Finite Group Databasebeta

← C2×#11720837 #13149810

Subgroups — C2×#11720837 #13149810

Order 1728.

1119 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1119 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×#88655C211yesmaximal, characteristic
2C2×#88651C211yesmaximal, characteristic, complemented
3C2×C8×#272C211yesmaximal, characteristic, complemented
4C2×#3744⋊C3C212yesmaximal, complemented
5C2×#88431C212yesmaximal, complemented
6#11720837C218yesmaximal, direct product factor
7C2×D3×#204—32nomaximal
8C22×((C3⋊D3)⋊C3)⋊C2—21no—
9C2×C4×#272C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×#634⋊C3C2212yescomplemented
12C2×#6199—21no—
13C2×#6198—21no—
14C2×#6196—21no—
15#88655—22no—
16#88655C2214yes—
17#88651—22no—
18#88651C2214yescomplemented
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20C2×#6180C2211yescharacteristic, complemented
21C8×#272C2214yescomplemented
22#3744⋊C3C2218yescomplemented
23#88431C2218yescomplemented
24C2×D3×D12—32no—
25C2×D3×Dic6—32no—
26C2×#3744D312yescomplemented
27C2×C3×#204—32no—
28C2×C3×D3×C8—32no—
29C2×C3×D3×C8—61no—
30C2×#3717—32no—
31C2×#3716—32no—
32D3×#204—316no—
33C22×#272D411yescharacteristic
34C22×(C3⋊D3)⋊C3—22no—
35C2×((C3⋊D3)⋊C3)⋊C2—24no—
36C2×((C3⋊D3)⋊C3)⋊C2—44no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084D411yescharacteristic
39C4×#272D412yes—
40C4×#272C2312yes—
41C2×(C3⋊Dic3)⋊C3—22no—
42C4⋊((C3⋊D3)⋊C3)—22no—
43C4⋊((C3⋊D3)⋊C3)C2314yes—
44#634⋊C3—22no—
45#634⋊C3C2314yescomplemented
46#6199—24no—
47#6198—24no—
48#6196—24no—
49C8×H(3,𝔽3)C2312yescomplemented
50#6180C2312yescomplemented
51C2×D3×SD16—92no—
52C22×#204—91no—
53C22×D32—62no—
54C2×C4⋊(C3⋊D3)C2×D312yes—
55C2×#634C2×D312yescomplemented
56C2×C3×D12—32no—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—32no—
60C2×#627—62no—
61C2×#626—62no—
62C2×D3×Dic3—62no—
63D3×D12—38no—
64D3×D12—64no—
65D3×Dic6—38no—
66D3×Dic6—64no—
67C2×C32×C8—21no—
68C2×C32×C8C2×D312yescomplemented
69#3744C2×D318yescomplemented
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—61no—
72C3×#204—38no—
73C3×D3×C8—38no—
74C3×D3×C8—64no—
75#3717—38no—
76#3716—38no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×#272SD1614yescomplemented
79C2×#272C2×D412yes—
80C2×(C3⋊D3)⋊C3—24no—
81C2×(C3⋊D3)⋊C3—44no—
82((C3⋊D3)⋊C3)⋊C2—48no—
83C4×H(3,𝔽3)C2×D411yescharacteristic
84C4×H(3,𝔽3)C2411yescharacteristic
85#1084C2×D412yes—
86(C3⋊Dic3)⋊C3—24no—
87C2×D3×Q8—92no—
88C2×D3×D4—92no—
89C22×D12—91no—
90C22×Dic6—91no—
91C2×C3×SD16—92no—
92C22×C3×C8—91no—
93C2×#215—92no—
94C2×#214—92no—
95D3×SD16—916no—
96C2×#204—32no—
97C2×#204—92no—
98C2×#204—98no—
99C2×#204—92no—
100C2×D3×C8—32no—
101C2×D3×C8—61no—
102C22×C3⋊D3—22no—
103C22×C3×D3—32no—
104C22×C3×D3—61no—
105C22×C3×D3—62no—
106C2×D32—68no—
107C2×D32—128no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D312yes—
