Finite Group Databasebeta

← C2×#11721163 #13150884

Subgroups — C2×#11721163 #13150884

Order 1728.

2553 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 2553 conjugacy classes of subgroups — too many to draw. Only its 219 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, complemented
2C22×C4⋊#272C211yesmaximal, characteristic, complemented
3C22×#6200C212yesmaximal, complemented
4C2×#3752⋊C3C212yesmaximal, complemented
5C2×#88451C211yesmaximal, characteristic, complemented
6#11721163C218yesmaximal, direct product factor
7C2×#63922—32nomaximal
8C23×#272C2211yescharacteristic
9C23×#272C2211yescharacteristic, complemented
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×((C3⋊D3)⋊C3)⋊C2—24no—
12C22×((C3⋊D3)⋊C3)⋊C2—28no—
13C22×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
14C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×C4⋊#272—24no—
16C2×C4⋊#272C2214yescomplemented
17C22×(C3⋊Dic3)⋊C3C2212yescomplemented
18C2×#6200—28no—
19C2×#6200C2218yescomplemented
20#3752⋊C3C2218yescomplemented
21#88451C2214yescomplemented
22C23×D32—32no—
23C22×C3×D12—32no—
24C22×C3×D12—61no—
25C22×#626—32no—
26C22×#625—32no—
27C2×#3752D312yescomplemented
28C2×C3×(C2×D3)⋊C4—32no—
29C2×(C2×D3)⋊Dic3—32no—
30#63922—316no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×#272—24no—
33C22×#272—22no—
34C22×#272—24no—
35C22×#272D414yescomplemented
36C22×#272C2312yes—
37C22×#272C2312yescomplemented
38C22×(C3⋊D3)⋊C3—24no—
39C22×(C3⋊D3)⋊C3—24no—
40C22×(C3⋊D3)⋊C3D418yescomplemented
41C22×(C3⋊D3)⋊C3C2314yes—
42C2×((C3⋊D3)⋊C3)⋊C2—232no—
43C2×((C3⋊D3)⋊C3)⋊C2—28no—
44C2×((C3⋊D3)⋊C3)⋊C2—216no—
45C2×C4×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2312yescomplemented
47C4⋊#272—28no—
48C2×(C3⋊Dic3)⋊C3—24no—
49C2×(C3⋊Dic3)⋊C3C2314yescomplemented
50#6200—216no—
51C2×#937—91no—
52C2×#882—92no—
53C23×C3⋊D3C2×D312yes—
54C23×C3×D3—32no—
55C23×C3×D3—32no—
56C23×C3×D3—32no—
57C23×C3×D3—61no—
58C23×C3×D3—61no—
59C22×D32—38no—
60C22×D32—68no—
61C22×D32—68no—
62C22×D32—68no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D312yescomplemented
65C22×C3⋊Dic3C2×D312yescomplemented
66C22×C3×Dic3—32no—
67C2×C3×D12—38no—
68C2×C3×D12—68no—
69C2×C3×D12—64no—
70C2×C3×D12—68no—
71C2×#626—38no—
72C2×#626—68no—
73C2×#625—38no—
74C2×#625—68no—
75#3752C2×D318yescomplemented
76C3×(C2×D3)⋊C4—38no—
77(C2×D3)⋊Dic3—38no—
78C22×H(3,𝔽3)C2×D414yes—
79C22×H(3,𝔽3)C2×D412yes—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×#272—28no—
82C2×#272—24no—
83C2×#272—22no—
84C2×#272—24no—
85C2×#272—44no—
86C2×#272C2×D414yescomplemented
87C2×(C3⋊D3)⋊C3—216no—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3C2×D418yescomplemented
91((C3⋊D3)⋊C3)⋊C2—232no—
92C4×H(3,𝔽3)—22no—
93(C3⋊Dic3)⋊C3—24no—
94C24×D3—91no—
95C24×D3—92no—
96C22×C3×D4—91no—
97C22×#67—92no—
98C22×#67—92no—
99C22×D12—32no—
100C22×D12—61no—
101C22×D12—92no—
102C2×C3×C22⋊C4—92no—
103C2×#217—91no—
104#937—98no—
105C2×(C2×D3)⋊C4—32no—
106C2×(C2×D3)⋊C4—92no—
107C2×(C2×D3)⋊C4—92no—
108#882—916no—
109C23×C32—21no—
110C23×C32C22×D312yes—
111C22×C3⋊D3—24no—
112C22×C3⋊D3—24no—
113C22×C3⋊D3D1218yescomplemented
114C22×C3⋊D3C22×D314yes—
115C22×C3×D3—38no—
116C22×C3×D3—38no—
117C22×C3×D3—34no—
118C22×C3×D3—34no—
119C22×C3×D3—34no—
120C22×C3×D3—64no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—64no—
124C22×C3×D3—64no—
125C22×C3×D3—64no—
126C22×C3×D3—64no—
127C22×C3×D3—64no—
128C22×C3×D3—64no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—124no—
