Finite Group Databasebeta

← C2×#11721161 #13150867

Subgroups — C2×#11721161 #13150867

Order 1728.

2595 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 2595 conjugacy classes of subgroups — too many to draw. Only its 283 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
1C23×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C22×C4×#272C211yesmaximal, characteristic, complemented
3C22×#6199C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C212yesmaximal, complemented
5C2×#88450C212yesmaximal, complemented
6#11721161C218yesmaximal, direct product factor
7C2×D3×(C2×D3)⋊C4—32nomaximal
8C23×#272C2211yescharacteristic
9C23×(C3⋊D3)⋊C3C2212yes—
10C22×((C3⋊D3)⋊C3)⋊C2—24no—
11C22×((C3⋊D3)⋊C3)⋊C2—24no—
12C22×((C3⋊D3)⋊C3)⋊C2C418yescomplemented
13C22×((C3⋊D3)⋊C3)⋊C2C2214yes—
14C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C22×#1084C2211yescharacteristic, complemented
16C2×C4×#272—24no—
17C2×C4×#272C2214yescomplemented
18C22×(C3⋊Dic3)⋊C3C2212yescomplemented
19C2×#6199—24no—
20C2×#6199C2214yescomplemented
21#3752⋊C3C2218yescomplemented
22#88450C2218yescomplemented
23C23×D32—32no—
24C22×C3×C4×D3—32no—
25C22×C3×C4×D3—61no—
26C22×D3×Dic3—32no—
27C2×#3752D312yescomplemented
28C2×C3×(C2×D3)⋊C4—32no—
29C2×#3722—32no—
30C2×(C2×D3)⋊Dic3—32no—
31D3×(C2×D3)⋊C4—316no—
32C23×H(3,𝔽3)C2311yescharacteristic
33C22×#272C2×C414yes—
34C22×#272D414yes—
35C22×#272D414yes—
36C22×#272C2312yes—
37C22×(C3⋊D3)⋊C3—24no—
38C22×(C3⋊D3)⋊C3—24no—
39C22×(C3⋊D3)⋊C3C2×C418yescomplemented
40C22×(C3⋊D3)⋊C3C2314yes—
41C2×((C3⋊D3)⋊C3)⋊C2—232no—
42C2×((C3⋊D3)⋊C3)⋊C2—28no—
43C2×((C3⋊D3)⋊C3)⋊C2—28no—
44C2×((C3⋊D3)⋊C3)⋊C2C2×C4116yescomplemented
45C2×C4×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2312yescomplemented
47C2×#1084—22no—
48C2×#1084C2312yescomplemented
49C4×#272—28no—
50C2×(C3⋊Dic3)⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3C2314yescomplemented
52#6199—28no—
53C22×(C2×D3)⋊C4—91no—
54C2×D3×C22⋊C4—92no—
55C23×C3⋊D3C2×D312yes—
56C23×C3×D3—32no—
57C23×C3×D3—32no—
58C23×C3×D3—61no—
59C22×D32—316no—
60C22×D32—38no—
61C22×D32—68no—
62C22×D32—68no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D312yescomplemented
65C22×C3⋊Dic3C2×D312yescomplemented
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—61no—
69C2×C3×C4×D3—38no—
70C2×C3×C4×D3—68no—
71C2×C3×C4×D3—64no—
72C2×C3×C4×D3—68no—
73C2×D3×Dic3—38no—
74C2×D3×Dic3—68no—
75#3752C2×D318yescomplemented
76C3×(C2×D3)⋊C4—38no—
77#3722—38no—
78(C2×D3)⋊Dic3—38no—
79C22×H(3,𝔽3)C22×C412yes—
80C22×H(3,𝔽3)C2×D412yes—
81C22×H(3,𝔽3)C2×D412yes—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×#272C22⋊C4116yescomplemented
84C2×#272C22×C414yes—
