Finite Group Databasebeta

← C2×#11723345 #13176674

Subgroups — C2×#11723345 #13176674

Order 1728.

1900 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 1900 conjugacy classes of subgroups — too many to draw. Only its 268 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×#89017C212yesmaximal, complemented
2C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×C3×#3745C211yesmaximal, characteristic, complemented
4C2×C32×Dic12C211yesmaximal, characteristic, complemented
5C2×#88793C212yesmaximal, complemented
6#11723345C218yesmaximal, direct product factor
7C2×#64128—31nomaximal
8C2×#63810—34nomaximal
9C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C32×Dic6C2212yes—
12C2×#6292—22no—
13C2×Dic3×C3⋊D3—22no—
14#89017—24no—
15#89017C2218yescomplemented
16C2×C33×C8C2211yescharacteristic
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C8×C3⋊(C3⋊D3)C2214yescomplemented
19C3×#3745C2214yescomplemented
20C32×Dic12C2214yescomplemented
21#88793C2218yescomplemented
22C2×#3832—32no—
23C2×#3796—38no—
24C2×C32×Q16—31no—
25C2×#3755—32no—
26#64128—38no—
27C2×#3745D311yescharacteristic, complemented
28C2×C8×C3⋊D3—31no—
29C2×C8×C3⋊D3—34no—
30C2×C8×C3⋊D3—64no—
31C2×C3×Dic12D314yescomplemented
32C2×C3×D24—34no—
33C2×#3715—38no—
34#63810—332no—
35C22×C3⋊(C3⋊D3)D411yescharacteristic
36C22×C3×C3⋊D3—22no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C4×C3⋊(C3⋊D3)D412yes—
40C4×C3⋊(C3⋊D3)C2312yes—
41C3×C4⋊(C3⋊D3)—22no—
42C3×C4⋊(C3⋊D3)C2314yes—
43C2×C32×Dic3—22no—
44C32×Dic6—22no—
45C32×Dic6C2314yes—
46#6292—28no—
47Dic3×C3⋊D3—28no—
48C33×C8C2312yes—
49C3⋊(C3⋊(C3⋊C8))C2312yes—
50C2×#919—94no—
51C2×#912—91no—
52C2×C32×Q8—32no—
53#3832—38no—
54#3832—64no—
55C2×C4⋊(C3⋊D3)—62no—
56C2×C4⋊(C3⋊D3)C2×D312yes—
57C2×C4×C3⋊D3—31no—
58C2×C4×C3⋊D3—34no—
59C2×C4×C3⋊D3—62no—
60C2×C4×C3⋊D3—64no—
61C2×C3×D12—38no—
62C2×C3×Dic6C2×D318yes—
63C2×#626—68no—
64C2×D3×Dic3—68no—
65#3796—332no—
66#3796—616no—
67C32×Q16—34no—
68C2×C32×C8—24no—
69C2×C32×C8C2×D314yes—
70C2×C32×C8C2×D311yescharacteristic
71#3755—38no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75#3745C2×D314yescomplemented
76C8×C3⋊D3—34no—
77C8×C3⋊D3—316no—
78C8×C3⋊D3—616no—
79C3×Dic12C2×D3116yescomplemented
80C3×D24—316no—
81#3715—332no—
82C22×C33C2×D411yescharacteristic
83C2×C3⋊(C3⋊D3)—22no—
84C2×C3⋊(C3⋊D3)C2×D412yes—
85C2×C3×C3⋊D3—24no—
86C2×C3×C3⋊D3—44no—
87C33×C4C2×D411yescharacteristic
88C33×C4C2411yescharacteristic
89C3⋊(C3⋊Dic3)C2×D412yes—
90C32×Dic3—24no—
91C2×#239—98no—
92C2×#237—92no—
93C2×C3×Q16—34no—
94C2×C3×D8—91no—
95C2×#215—98no—
96C2×#214—92no—
97#919—932no—
98#912—98no—