132C22×C3×D3—124no—
133C2×D32—616no—
134C2×D32—616no—
135C2×D32—664no—
136C2×D32—616no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—22no—
140C2×C32×C4C22×D314yescomplemented
141C2×C3⋊Dic3—24no—
142C2×C3⋊Dic3C22×D314yescomplemented
143C2×C3×Dic3—34no—
144C2×C3×Dic3—64no—
145C3×D12—616no—
146C3×D12—616no—
147#626—616no—
148#625—616no—
149C2×#120—271noSylow
150C2×H(3,𝔽3)#12014yescomplemented
151C2×H(3,𝔽3)C22×D412yes—
152C2×H(3,𝔽3)C22×D411yescharacteristic
153#272—24no—
154#272—42no—
155(C3⋊D3)⋊C3—28no—
156C24×C3—92no—
157C24×C3—91no—
158C24×C3—91no—
159C23×D3—32no—
160C23×D3—32no—
161C23×D3—32no—
162C23×D3—61no—
163C23×D3—61no—
164C23×D3—92no—
165C23×D3—94no—
166C23×D3—98no—
167C23×D3—188no—
168C23×D3—188no—
169C23×D3—184no—
170C23×D3—188no—
171C23×D3—188no—
172C2×C3×D4—94no—
173C2×C3×D4—184no—
174C22×C3×C4—32no—
175C22×C3×C4—61no—
176C22×C3×C4—92no—
177C22×C3×C4D3211yescharacteristic
178C2×#67—98no—
179C2×#67—98no—
180C2×#67—188no—
181C2×#67—188no—
182C22×Dic3—32no—
183C22×Dic3—92no—
184C2×D12—38no—
185C2×D12—64no—
186C2×D12—68no—
187C2×D12—68no—
188C2×D12—98no—
189C2×D12—188no—
190C3×C22⋊C4—98no—
191#217—94no—
192(C2×D3)⋊C4—38no—
193(C2×D3)⋊C4—98no—
194(C2×D3)⋊C4—98no—
195C22×C32—21no—
196C22×C32—24no—
197C22×C32—22no—
198C22×C32C2×D1214yes—
199C22×C32D3×D414yes—
200C22×C32D3×D414yes—
201C22×C32C23×D312yes—
202C2×C3⋊D3—216no—
203C2×C3⋊D3—24no—
204C2×C3⋊D3—24no—
205C2×C3⋊D3C2×D1218yescomplemented
206C2×C3×D3—38no—
207C2×C3×D3—38no—
208C2×C3×D3—64no—
209C2×C3×D3—616no—
210C2×C3×D3—616no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—64no—
219C2×C3×D3—124no—
220C2×C3×D3—128no—
221C2×C3×D3—124no—
222C2×C3×D3—128no—
223C2×C3×D3—128no—
224D32—664no—
225C32×C4—24no—
226C32×C4—24no—
227C3⋊Dic3—24no—
228C3×Dic3—64no—
229C25—271no—
230C22×D4—91no—
231C22×D4—272no—
232#120—278no—
233C2×C22⋊C4—92no—
234C2×C22⋊C4—271no—
235H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
236C23×C3—32no—
237C23×C3—61no—
238C23×C3—94no—
239C23×C3—98no—
240C23×C3—92no—
241C23×C3—94no—
242C23×C3—92no—
243C23×C3—184no—
244C23×C3—184no—
245C23×C3—184no—
246C23×C3—184no—
247C23×C3—182no—
248C23×C3C2×D3211yescharacteristic
249C22×D3—34no—
250C22×D3—38no—
251C22×D3—34no—
252C22×D3—38no—
253C22×D3—34no—
254C22×D3—62no—
255C22×D3—64no—
256C22×D3—62no—
257C22×D3—64no—
258C22×D3—64no—
259C22×D3—64no—
260C22×D3—64no—
261C22×D3—64no—
262C22×D3—64no—
263C22×D3—64no—
264C22×D3—64no—
265C22×D3—98no—
266C22×D3—94no—
267C22×D3—124no—
268C22×D3—124no—
269C22×D3—184no—
270C22×D3—184no—
271C22×D3—1816no—
272C22×D3—1816no—
273C22×D3—188no—
274C22×D3—1832no—
275C22×D3—1816no—
276C22×D3—1816no—
277C22×D3—1864no—
278C3×D4—188no—
279C2×C3×C4—22no—
280C2×C3×C4—34no—
281C2×C3×C4—64no—
282C2×C3×C4—64no—
283C2×C3×C4—62no—
284C2×C3×C4—94no—
285C2×C3×C4—184no—
286C2×C3×C4C2×D3212yes—
287#67—1816no—
288#67—1816no—
289C2×Dic3—34no—
290C2×Dic3—64no—
291C2×Dic3—94no—
292C2×Dic3—184no—
293D12—616no—
294D12—616no—
295D12—1816no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32—24no—
299C2×C32#88218yescomplemented
300C2×C32C22×D1212yes—
301C2×C32C2×D3×D412yes—
302C2×C32C2×D3×D412yes—
303C3⋊D3—28no—
304C3×D3—68no—
305C3×D3—68no—
306C3×D3—124no—
307C3×D3—124no—
308C3×D3—124no—
309C24—92no—
310C24—91no—
311C24—91no—
312C24—274no—
313C24—548no—
314C24—544no—
315C2×D4—94no—
316C2×D4—184no—
317C2×D4—278no—
318C2×D4—548no—
319C22×C4—92no—
