Finite Group Databasebeta

← C2×#11721071 #13150647

Subgroups — C2×#11721071 #13150647

Order 1728.

1207 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1207 conjugacy classes of subgroups — too many to draw. Only its 179 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×#88740C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4C2×#88438C212yesmaximal, complemented
5#11721071C218yesmaximal, direct product factor
6C2×#88436C212yesmaximal, complemented
7C2×#63825—31nomaximal
8C2×#63823—31nomaximal
9C23×(C3⋊D3)⋊C3—21no—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12#88740—22no—
13#88740C2214yescomplemented
14C2×C4⋊#272C2211yescharacteristic, complemented
15C2×C4×#272—21no—
16C2×#6229C2211yescharacteristic, complemented
17C2×C4⋊((C3⋊D3)⋊C3)—22no—
18C2×C4⋊((C3⋊D3)⋊C3)—21no—
19C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
20C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
21#3748⋊C3C2214yescomplemented
22C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
23#88438C2218yescomplemented
24#88436C2218yescomplemented
25C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
26C2×C3×#235—31no—
27C2×C3×#235—31no—
28C2×C3×#235—61no—
29C22×C3×D12—31no—
30C2×#3748D311yescharacteristic, complemented
31C2×C3×#208—31no—
32C2×#3717—32no—
33C2×#3715—32no—
34#63825—38no—
35C2×#3714—32no—
36#63823—38no—
37C2×#3713—32no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—44no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)D412yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)C2312yes—
47#6235—24no—
48C2×#1084—21no—
49C4⋊#272—21no—
50C4⋊#272C2312yescomplemented
51C4×#272—24no—
52#6229—21no—
53#6229C2312yescomplemented
54C4⋊((C3⋊D3)⋊C3)—22no—
55C4⋊((C3⋊D3)⋊C3)—24no—
56C4⋊((C3⋊D3)⋊C3)C2314yes—
57(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
58C2×#949—91no—
59C2×#932—91no—
60C2×#908—91no—
61C23×C3⋊D3—21no—
62C23×C3×D3—61no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C2×C4⋊(C3⋊D3)—22no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)C2×D312yes—
69C2×C4⋊(C3⋊D3)C2×D314yescomplemented
70C2×C3×#67—62no—
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C3×#235—34no—
74C3×#235—34no—
75C3×#235—64no—
76C3×#235—64no—
77C3×#235—62no—
78C3×#235—62no—
79C2×C3×D12—31no—
80C2×C3×D12—32no—
81C2×C3×D12—34no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C2×C3×D12—61no—
85C2×C3×D12—62no—
86C2×C3×C4×D3—62no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—61no—
92#3748C2×D314yescomplemented
93C2×C3⋊(C3⋊C8)C2×D312yescomplemented
94C3×#208—34no—
95C2×C3×C3⋊C8—32no—
96#3717—38no—
97#3715—38no—
98#3714—38no—
99#3713—38no—
100C22×H(3,𝔽3)—22no—
101C22×H(3,𝔽3)C2×D412yes—
102C22×H(3,𝔽3)C2×D411yescharacteristic
103C2×#272—22no—
104C2×#272—42no—
105C2×(C3⋊D3)⋊C3—24no—
106C2×(C3⋊D3)⋊C3—48no—
107C2×(C3⋊D3)⋊C3—44no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D412yes—
110C4×H(3,𝔽3)C2411yescharacteristic
111#1084—22no—
112C2×C3×#41—91no—
113C22×C3×D4—91no—
114C2×#235—31no—
115C2×#235—31no—
116C2×#235—61no—
117C22×D12—31no—
118C22×D12—91no—
119C2×C3×M16—91no—
120#949—98no—
121C2×#215—92no—
122C2×#214—92no—
123#932—98no—
124C2×#213—92no—
125C2×#213—92no—
126C2×#208—31no—
127C2×#208—91no—
128#908—98no—
129C2×D24—92no—
130C2×#204—92no—
131C23×C32—21no—
132C23×C32D1211yescharacteristic
133C23×C32#6711yescharacteristic
134C22×C3⋊D3—22no—
135C22×C3⋊D3—24no—
136C22×C3⋊D3—44no—
137C22×C3×D3—62no—
138C22×C3×D3—64no—
139C22×C3×D3—62no—
140C22×C3×D3—61no—
141C22×C3×D3—61no—
142C22×C3×D3—124no—
143C2×C32×C4—21no—
144C2×C32×C4—22no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4D1211yescharacteristic
148C2×C32×C4D1212yes—
149C2×C32×C4#6711yescharacteristic
150C2×C32×C4#6712yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D312yes—