85C2×#272C2×D414yes—
86C2×#272C2×D414yes—
87C2×(C3⋊D3)⋊C3—216no—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3C22×C418yescomplemented
91((C3⋊D3)⋊C3)⋊C2—232no—
92C4×H(3,𝔽3)—22no—
93#1084—22no—
94(C3⋊Dic3)⋊C3—24no—
95C24×D3—91no—
96C24×D3—92no—
97C23×C3×C4—91no—
98C23×Dic3—91no—
99C22×C4×D3—32no—
100C22×C4×D3—61no—
101C22×C4×D3—92no—
102C2×C3×C22⋊C4—92no—
103C2×#217—92no—
104C2×(C2×D3)⋊C4—32no—
105C2×(C2×D3)⋊C4—92no—
106C2×(C2×D3)⋊C4—98no—
107C2×(C2×D3)⋊C4—92no—
108C2×(C2×D3)⋊C4—92no—
109D3×C22⋊C4—916no—
110C23×C32—21no—
111C23×C32C22×D312yes—
112C22×C3⋊D3—24no—
113C22×C3⋊D3—24no—
114C22×C3⋊D3C4×D318yescomplemented
115C22×C3⋊D3C22×D314yes—
116C22×C3×D3—38no—
117C22×C3×D3—38no—
118C22×C3×D3—38no—
119C22×C3×D3—34no—
120C22×C3×D3—38no—
121C22×C3×D3—34no—
122C22×C3×D3—64no—
123C22×C3×D3—64no—
124C22×C3×D3—64no—
125C22×C3×D3—62no—
126C22×C3×D3—64no—
127C22×C3×D3—64no—
128C2×D32—332no—
129C2×D32—664no—
130C2×D32—616no—
131C2×D32—616no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—22no—
135C2×C32×C4C22×D314yescomplemented
136C2×C3⋊Dic3—24no—
137C2×C3⋊Dic3C22×D314yescomplemented
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C3×C4×D3—616no—
145C3×C4×D3—616no—
146D3×Dic3—616no—
147C22×C22⋊C4—271noSylow
148C2×H(3,𝔽3)C2×C22⋊C414yescomplemented
149C2×H(3,𝔽3)C23×C411yescharacteristic
150C2×H(3,𝔽3)C22×D411yescharacteristic
151C2×H(3,𝔽3)C22×D411yescharacteristic
152#272C2×C22⋊C418yescomplemented
153(C3⋊D3)⋊C3—28no—
154C24×C3—92no—
155C24×C3—91no—
156C23×D3—32no—
157C23×D3—32no—
158C23×D3—61no—
159C23×D3—92no—
160C23×D3—94no—
161C23×D3—98no—
162C23×D3—98no—
163C23×D3—916no—
164C23×D3—188no—
165C23×D3—188no—
166C23×D3—184no—
167C23×D3—184no—
168C22×C3×C4—32no—
169C22×C3×C4—61no—
170C22×C3×C4—92no—
171C22×C3×C4—91no—
172C22×C3×C4—94no—
173C22×C3×C4—184no—
174C22×C3×C4D3211yescharacteristic
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—61no—
178C22×Dic3—92no—
179C22×Dic3—94no—
180C22×Dic3—184no—
181C2×C4×D3—38no—
182C2×C4×D3—64no—
183C2×C4×D3—68no—
184C2×C4×D3—68no—
185C2×C4×D3—98no—
186C2×C4×D3—188no—
187C3×C22⋊C4—98no—
188#217—98no—
189(C2×D3)⋊C4—38no—
190(C2×D3)⋊C4—98no—
191(C2×D3)⋊C4—98no—
192(C2×D3)⋊C4—98no—
193C22×C32—21no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32C2×C4×D314yes—
198C22×C32D3×D414yes—
199C22×C32D3×D414yes—
200C22×C32C23×D312yes—
201C2×C3⋊D3—216no—
202C2×C3⋊D3—24no—
203C2×C3⋊D3—24no—
204C2×C3⋊D3C2×C4×D318yescomplemented
205C2×C3×D3—332no—
206C2×C3×D3—38no—
207C2×C3×D3—38no—
208C2×C3×D3—38no—
209C2×C3×D3—38no—
210C2×C3×D3—616no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—64no—
214C2×C3×D3—616no—
215C2×C3×D3—64no—
216C2×C3×D3—64no—
217D32—664no—
218C32×C4—24no—