320C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
321C22⋊C4—98no—
322C22⋊C4—274no—
323C22×C3—32no—
324C22×C3—34no—
325C22×C3—34no—
326C22×C3—34no—
327C22×C3—61no—
328C22×C3—64no—
329C22×C3—62no—
330C22×C3—98no—
331C22×C3—94no—
332C22×C3—184no—
333C22×C3—1816no—
334C22×C3—184no—
335C22×C3—184no—
336C22×C3—188no—
337C22×C3—184no—
338C22×C3—182no—
339C22×C3—364no—
340C22×C3D3×D1214yes—
341C22×C3#381112yes—
342C22×C3C22×D3211yescharacteristic
343C2×D3—38no—
344C2×D3—38no—
345C2×D3—64no—
346C2×D3—616no—
347C2×D3—64no—
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—64no—
351C2×D3—616no—
352C2×D3—64no—
353C2×D3—64no—
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—98no—
357C2×D3—128no—
358C2×D3—124no—
359C2×D3—124no—
360C2×D3—128no—
361C2×D3—128no—
362C2×D3—184no—
363C2×D3—1816no—
364C2×D3—184no—
365C2×D3—1832no—
366C2×D3—1864no—
367C3×C4—22no—
368C3×C4—64no—
369C3×C4—64no—
370C3×C4—184no—
371Dic3—64no—
372Dic3—184no—
373C32—21no—
374C32C2×#88212yescomplemented
375C23—94no—
376C23—92no—
377C23—98no—
378C23—94no—
379C23—92no—
380C23—184no—
381C23—184no—
382C23—184no—
383C23—184no—
384C23—182no—
385C23—548no—
386C23—5416no—
387C23—5432no—
388C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
389D4—188no—
390D4—5416no—
391C2×C4—22no—
392C2×C4—94no—
393C2×C4—184no—
394C2×C4C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
395C2×C3—32no—
396C2×C3—38no—
397C2×C3—32no—
398C2×C3—32no—
399C2×C3—61no—
400C2×C3—64no—
401C2×C3—62no—
402C2×C3—188no—
403C2×C3—184no—
404C2×C3—362no—
405C2×C3#6392314yes—
406C2×C3C2×D3×D1212yes—
407C2×C3C2×#381111yescharacteristic
408D3—68no—
409D3—68no—
410D3—124no—
411D3—124no—
412D3—124no—
413D3—188no—
414C22—94no—
415C22—98no—
416C22—184no—
417C22—188no—
418C22—1816no—
419C22—184no—
420C22—184no—
421C22—182no—
422C22—184no—
423C22—364no—
424C22—5432no—
425C22#8865414yes—
426C22#8867712yes—
427C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C4—22no—
429C4—184no—
430C3—32no—
431C3—61no—
432C3C2×#6392311yescharacteristic
433C2—184no—
434C2—188no—
435C2—362no—
436C2#1172116314yesdirect product factor
437C2C2×#8865412yes—
438C2C2×#8867711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#120
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C23×((C3⋊D3)⋊C3)⋊C2C2semidirect
2. C22×C4⋊#272C2semidirect
3. C22×#6200C2semidirect
4. C2×#3752⋊C3C2semidirect
5. C2×#88451C2semidirect
6. #11721163C2direct (N × H)
9. C23×#272C22semidirect
13. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
14. C22×C4×H(3,𝔽3)C22semidirect
16. C2×C4⋊#272C22semidirect
17. C22×(C3⋊Dic3)⋊C3C22semidirect
19. C2×#6200C22semidirect
20. #3752⋊C3C22semidirect
21. #88451C22semidirect
27. C2×#3752D3semidirect
35. C22×#272D4semidirect
37. C22×#272C23semidirect
40. C22×(C3⋊D3)⋊C3D4semidirect
46. C2×C4×H(3,𝔽3)C23semidirect
49. C2×(C3⋊Dic3)⋊C3C23semidirect
64. C22×C32×C4C2×D3semidirect
65. C22×C3⋊Dic3C2×D3semidirect
75. #3752C2×D3semidirect
86. C2×#272C2×D4semidirect
90. C2×(C3⋊D3)⋊C3C2×D4semidirect
113. C22×C3⋊D3D12semidirect
140. C2×C32×C4C22×D3semidirect
142. C2×C3⋊Dic3C22×D3semidirect
150. C2×H(3,𝔽3)#120semidirect
205. C2×C3⋊D3C2×D12semidirect
235. H(3,𝔽3)C2×#120semidirect
299. C2×C32#882semidirect
320. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
374. C32C2×#882semidirect
394. C2×C4C2×((C3⋊D3)⋊C3)⋊C2semidirect
436. C2#11721163direct (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×#62002
C2×#3752⋊C32
C2×#884512
#117211632
C2×#639223