110C2×C3⋊Dic3—22no—
111C4⋊(C3⋊D3)—22no—
112C4⋊(C3⋊D3)C22×D314yes—
113#634—22no—
114#634C22×D314yescomplemented
115C2×C3×Dic3—32no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C3×D12—34no—
119C3×D12—62no—
120C3×C4×D3—34no—
121C3×C4×D3—34no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×Dic6—34no—
125C3×Dic6—62no—
126#627—68no—
127#626—68no—
128D3×Dic3—68no—
129C32×C8—22no—
130C32×C8C22×D314yescomplemented
131C3×C3⋊C8—34no—
132C3×C3⋊C8—62no—
133C22×SD16—271noSylow
134C2×H(3,𝔽3)C2×SD1612yescomplemented
135C2×H(3,𝔽3)C22×D411yescharacteristic
136#272C2×SD1614yescomplemented
137(C3⋊D3)⋊C3—44no—
138C23×D3—182no—
139C23×D3—181no—
140C2×C3×Q8—92no—
141C2×C3×D4—92no—
142C22×C3×C4—91no—
143C2×#67—182no—
144C22×Dic3—181no—
145D3×Q8—98no—
146D3×Q8—184no—
147D3×D4—98no—
148D3×D4—184no—
149C2×D12—32no—
150C2×D12—92no—
151C2×D12—94no—
152C2×D12—182no—
153C2×D12—181no—
154C2×C4×D3—32no—
155C2×C4×D3—61no—
156C2×C4×D3—182no—
157C2×Dic6—32no—
158C2×Dic6—92no—
159C2×Dic6—94no—
160C2×Dic6—182no—
161C2×Dic6—182no—
162C2×Dic6—181no—
163C3×SD16—98no—
164C2×C3×C8—32no—
165C2×C3×C8—61no—
166C2×C3×C8—91no—
167C2×C3×C8—94no—
168C2×C3×C8D3211yescharacteristic
169#215—98no—
170#214—98no—
171C2×C3⋊C8—32no—
172C2×C3⋊C8—61no—
173#204—38no—
174#204—98no—
175#204—98no—
176D3×C8—38no—
177D3×C8—64no—
178C22×C32—21no—
179C22×C32D3×D412yes—
180C2×C3⋊D3—24no—
181C2×C3⋊D3—44no—
182C2×C3×D3—38no—
183C2×C3×D3—34no—
184C2×C3×D3—64no—
185C2×C3×D3—62no—
186C2×C3×D3—64no—
187C2×C3×D3—124no—
188D32—1216no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4D3×D412yes—
192C32×C4C23×D312yes—
193C3⋊Dic3—24no—
194C3×Dic3—34no—
195C3×Dic3—62no—
196C3×Dic3—64no—
197C22×Q8—271no—
198C22×D4—271no—
199C2×SD16—92no—
200C2×SD16—278no—
201C2×SD16—272no—
202C22×C8—91no—
203H(3,𝔽3)C22×SD1611yescharacteristic, complemented, Sylow
204C23×C3—91no—
205C23×C3—182no—
206C22×D3—32no—
207C22×D3—61no—
208C22×D3—62no—
209C22×D3—182no—
210C22×D3—188no—
211C22×D3—184no—
212C22×D3—368no—
213C22×D3—364no—
214C3×Q8—94no—
215C3×Q8—182no—
216C3×D4—94no—
217C3×D4—182no—
218C2×C3×C4—32no—
219C2×C3×C4—61no—
220C2×C3×C4—91no—
221C2×C3×C4—92no—
222C2×C3×C4—92no—
223C2×C3×C4—182no—
224C2×C3×C4C2×D3211yescharacteristic
225#67—188no—
226C2×Dic3—32no—
227C2×Dic3—61no—
228C2×Dic3—62no—
229C2×Dic3—182no—
230C2×Dic3—184no—
231D12—34no—
232D12—62no—
233D12—94no—
234D12—182no—
235D12—184no—
236C4×D3—34no—
237C4×D3—34no—
238C4×D3—62no—
239C4×D3—62no—
240C4×D3—188no—
241Dic6—34no—
242Dic6—62no—
243Dic6—94no—
244Dic6—182no—
245Dic6—188no—
246Dic6—184no—
247C3×C8—34no—
248C3×C8—62no—
249C3×C8—92no—
250C3×C8C2×D3212yes—
251C3⋊C8—34no—
252C3⋊C8—62no—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32D3×SD1614yescomplemented
256C2×C32C2×D3×D412yes—
257C3⋊D3—44no—
258C3×D3—38no—
259C3×D3—64no—
260C3×D3—124no—
261C24—541no—