219C32×C4—24no—
220C3⋊Dic3—24no—
221C3×Dic3—64no—
222C3×Dic3—64no—
223C3×Dic3—64no—
224C25—271no—
225C23×C4—91no—
226C23×C4—271no—
227C2×C22⋊C4—92no—
228C2×C22⋊C4—278no—
229C2×C22⋊C4—272no—
230H(3,𝔽3)C22×C22⋊C411yescharacteristic, complemented, Sylow
231C23×C3—32no—
232C23×C3—61no—
233C23×C3—94no—
234C23×C3—98no—
235C23×C3—92no—
236C23×C3—94no—
237C23×C3—94no—
238C23×C3—94no—
239C23×C3—184no—
240C23×C3—184no—
241C23×C3C2×D3211yescharacteristic
242C22×D3—34no—
243C22×D3—38no—
244C22×D3—38no—
245C22×D3—38no—
246C22×D3—34no—
247C22×D3—38no—
248C22×D3—62no—
249C22×D3—64no—
250C22×D3—64no—
251C22×D3—64no—
252C22×D3—64no—
253C22×D3—64no—
254C22×D3—98no—
255C22×D3—94no—
256C22×D3—916no—
257C22×D3—932no—
258C22×D3—184no—
259C22×D3—184no—
260C22×D3—1816no—
261C22×D3—1816no—
262C22×D3—1864no—
263C22×D3—188no—
264C22×D3—188no—
265C22×D3—1832no—
266C2×C3×C4—22no—
267C2×C3×C4—34no—
268C2×C3×C4—64no—
269C2×C3×C4—64no—
270C2×C3×C4—62no—
271C2×C3×C4—94no—
272C2×C3×C4—92no—
273C2×C3×C4—188no—
274C2×C3×C4—184no—
275C2×C3×C4—182no—
276C2×C3×C4C2×D3212yes—
277C2×Dic3—34no—
278C2×Dic3—34no—
279C2×Dic3—62no—
280C2×Dic3—64no—
281C2×Dic3—64no—
282C2×Dic3—64no—
283C2×Dic3—94no—
284C2×Dic3—184no—
285C2×Dic3—188no—
286C4×D3—616no—
287C4×D3—616no—
288C4×D3—1816no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—24no—
293C2×C32D3×C22⋊C418yescomplemented
294C2×C32C22×C4×D312yes—
295C2×C32C2×D3×D412yes—
296C2×C32C2×D3×D412yes—
297C3⋊D3—28no—
298C3×D3—316no—
299C3×D3—68no—
300C3×D3—68no—
301C24—91no—
302C24—92no—
303C24—274no—
304C24—278no—
305C24—544no—
306C24—544no—
307C22×C4—92no—
308C22×C4—91no—
309C22×C4—94no—
310C22×C4—184no—
311C22×C4—274no—
312C22×C4—544no—
313C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
314C22⋊C4—98no—
315C22⋊C4—278no—
316C22×C3—32no—
317C22×C3—34no—
318C22×C3—34no—
319C22×C3—34no—
320C22×C3—61no—
321C22×C3—62no—
322C22×C3—62no—
323C22×C3—62no—
324C22×C3—98no—
325C22×C3—94no—
326C22×C3—94no—
327C22×C3—94no—
328C22×C3—916no—
329C22×C3—184no—
330C22×C3—1816no—
331C22×C3—184no—
332C22×C3C4×D3212yes—
333C22×C3#380212yes—
334C22×C3#381112yes—
335C22×C3C22×D3211yescharacteristic
336C2×D3—38no—
337C2×D3—38no—
338C2×D3—38no—
339C2×D3—332no—
340C2×D3—38no—
341C2×D3—64no—
342C2×D3—616no—
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—616no—
348C2×D3—98no—
349C2×D3—184no—
350C2×D3—1816no—
351C2×D3—184no—
352C2×D3—1864no—
353C2×D3—1832no—
354C3×C4—22no—
355C3×C4—64no—
356C3×C4—64no—
357C3×C4—184no—
358C3×C4—182no—
359Dic3—64no—
360Dic3—64no—
361Dic3—64no—
362Dic3—184no—
363C32—21no—
364C32C2×D3×C22⋊C412yescomplemented
365C23—92no—
366C23—94no—
367C23—94no—
368C23—94no—
369C23—94no—
