—
146C3×#41—92no—
147C2×C3×Q8—21no—
148C2×C3×Q8D3211yescharacteristic
149C2×C3×Q8C2×C3⋊D311yescharacteristic
150C2×C3×D4—91no—
151C2×C3×D4—91no—
152C22×C3×C4—91no—
153C2×#67—181no—
154#239—32no—
155#239C2×C3⋊D312yescomplemented
156D3×D4—94no—
157D3×D4—182no—
158C2×D12—31no—
159C2×D12—31no—
160C2×D12—94no—
161C2×D12—91no—
162C2×D12—91no—
163C2×D12—181no—
164C2×D12—182no—
165C2×D12—181no—
166C2×D12—363no—
167C2×C4×D3—31no—
168#41⋊C3—36no—
169C2×SL(2,𝔽3)—23no—
170C2×SL(2,𝔽3)D3213yescomplemented
171GL(2,𝔽3)—36no—
172GL(2,𝔽3)—96no—
173GL(2,𝔽3)—186no—
174C3×SD16—92no—
175C3×SD16—92no—
176C2×C3×C8—91no—
177#215—32no—
178#215—32no—
179#215—92no—
180#215—92no—
181#215—92no—
182#215—92no—
183#215—182no—
184#215—182no—
185#213—94no—
186#213—94no—
187#213—94no—
188#208—94no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—91no—
192C2×C3⋊C8—181no—
193D24—94no—
194ΓL(1,𝔽25)—94no—
195C22×C32—41no—
196C22×C32—43no—
197C22×C32—83no—
198C22×C32—81no—
199C22×C32C2×S411yescharacteristic
200C2×C3⋊D3—42no—
201C2×C3⋊D3—62no—
202C2×C3⋊D3—82no—
203C2×C3⋊D3—122no—
204C2×C3⋊D3—126no—
205C2×C3⋊D3—122no—
206C2×C3⋊D3—242no—
207C2×C3⋊D3—246no—
208C2×C3⋊D3—246no—
209C2×C3⋊D3—242no—
210C2×C3⋊D3—244no—
211C2×C3⋊D3—2412no—
212C2×C3×D3—32no—
213C2×C3×D3—62no—
214C2×C3×D3—62no—
215C2×C3×D3—126no—
216C2×C3×D3—122no—
217C2×C3×D3—246no—
218D32—128no—
219C32×C4—31no—
220C32×C4—31no—
221C32×C4—122no—
222C3×Dic3—46no—
223C3×Dic3C2×S412yes—
224C2×#41—31no—
225C22×D4—271no—
226AGL(1,ℤ/8)—274no—
227AGL(1,ℤ/8)—274no—
228C2×SD16—91no—
229C2×SD16—271no—
230C2×D8—271no—
231C2×D8—271no—
232C2×M16—271no—
233C33—41noSylow
234C23×C3—91no—
235C23×C3—181no—
236C22×D3—31no—
237C22×D3—61no—
238C22×D3—123no—
239C22×D3—184no—
240C22×D3—184no—
241C22×D3—181no—
242C22×D3—181no—
243C22×D3—361no—
244C22×D3—363no—
245C22×D3—364no—
246C22×D3—364no—
247C22×D3—723no—
248C3×Q8—21no—
249C3×Q8—31no—
250C3×Q8—31no—
251C3×Q8—61no—
252C3×Q8C2×D3211yescharacteristic
253C3×Q8C22×C3⋊D311yescharacteristic
254C3×D4—92no—
255C3×D4—92no—
256C3×D4—181no—
257C3×D4—181no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—61no—
262C2×C3×C4—92no—
263C2×C3×C4—92no—
264C2×C3×C4—123no—
265#67—184no—
266C2×Dic3C3⋊S411yescharacteristic
267D12—32no—
268D12—32no—
269D12—61no—
270D12—61no—
271D12—92no—
272D12—92no—
273D12—182no—
274D12—184no—
275D12—181no—
276D12—181no—
277D12—361no—
278D12—3612no—
279C4×D3—32no—
280C4×D3—32no—
281SL(2,𝔽3)—23no—
282SL(2,𝔽3)C2×D3213yescomplemented
283C3×C8—92no—
284C3⋊C8—32no—
285C3⋊C8—92no—
286C3⋊C8—92no—
287C3⋊C8—182no—
288C2×C32—41no—
289C2×C32—42no—
290C2×C32—46no—
291C2×C32—43no—
292C2×C32—83no—
293C2×C32—86no—
294C2×C32—81no—
295C2×C32—82no—
296C2×C32#98612yescomplemented
297C2×C32C22×S411yescharacteristic
298C3⋊D3—82no—
299C3⋊D3—122no—
300C3⋊D3—242no—
301C3⋊D3—246no—
302C3⋊D3—244no—
303C3⋊D3—2412no—
304C3×D3—62no—
305C3×D3—122no—
306C3×D3—246no—
307C24—541no—
308#41—32no—
309#41—92no—
310C2×Q8D3×C3⋊D311yescharacteristic
311C2×D4—91no—
312C2×D4—91no—
313C2×D4—271no—
314C2×D4—274no—
315C2×D4—541no—
316C2×D4—542no—
317C22×C4—91no—
318SD16—92no—
