Finite Group Databasebeta

← C2×#11721190 #13151389

Subgroups — C2×#11721190 #13151389

Order 1728.

1419 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×#272C211yesmaximal, characteristic, complemented
2C2×#88655C211yesmaximal, characteristic, complemented
3C2×#3753⋊C3C212yesmaximal, complemented
4#11721190C218yesmaximal, direct product factor
5C2×#88436C212yesmaximal, complemented
6C2×#88434C211yesmaximal, characteristic, complemented
7C2×D3×#213—32nomaximal
8C23×#272—21no—
9C22×((C3⋊D3)⋊C3)⋊C2—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×#6235—21no—
12D4×#272—22no—
13D4×#272C2214yescomplemented
14C2×C4⋊#272C2211yescharacteristic, complemented
15C2×C4×#272C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
17C2×#6198—21no—
18#88655—22no—
19#88655C2214yescomplemented
20#3753⋊C3C2218yescomplemented
21C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
22#88436C2218yescomplemented
23#88434C2214yescomplemented
24C2×C3×D3×D4—32no—
25C2×C3×D3×D4—61no—
26C2×D3×D12—32no—
27C2×#3753D312yescomplemented
28C2×C3×#213—32no—
29D3×#213—316no—
30C2×#3714—32no—
31C2×#3713—32no—
32C2×D3×C3⋊C8—32no—
33C23×H(3,𝔽3)—21no—
34C22×#272—21no—
35C22×#272—24no—
36C22×#272—44no—
37C22×#272D411yescharacteristic
38C22×(C3⋊D3)⋊C3—22no—
39C2×((C3⋊D3)⋊C3)⋊C2—24no—
40C2×((C3⋊D3)⋊C3)⋊C2—44no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2312yescomplemented
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#6235—24no—
45C2×#1084D411yescharacteristic
46C4⋊#272—21no—
47C4⋊#272C2312yescomplemented
48C4×#272D412yes—
49C4×#272C2312yes—
50C4⋊((C3⋊D3)⋊C3)—22no—
51C4⋊((C3⋊D3)⋊C3)C2314yescomplemented
52#6198—24no—
53(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
54C22×#213—91no—
55C2×D3×D8—92no—
56C23×C3×D3—62no—
57C23×C3×D3—62no—
58C22×D32—62no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D312yescomplemented
61C2×C4⋊(C3⋊D3)C2×D312yescomplemented
62C2×C3×#67—62no—
63C2×C3×#67—62no—
64C3×D3×D4—38no—
65C3×D3×D4—64no—
66C3×D3×D4—64no—
67C3×D3×D4—64no—
68C2×C3×D12—32no—
69C2×C3×D12—32no—
70C2×C3×D12—61no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—61no—
73C2×#626—62no—
74D3×D12—38no—
75D3×D12—64no—
76#3753C2×D318yescomplemented
77C2×C3⋊(C3⋊C8)C2×D312yescomplemented
78C3×#213—38no—
79C2×C3×C3⋊C8—32no—
80#3714—38no—
81#3713—38no—
82D3×C3⋊C8—38no—
83C22×H(3,𝔽3)—22no—
84C22×H(3,𝔽3)—42no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×#272—22no—
87C2×#272—42no—
88C2×#272—48no—
89C2×#272D814yescomplemented
90C2×#272C2×D412yes—
91C2×(C3⋊D3)⋊C3—24no—
92C2×(C3⋊D3)⋊C3—44no—
93((C3⋊D3)⋊C3)⋊C2—48no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)C2411yescharacteristic
96#1084C2×D412yes—
97C22×C3×D4—91no—
98C2×D3×D4—32no—
99C2×D3×D4—61no—
100C2×D3×D4—92no—
101C22×D12—91no—
102C2×C3×D8—92no—
103C2×#213—32no—
104C2×#213—92no—
105C2×#213—92no—
106C2×#213—98no—
107C2×#213—92no—
108C22×C3⋊C8—91no—
109D3×D8—916no—
110C2×D24—92no—
111C2×D3×C8—92no—
112C23×C32—22no—
113C23×C32—22no—
114C22×C3⋊D3—22no—
115C22×C3×D3—32no—
116C22×C3×D3—61no—
117C22×C3×D3—68no—
118C22×C3×D3—62no—
119C22×C3×D3—68no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—128no—
123C22×C3×D3—128no—
124C2×D32—68no—
125C2×D32—128no—
126C32×D4—22no—
127C32×D4—22no—
128C32×D4—22no—
129C32×D4C22×D314yescomplemented
130C2×C32×C4—21no—
131C2×C32×C4C22×D312yes—