155C4⋊(C3⋊D3)—22no—
156C4⋊(C3⋊D3)—24no—
157C4⋊(C3⋊D3)C22×D314yes—
158C3×#67—68no—
159C3×#67—64no—
160C3×#67—64no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C3×D12—34no—
165C3×D12—32no—
166C3×D12—32no—
167C3×D12—62no—
168C3×D12—62no—
169C3×D12—62no—
170C3×D12—64no—
171C3×D12—61no—
172C3×D12—61no—
173C3×C4×D3—68no—
174C3×C4×D3—64no—
175C3×C4×D3—64no—
176C3×Dic6—32no—
177C3×Dic6—32no—
178C3×Dic6—62no—
179C3×Dic6—62no—
180C3×Dic6—61no—
181C3×Dic6—61no—
182C3⋊(C3⋊C8)C22×D314yescomplemented
183C3×C3⋊C8—34no—
184C2×AGL(1,ℤ/8)—271noSylow
185C2×H(3,𝔽3)—22no—
186C2×H(3,𝔽3)AGL(1,ℤ/8)12yescomplemented
187C2×H(3,𝔽3)C22×D411yescharacteristic
188#272—42no—
189(C3⋊D3)⋊C3—44no—
190C24×C3—181no—
191C23×D3—61no—
192C23×D3—181no—
193C3×#41—94no—
194C3×#41—182no—
195C2×C3×Q8—91no—
196C2×C3×D4—92no—
197C2×C3×D4—94no—
198C2×C3×D4—91no—
199C2×C3×D4—181no—
200C2×C3×D4—182no—
201C2×C3×D4—181no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—61no—
205C22×C3×C4—181no—
206C22×C3×C4D3211yescharacteristic
207C2×#67—61no—
208C2×#67—61no—
209C2×#67—62no—
210#235—34no—
211#235—34no—
212#235—64no—
213#235—62no—
214#235—62no—
215#235—64no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—32no—
219C2×D12—34no—
220C2×D12—61no—
221C2×D12—61no—
222C2×D12—62no—
223C2×D12—92no—
224C2×D12—94no—
225C2×D12—181no—
226C2×D12—182no—
227C2×C4×D3—61no—
228C2×C4×D3—61no—
229C2×C4×D3—62no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—61no—
233C3×M16—94no—
234C2×C3×C8—92no—
235#215—98no—
236#214—98no—
237#213—98no—
238#213—98no—
239#208—34no—
240#208—94no—
241C2×C3⋊C8—32no—
242C2×C3⋊C8—92no—
243D24—98no—
244#204—98no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—42no—
250C22×C32C2×D1212yes—
251C22×C32C2×D1211yescharacteristic
252C22×C32C2×#6712yes—
253C22×C32C2×#6711yescharacteristic
254C2×C3⋊D3—24no—
255C2×C3⋊D3—48no—
256C2×C3⋊D3—44no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—128no—
262C2×C3×D3—124no—
263C2×C3×D3—124no—
264C2×C3×D3—122no—
265C2×C3×D3—122no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—22no—
269C32×C4C2×D1211yescharacteristic
270C32×C4C2×D1212yes—
271C32×C4C2×#6711yescharacteristic
272C32×C4C2×#6712yes—
273C32×C4C23×D311yescharacteristic
274C32×C4C23×D311yescharacteristic
275C3×Dic3—64no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C2×#41—91no—
279C22×D4—91no—
280AGL(1,ℤ/8)—278no—
281C2×SD16—272no—
282C2×D8—272no—
283C2×M16—91no—
284H(3,𝔽3)C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
285C23×C3—31no—
286C23×C3—31no—
287C23×C3—61no—
288C23×C3—184no—
289C23×C3—182no—
290C23×C3—181no—
291C23×C3—364no—
292C23×C3#62611yescharacteristic
293C22×D3—61no—
294C22×D3—61no—
295C22×D3—62no—
296C22×D3—64no—
297C22×D3—62no—
298C22×D3—124no—
299C22×D3—184no—
300C22×D3—182no—
301C22×D3—364no—
302C3×Q8—92no—
303C3×Q8—181no—
304C3×D4—94no—
305C3×D4—92no—
306C3×D4—182no—
307C3×D4—184no—
308C3×D4—184no—
309C3×D4—181no—
310C2×C3×C4—31no—
311C2×C3×C4—32no—
312C2×C3×C4—31no—
313C2×C3×C4—32no—
314C2×C3×C4—31no—
315C2×C3×C4—32no—
316C2×C3×C4—31no—
317C2×C3×C4—32no—
318C2×C3×C4—61no—
319C2×C3×C4—62no—
320C2×C3×C4—61no—
321C2×C3×C4—62no—
322C2×C3×C4—181no—
323C2×C3×C4—184no—
324C2×C3×C4#62611yescharacteristic
325C2×C3×C4#62612yes—
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3212yes—
328#67—64no—
329#67—64no—
330#67—68no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—62no—
334D12—32no—
335D12—32no—
336D12—34no—
337D12—62no—
338D12—61no—
339D12—61no—
340D12—62no—
341D12—64no—
342D12—62no—
343D12—94no—
344D12—182no—
345D12—184no—
346C4×D3—64no—
347C4×D3—64no—
348C4×D3—68no—
349Dic6—32no—
350Dic6—32no—
351Dic6—62no—
352Dic6—61no—
353Dic6—61no—
354Dic6—62no—
355C3×C8—94no—
356C3⋊C8—34no—
357C3⋊C8—94no—
358C2×C32—22no—