262C2×Q8—92no—
263C2×Q8—274no—
264C2×Q8—541no—
265C2×Q8—542no—
266C2×D4—92no—
267C2×D4—274no—
268C2×D4—542no—
269C2×D4—541no—
270C22×C4—91no—
271C22×C4—541no—
272SD16—98no—
273SD16—278no—
274C2×C8—91no—
275C2×C8—94no—
276C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
277C22×C3—32no—
278C22×C3—61no—
279C22×C3—92no—
280C22×C3—94no—
281C22×C3—184no—
282C22×C3—364no—
283C22×C3#380211yescharacteristic
284C2×D3—34no—
285C2×D3—38no—
286C2×D3—62no—
287C2×D3—64no—
288C2×D3—64no—
289C2×D3—124no—
290C2×D3—184no—
291C2×D3—364no—
292C2×D3—3616no—
293C2×D3—368no—
294C3×C4—32no—
295C3×C4—32no—
296C3×C4—61no—
297C3×C4—61no—
298C3×C4—92no—
299C3×C4—184no—
300C3×C4#380211yescharacteristic
301C3×C4C22×D3211yescharacteristic
302Dic3—34no—
303Dic3—62no—
304Dic3—64no—
305Dic3—184no—
306C32—21no—
307C32C2×D3×SD1612yescomplemented
308C23—91no—
309C23—182no—
310C23—544no—
311C23—1084no—
312Q8—94no—
313Q8—182no—
314Q8—544no—
315D4—94no—
316D4—182no—
317D4—544no—
318C2×C4—91no—
319C2×C4—92no—
320C2×C4—92no—
321C2×C4—182no—
322C2×C4—544no—
323C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C8—92no—
325C8C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
326C2×C3—34no—
327C2×C3—32no—
328C2×C3—62no—
329C2×C3—61no—
330C2×C3—94no—
331C2×C3—364no—
332C2×C3#6379612yes—
333C2×C3C2×#380211yescharacteristic
334D3—38no—
335D3—64no—
336D3—124no—
337D3—364no—
338C22—94no—
339C22—92no—
340C22—184no—
341C22—364no—
342C22—1088no—
343C22#8865511yescharacteristic
344C4—92no—
345C4—184no—
346C4#8865511yescharacteristic
347C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
348C3—32no—
349C3—61no—
350C3C2×#6379611yescharacteristic
351C2—94no—
352C2—364no—
353C2#1172083712yesdirect product factor
354C2C2×#8865511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88655C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88651C2semidirect
3. C2×C8×#272C2semidirect
4. C2×#3744⋊C3C2semidirect
5. C2×#88431C2semidirect
6. #11720837C2direct (N × H)
11. C2×#634⋊C3C22semidirect
18. #88651C22semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
20. C2×#6180C22semidirect
21. C8×#272C22semidirect
22. #3744⋊C3C22semidirect
23. #88431C22semidirect
26. C2×#3744D3semidirect
45. #634⋊C3C23semidirect
49. C8×H(3,𝔽3)C23semidirect
50. #6180C23semidirect
55. C2×#634C2×D3semidirect
68. C2×C32×C8C2×D3semidirect
69. #3744C2×D3semidirect
78. C2×#272SD16semidirect
114. #634C22×D3semidirect
130. C32×C8C22×D3semidirect
134. C2×H(3,𝔽3)C2×SD16semidirect
136. #272C2×SD16semidirect
203. H(3,𝔽3)C22×SD16semidirect
255. C2×C32D3×SD16semidirect
276. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
307. C32C2×D3×SD16semidirect
325. C8C2×((C3⋊D3)⋊C3)⋊C2semidirect
353. C2#11720837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886512
C2×C8×#2722
C2×#3744⋊C32
C2×#884312
#117208372
C2×D3×#2043