132C4⋊(C3⋊D3)—22no—
133C4⋊(C3⋊D3)C22×D314yescomplemented
134C3×#67—68no—
135C3×#67—68no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—61no—
138C3×D12—34no—
139C3×D12—34no—
140C3×D12—62no—
141C3×D12—62no—
142C3×D12—62no—
143C3×D12—62no—
144C3×C4×D3—34no—
145C3×C4×D3—34no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148#626—68no—
149C3⋊(C3⋊C8)C22×D314yescomplemented
150C3×C3⋊C8—34no—
151C22×D8—271noSylow
152C2×H(3,𝔽3)—42no—
153C2×H(3,𝔽3)C2×D812yescomplemented
154C2×H(3,𝔽3)C22×D411yescharacteristic
155#272—42no—
156#272C2×D814yescomplemented
157(C3⋊D3)⋊C3—44no—
158C24×C3—181no—
159C23×D3—62no—
160C23×D3—62no—
161C23×D3—182no—
162C23×D3—181no—
163C2×C3×D4—32no—
164C2×C3×D4—61no—
165C2×C3×D4—92no—
166C2×C3×D4—94no—
167C2×C3×D4—91no—
168C2×C3×D4—182no—
169C2×C3×D4—181no—
170C2×C3×D4D3211yescharacteristic
171C22×C3×C4—91no—
172C2×#67—62no—
173C2×#67—62no—
174C2×#67—182no—
175D3×D4—38no—
176D3×D4—64no—
177D3×D4—64no—
178D3×D4—64no—
179D3×D4—98no—
180D3×D4—184no—
181C2×D12—32no—
182C2×D12—32no—
183C2×D12—61no—
184C2×D12—92no—
185C2×D12—94no—
186C2×D12—182no—
187C2×D12—181no—
188C2×C4×D3—32no—
189C2×C4×D3—61no—
190C3×D8—98no—
191C2×C3×C8—92no—
192#213—38no—
193#213—98no—
194#213—98no—
195#213—98no—
196C2×C3⋊C8—32no—
197C2×C3⋊C8—92no—
198C2×C3⋊C8—94no—
199D24—98no—
200D3×C8—98no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—21no—
204C22×C32—44no—
205C22×C32—44no—
206C22×C32D3×D412yes—
207C2×C3⋊D3—24no—
208C2×C3⋊D3—44no—
209C2×C3×D3—38no—
210C2×C3×D3—34no—
211C2×C3×D3—64no—
212C2×C3×D3—62no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—1216no—
217C2×C3×D3—1216no—
218C2×C3×D3—124no—
219C2×C3×D3—124no—
220C2×C3×D3—124no—
221D32—1216no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4D3×D412yes—
225C32×C4C23×D312yes—
226C3×Dic3—34no—
227C3×Dic3—62no—
228C22×D4—91no—
229C22×D4—271no—
230C2×D8—92no—
231C2×D8—272no—
232C2×D8—278no—
233C22×C8—271no—
234H(3,𝔽3)C22×D811yescharacteristic, complemented, Sylow
235C23×C3—21no—
236C23×C3—62no—
237C23×C3—62no—
238C23×C3—91no—
239C23×C3—184no—
240C23×C3—182no—
241C23×C3—181no—
242C23×C3—364no—
243C22×D3—32no—
244C22×D3—61no—
245C22×D3—68no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—68no—
249C22×D3—62no—
250C22×D3—128no—
251C22×D3—128no—
252C22×D3—182no—
253C22×D3—188no—
254C22×D3—184no—
255C22×D3—368no—
256C22×D3—364no—
257C3×D4—21no—
258C3×D4—34no—
259C3×D4—62no—
260C3×D4—62no—
261C3×D4—62no—
262C3×D4—94no—
263C3×D4—92no—
264C3×D4—182no—
265C3×D4—184no—
266C3×D4—181no—
267C3×D4C2×D3212yes—
268C2×C3×C4—32no—
269C2×C3×C4—61no—
270C2×C3×C4—91no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4C2×D3211yescharacteristic
274#67—68no—
275#67—68no—
276#67—188no—
277C2×Dic3—32no—
278C2×Dic3—61no—
279D12—34no—
280D12—34no—
281D12—62no—
282D12—62no—
283D12—62no—
284D12—62no—
285D12—94no—
286D12—182no—
287D12—184no—
288C4×D3—34no—
289C4×D3—34no—
290C4×D3—62no—
291C4×D3—62no—
292C3×C8—94no—
293C3⋊C8—34no—
294C3⋊C8—94no—
295C2×C32—21no—
296C2×C32—22no—
297C2×C32—44no—
298C2×C32—44no—
299C2×C32D3×D814yescomplemented
300C2×C32C2×D3×D412yes—
301C3⋊D3—44no—
302C3×D3—38no—
303C3×D3—64no—
304C3×D3—124no—
305C3×D3—124no—
306C3×D3—124no—
307C24—181no—
308C24—541no—
309C2×D4—94no—
310C2×D4—92no—
311C2×D4—91no—
312C2×D4—182no—
313C2×D4—181no—
314C2×D4—274no—
315C2×D4—542no—