319SD16—92no—
320SD16—272no—
321SD16—272no—
322D8—274no—
323D8—274no—
324M16—274no—
325C2×C8—91no—
326C2×C8—271no—
327C22×C3—21no—
328C22×C3—43no—
329C22×C3—83no—
330C22×C3—92no—
331C22×C3—182no—
332C22×C3—182no—
333C22×C3—362no—
334C22×C3D3×S411yescharacteristic
335C22×C3C2×C3⋊S411yescharacteristic
336C2×D3—32no—
337C2×D3—62no—
338C2×D3—62no—
339C2×D3—126no—
340C2×D3—122no—
341C2×D3—182no—
342C2×D3—182no—
343C2×D3—246no—
344C2×D3—362no—
345C2×D3—366no—
346C2×D3—368no—
347C2×D3—368no—
348C2×D3—362no—
349C2×D3—362no—
350C2×D3—726no—
351C2×D3—722no—
352C2×D3—726no—
353C2×D3—7212no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—32no—
357C3×C4—31no—
358C3×C4—31no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—126no—
362Dic3C2×C3⋊S412yes—
363C32—41no—
364C32—43no—
365C32—83no—
366C32—81no—
367C32C2×#98611yescharacteristic, complemented
368C23—91no—
369C23—181no—
370C23—544no—
371C23—541no—
372C23—1084no—
373Q8—31no—
374Q8C2×D3×C3⋊D311yescharacteristic
375D4—92no—
376D4—92no—
377D4—181no—
378D4—181no—
379D4—272no—
380D4—544no—
381D4—541no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—92no—
385C2×C4—92no—
386C8—92no—
387C8—272no—
388C2×C3—22no—
389C2×C3—21no—
390C2×C3—46no—
391C2×C3—43no—
392C2×C3—86no—
393C2×C3—83no—
394C2×C3—182no—
395C2×C3—362no—
396C2×C3#6419912yescomplemented
397C2×C3#6426712yescomplemented
398C2×C3C2×D3×S411yescharacteristic
399C2×C3C22×C3⋊S411yescharacteristic
400D3—62no—
401D3—122no—
402D3—246no—
403D3—362no—
404D3—362no—
405D3—722no—
406D3—726no—
407D3—7212no—
408C22—92no—
409C22—182no—
410C22—182no—
411C22—362no—
412C22—542no—
413C22—1088no—
414C22—1082no—
415C22D3×C3⋊S411yescharacteristic
416C4—31no—
417C4—31no—
418C4—32no—
419C3—21no—
420C3—43no—
421C3—83no—
422C3C2×#6419911yescharacteristic, complemented
423C3C2×#6426711yescharacteristic, complemented
424C2—182no—
425C2—362no—
426C2—1082no—
427C2#1172419412yesdirect product factor
428C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C2×D3×C3⋊S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×C3⋊(#41⋊C3)C2semidirect
2. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
4. #11724194C2direct (N × H)
8. C2×C32×SL(2,𝔽3)C22semidirect
10. C3×C3⋊(#41⋊C3)C22semidirect
11. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
12. C3×C3⋊GL(2,𝔽3)C22semidirect
13. C2×C3×#239D3semidirect
16. C2×C3⋊(#41⋊C3)D3semidirect
22. C2×C3⋊GL(2,𝔽3)D3semidirect
35. C32×SL(2,𝔽3)C23semidirect
45. C3×#239C2×D3semidirect
52. C2×C3×SL(2,𝔽3)C2×D3semidirect
53. C2×C3×SL(2,𝔽3)C2×D3semidirect
58. C3⋊(#41⋊C3)C2×D3semidirect
63. C3⋊GL(2,𝔽3)C2×D3semidirect
81. C2×#239C3⋊D3semidirect
132. C3×SL(2,𝔽3)C22×D3semidirect
133. C3×SL(2,𝔽3)C22×D3semidirect
155. #239C2×C3⋊D3semidirect
170. C2×SL(2,𝔽3)D32semidirect
282. SL(2,𝔽3)C2×D32semidirect
296. C2×C32#986semidirect
367. C32C2×#986semidirect
396. C2×C3#64199semidirect
397. C2×C3#64267semidirect
422. C3C2×#64199semidirect
423. C3C2×#64267semidirect
427. C2#11724194direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C3×C3⋊GL(2,𝔽3)2
#117241942
C2×#642673
C2×#641993
C2×#639393
C2×#62924