370C23—98no—
371C23—184no—
372C23—184no—
373C23—2716no—
374C23—548no—
375C23—5432no—
376C23—548no—
377C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C2×C4—22no—
379C2×C4—94no—
380C2×C4—92no—
381C2×C4—188no—
382C2×C4—184no—
383C2×C4—182no—
384C2×C4—548no—
385C2×C4C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
386C2×C3—32no—
387C2×C3—38no—
388C2×C3—32no—
389C2×C3—32no—
390C2×C3—61no—
391C2×C3—64no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—98no—
395C2×C3—188no—
396C2×C3#6392114yes—
397C2×C3C2×C4×D3211yescharacteristic
398C2×C3C2×#380211yescharacteristic
399C2×C3C2×#381111yescharacteristic
400D3—316no—
401D3—68no—
402D3—68no—
403D3—188no—
404C22—94no—
405C22—916no—
406C22—94no—
407C22—94no—
408C22—98no—
409C22—184no—
410C22—1816no—
411C22—184no—
412C22—5432no—
413C22C4×((C3⋊D3)⋊C3)⋊C212yes—
414C22#8865512yes—
415C22#8867712yes—
416C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C4—22no—
418C4—182no—
419C4—184no—
420C3—32no—
421C3—61no—
422C3C2×#6392111yescharacteristic
423C2—98no—
424C2—188no—
425C2#1172116114yesdirect product factor
426C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
427C2C2×#8865511yescharacteristic
428C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×((C3⋊D3)⋊C3)⋊C2C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×C22⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×#272C2semidirect
3. C22×#6199C2semidirect
4. C2×#3752⋊C3C2semidirect
5. C2×#88450C2semidirect
6. #11721161C2direct (N × H)
12. C22×((C3⋊D3)⋊C3)⋊C2C4semidirect
14. C22×C4×H(3,𝔽3)C22semidirect
15. C22×#1084C22semidirect
17. C2×C4×#272C22semidirect
18. C22×(C3⋊Dic3)⋊C3C22semidirect
20. C2×#6199C22semidirect
21. #3752⋊C3C22semidirect
22. #88450C22semidirect
27. C2×#3752D3semidirect
39. C22×(C3⋊D3)⋊C3C2×C4semidirect
44. C2×((C3⋊D3)⋊C3)⋊C2C2×C4semidirect
46. C2×C4×H(3,𝔽3)C23semidirect
48. C2×#1084C23semidirect
51. C2×(C3⋊Dic3)⋊C3C23semidirect
64. C22×C32×C4C2×D3semidirect
65. C22×C3⋊Dic3C2×D3semidirect
75. #3752C2×D3semidirect
83. C2×#272C22⋊C4semidirect
90. C2×(C3⋊D3)⋊C3C22×C4semidirect
114. C22×C3⋊D3C4×D3semidirect
135. C2×C32×C4C22×D3semidirect
137. C2×C3⋊Dic3C22×D3semidirect
148. C2×H(3,𝔽3)C2×C22⋊C4semidirect
152. #272C2×C22⋊C4semidirect
204. C2×C3⋊D3C2×C4×D3semidirect
230. H(3,𝔽3)C22×C22⋊C4semidirect
293. C2×C32D3×C22⋊C4semidirect
313. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
364. C32C2×D3×C22⋊C4semidirect
385. C2×C4C2×((C3⋊D3)⋊C3)⋊C2semidirect
425. C2#11721161direct (N × H)

Maximal subgroups

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

subgroupindex
C23×((C3⋊D3)⋊C3)⋊C22
C22×C4×#2722
C22×#61992
C2×#3752⋊C32
C2×#884502
#117211612
C2×D3×(C2×D3)⋊C43