316C2×D4—541no—
317C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
318C22×C4—91no—
319D8—98no—
320D8—278no—
321C2×C8—92no—
322C2×C8—274no—
323C22×C3—22no—
324C22×C3—32no—
325C22×C3—42no—
326C22×C3—64no—
327C22×C3—64no—
328C22×C3—61no—
329C22×C3—92no—
330C22×C3—94no—
331C22×C3—124no—
332C22×C3—124no—
333C22×C3—184no—
334C22×C3—182no—
335C22×C3—364no—
336C22×C3—368no—
337C22×C3—362no—
338C22×C3#381111yescharacteristic
339C2×D3—34no—
340C2×D3—38no—
341C2×D3—62no—
342C2×D3—64no—
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—1216no—
347C2×D3—1216no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C2×D3—184no—
352C2×D3—364no—
353C2×D3—3616no—
354C2×D3—368no—
355C3×C4—32no—
356C3×C4—32no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—92no—
360C3×C4#381111yescharacteristic
361C3×C4C22×D3211yescharacteristic
362Dic3—34no—
363Dic3—62no—
364C32—21no—
365C32C2×D3×D812yescomplemented
366C23—21no—
367C23—91no—
368C23—184no—
369C23—181no—
370C23—182no—
371C23—364no—
372C23—544no—
373C23—1084no—
374D4—21no—
375D4—94no—
376D4—92no—
377D4—184no—
378D4—181no—
379D4—182no—
380D4—544no—
381D4C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
382C2×C4—91no—
383C2×C4—92no—
384C2×C4—92no—
385C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
386C8—94no—
387C2×C3—32no—
388C2×C3—34no—
389C2×C3—42no—
390C2×C3—61no—
391C2×C3—62no—
392C2×C3—94no—
393C2×C3—124no—
394C2×C3—124no—
395C2×C3—364no—
396C2×C3—362no—
397C2×C3#6392612yes—
398C2×C3C2×#381111yescharacteristic
399D3—38no—
400D3—64no—
401D3—124no—
402D3—124no—
403D3—124no—
404D3—364no—
405C22—22no—
406C22—42no—
407C22—92no—
408C22—94no—
409C22—182no—
410C22—184no—
411C22—368no—
412C22—362no—
413C22—364no—
414C22—1088no—
415C22#8867711yescharacteristic
416C4—92no—
417C4#8867711yescharacteristic
418C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
419C3—32no—
420C3—61no—
421C3C2×#6392611yescharacteristic
422C2—42no—
423C2—94no—
424C2—362no—
425C2—364no—
426C2#1172119012yesdirect product factor
427C2C2×#8867711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×D8
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×#272C2semidirect
2. C2×#88655C2semidirect
3. C2×#3753⋊C3C2semidirect
4. #11721190C2direct (N × H)
5. C2×#88436C2semidirect
6. C2×#88434C2semidirect
10. C2×D4×H(3,𝔽3)C22semidirect
13. D4×#272C22semidirect
14. C2×C4⋊#272C22semidirect
16. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. #88655C22semidirect
20. #3753⋊C3C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. #88436C22semidirect
23. #88434C22semidirect
27. C2×#3753D3semidirect
42. D4×H(3,𝔽3)C23semidirect
47. C4⋊#272C23semidirect
51. C4⋊((C3⋊D3)⋊C3)C23semidirect
53. (C3⋊(C3⋊C8))⋊C3C23semidirect
60. C2×C32×D4C2×D3semidirect
61. C2×C4⋊(C3⋊D3)C2×D3semidirect
76. #3753C2×D3semidirect
77. C2×C3⋊(C3⋊C8)C2×D3semidirect
89. C2×#272D8semidirect
129. C32×D4C22×D3semidirect
133. C4⋊(C3⋊D3)C22×D3semidirect
149. C3⋊(C3⋊C8)C22×D3semidirect
153. C2×H(3,𝔽3)C2×D8semidirect
156. #272C2×D8semidirect
234. H(3,𝔽3)C22×D8semidirect
299. C2×C32D3×D8semidirect
317. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
365. C32C2×D3×D8semidirect
381. D4C2×((C3⋊D3)⋊C3)⋊C2semidirect
426. C2#11721190direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×#886552
C2×#3753⋊C32
#117211902
C2×#884362
C2×#884342
C2×D3×#2133