99C2×Dic12C3⋊D311yescharacteristic, complemented
100C2×D24—34no—
101C2×D3×C8—94no—
102C2×D3×C8—91no—
103C2×D3×C8—184no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—31no—
106C22×C3⋊D3—34no—
107C22×C3⋊D3—64no—
108C22×C3×D3—68no—
109C32×Q8—34no—
110C32×Q8—62no—
111C2×C32×C4—24no—
112C2×C32×C4—62no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—64no—
118C4⋊(C3⋊D3)—22no—
119C4⋊(C3⋊D3)—68no—
120C4⋊(C3⋊D3)C22×D314yes—
121C4×C3⋊D3—32no—
122C4×C3⋊D3—32no—
123C4×C3⋊D3—38no—
124C4×C3⋊D3—38no—
125C4×C3⋊D3—68no—
126C4×C3⋊D3—68no—
127C4×C3⋊D3—68no—
128C2×C3×Dic3—28no—
129C3×D12—316no—
130C3×D12—68no—
131C3×Dic6—28no—
132C3×Dic6C22×D3116yes—
133#626—632no—
134D3×Dic3—632no—
135C32×C8—28no—
136C32×C8C22×D318yes—
137C32×C8C22×D312yes—
138C3⋊(C3⋊C8)—32no—
139C3⋊(C3⋊C8)—38no—
140C3⋊(C3⋊C8)—68no—
141C2×#135—271noSylow
142C2×C33#13512yescomplemented
143C2×C33C22×D411yescharacteristic
144C3⋊(C3⋊D3)—22no—
145C3×C3⋊D3—44no—
146C2×C3×Q8—38no—
147C2×C3×D4—92no—
148C2×#67—182no—
149C22×Dic3—182no—
150#239—932no—
151#239—1816no—
152#237—98no—
153#237—184no—
154C2×D12—38no—
155C2×D12—188no—
156C2×C4×D3—94no—
157C2×C4×D3—91no—
158C2×C4×D3—184no—
159C2×C4×D3—188no—
160C2×Dic6C2×C3⋊D312yes—
161C3×Q16—316no—
162C3×D8—94no—
163C2×C3×C8—24no—
164C2×C3×C8D3214yes—
165C2×C3×C8C2×C3⋊D311yescharacteristic
166#215—932no—
167#214—98no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
171Dic12C2×C3⋊D314yescomplemented
172D24—316no—
173D3×C8—916no—
174D3×C8—94no—
175D3×C8—1816no—
176C22×C32—24no—
177C22×C32D3×D414yes—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—24no—
180C2×C3⋊D3—32no—
181C2×C3⋊D3—38no—
182C2×C3⋊D3—44no—
183C2×C3⋊D3—68no—
184C2×C3⋊D3—616no—
185C2×C3⋊D3—62no—
186C2×C3⋊D3—68no—
187C2×C3×D3—616no—
188C2×C3×D3—1216no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—64no—
192C32×C4D3×D414yes—
193C32×C4D3×D411yescharacteristic
194C32×C4C23×D314yes—
195C32×C4C23×D311yescharacteristic
196C3⋊Dic3—32no—
197C3⋊Dic3—38no—
198C3⋊Dic3—68no—
199C3×Dic3—216no—
200C2×#41—272no—
201#135—278no—
202C2×Q16—31no—
203C2×SD16—272no—
204C2×D8—91no—
205C22×C8—271no—
206C33C2×#13511yescharacteristic, complemented, Sylow
207C23×C3—182no—
208C22×D3—68no—
209C22×D3—94no—
210C22×D3—91no—
211C22×D3—184no—
212C3×Q8—316no—
213C3×Q8—68no—
214C3×D4—94no—
215C3×D4—182no—
216C2×C3×C4—24no—
217C2×C3×C4—68no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220#67—188no—
221C2×Dic3—22no—
222C2×Dic3—94no—
223C2×Dic3—91no—
224C2×Dic3—184no—
225C2×Dic3—188no—
226D12—316no—
227D12—68no—
228D12—1832no—
229C4×D3—98no—
230C4×D3—92no—
231C4×D3—98no—
232C4×D3—92no—
233C4×D3—188no—
234C4×D3—188no—
235C4×D3—1832no—