359C2×C32—22no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32—42no—
363C2×C32#90812yescomplemented
364C2×C32#93212yescomplemented
365C2×C32C22×D1211yescharacteristic
366C2×C32C22×#6711yescharacteristic
367C3⋊D3—44no—
368C3×D3—124no—
369C3×D3—124no—
370C3×D3—122no—
371C3×D3—122no—
372C24—181no—
373#41—94no—
374#41—182no—
375C2×Q8—91no—
376C2×D4—91no—
377C2×D4—92no—
378C2×D4—94no—
379C2×D4—181no—
380C2×D4—182no—
381C2×D4—181no—
382C22×C4—181no—
383C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
384SD16—278no—
385D8—278no—
386M16—94no—
387C2×C8—92no—
388C22×C3—22no—
389C22×C3—32no—
390C22×C3—32no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—62no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—61no—
397C22×C3—122no—
398C22×C3—184no—
399C22×C3—182no—
400C22×C3—364no—
401C22×C3—368no—
402C22×C3—362no—
403C22×C3C2×#62612yes—
404C22×C3C2×#62611yescharacteristic
405C2×D3—62no—
406C2×D3—62no—
407C2×D3—64no—
408C2×D3—64no—
409C2×D3—128no—
410C2×D3—122no—
411C2×D3—122no—
412C2×D3—124no—
413C2×D3—124no—
414C2×D3—184no—
415C2×D3—364no—
416C2×D3—368no—
417C3×C4—31no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—32no—
422C3×C4—32no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—62no—
426C3×C4—182no—
427C3×C4C2×#62612yes—
428C3×C4C2×#62611yescharacteristic
429C3×C4C22×D3211yescharacteristic
430Dic3—62no—
431Dic3—62no—
432Dic3—64no—
433C32—21no—
434C32C2×#90811yescharacteristic, complemented
435C32C2×#93211yescharacteristic, complemented
436C23—184no—
437C23—181no—
438C23—182no—
439C23—364no—
440C23#620011yescharacteristic
441Q8—92no—
442Q8—181no—
443D4—92no—
444D4—94no—
445D4—184no—
446D4—184no—
447D4—182no—
448D4—181no—
449C2×C4—181no—
450C2×C4—184no—
451C2×C4#620012yes—
452C2×C4#620011yescharacteristic
453C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
454C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
455C8—94no—
456C2×C3—22no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—32no—
460C2×C3—32no—
461C2×C3—62no—
462C2×C3—62no—
463C2×C3—61no—
464C2×C3—62no—
465C2×C3—122no—
466C2×C3—364no—
467C2×C3—362no—
468C2×C3#6383012yes—
469C2×C3C22×#62611yescharacteristic
470D3—122no—
471D3—122no—
472D3—124no—
473D3—124no—
474D3—364no—
475C22—22no—
476C22—182no—
477C22—184no—
478C22—368no—
479C22—364no—
480C22—362no—
481C22C2×#620012yes—
482C22C2×#620011yescharacteristic
483C4—182no—
484C4C2×#620012yes—
485C4C2×#620011yescharacteristic
486C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
487C3—31no—
488C3—31no—
489C3—61no—
490C3C2×#6383011yescharacteristic
491C2—22no—
492C2—364no—
493C2—362no—
494C2#1172107112yesdirect product factor
495C2C22×#620011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×AGL(1,ℤ/8)
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88740C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C2×#3748⋊C3C2semidirect
4. C2×#88438C2semidirect
5. #11721071C2direct (N × H)
6. C2×#88436C2semidirect
13. #88740C22semidirect
14. C2×C4⋊#272C22semidirect
16. C2×#6229C22semidirect
20. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. #3748⋊C3C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #88438C22semidirect
24. #88436C22semidirect
25. C22×C4⋊(C3⋊D3)D3semidirect
30. C2×#3748D3semidirect
50. C4⋊#272C23semidirect
53. #6229C23semidirect
57. (C3⋊(C3⋊C8))⋊C3C23semidirect
69. C2×C4⋊(C3⋊D3)C2×D3semidirect
92. #3748C2×D3semidirect
93. C2×C3⋊(C3⋊C8)C2×D3semidirect
182. C3⋊(C3⋊C8)C22×D3semidirect
186. C2×H(3,𝔽3)AGL(1,ℤ/8)semidirect
284. H(3,𝔽3)C2×AGL(1,ℤ/8)semidirect
363. C2×C32#908semidirect
364. C2×C32#932semidirect
434. C32C2×#908semidirect
435. C32C2×#932semidirect
494. C2#11721071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887402
C22×C4⋊((C3⋊D3)⋊C3)2
C2×#3748⋊C32
C2×#884382
#117210712
C2×#884362
C2×#638253
C2×#638233