236Dic6—22no—
237Dic6C22×C3⋊D314yes—
238C3×C8—28no—
239C3×C8C2×D3218yes—
240C3×C8C22×C3⋊D312yes—
241C3⋊C8—98no—
242C3⋊C8—92no—
243C3⋊C8—188no—
244C2×C32—24no—
245C2×C32—28no—
246C2×C32#91212yescomplemented
247C2×C32#91918yescomplemented
248C2×C32C2×D3×D414yes—
249C2×C32C2×D3×D411yescharacteristic
250C3⋊D3—44no—
251C3⋊D3—616no—
252C3⋊D3—62no—
253C3⋊D3—68no—
254C3×D3—1216no—
255#41—278no—
256#41—544no—
257C2×Q8—32no—
258C2×D4—92no—
259C2×D4—542no—
260C22×C4—271no—
261C22×C4—542no—
262Q16—34no—
263SD16—278no—
264D8—94no—
265C2×C8—271no—
266C2×C8—274no—
267C2×C8D3×C3⋊D311yescharacteristic
268C22×C3—24no—
269C22×C3—184no—
270C22×C3—364no—
271C22×C3#380214yes—
272C22×C3D4×C3⋊D311yescharacteristic
273C2×D3—616no—
274C2×D3—98no—
275C2×D3—92no—
276C2×D3—1216no—
277C2×D3—188no—
278C2×D3—1816no—
279C2×D3—188no—
280C2×D3—182no—
281C3×C4—24no—
282C3×C4—24no—
283C3×C4—616no—
284C3×C4#380214yes—
285C3×C4D4×C3⋊D311yescharacteristic
286C3×C4C22×D3214yes—
287C3×C4C23×C3⋊D311yescharacteristic
288Dic3—24no—
289Dic3—98no—
290Dic3—92no—
291Dic3—188no—
292C32—24no—
293C32C2×#91211yescharacteristic, complemented
294C32C2×#91914yescomplemented
295C23—182no—
296C23—271no—
297Q8—34no—
298Q8—62no—
299D4—94no—
300D4—182no—
301D4—548no—
302C2×C4—62no—
303C2×C4—271no—
304C2×C4—272no—
305C2×C4—272no—
306C2×C4—548no—
307C2×C4C2×D3×C3⋊D311yescharacteristic
308C8—272no—
309C8C2×D3×C3⋊D312yes—
310C2×C3—24no—
311C2×C3—28no—
312C2×C3—364no—
313C2×C3#6381018yescomplemented
314C2×C3#6412812yescomplemented
315C2×C3C2×#380214yes—
316C2×C3C2×D4×C3⋊D311yescharacteristic
317D3—1216no—
318D3—1816no—
319D3—182no—
320D3—188no—
321C22—184no—
322C22—272no—
323C22—364no—
324C22—542no—
325C22#8902611yescharacteristic
326C4—64no—
327C4—272no—
328C4#8902611yescharacteristic
329C4C22×D3×C3⋊D311yescharacteristic
330C3—24no—
331C3C2×#6381014yescomplemented
332C3C2×#6412811yescharacteristic, complemented
333C2—364no—
334C2—542no—
335C2#1172334512yesdirect product factor
336C2C2×#8902611yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89017C2semidirect
2. C2×C8×C3⋊(C3⋊D3)C2semidirect
3. C2×C3×#3745C2semidirect
4. C2×C32×Dic12C2semidirect
5. C2×#88793C2semidirect
6. #11723345C2direct (N × H)
15. #89017C22semidirect
18. C8×C3⋊(C3⋊D3)C22semidirect
19. C3×#3745C22semidirect
20. C32×Dic12C22semidirect
21. #88793C22semidirect
27. C2×#3745D3semidirect
31. C2×C3×Dic12D3semidirect
75. #3745C2×D3semidirect
79. C3×Dic12C2×D3semidirect
99. C2×Dic12C3⋊D3semidirect
142. C2×C33#135semidirect
171. Dic12C2×C3⋊D3semidirect
206. C33C2×#135semidirect
246. C2×C32#912semidirect
247. C2×C32#919semidirect
293. C32C2×#912semidirect
294. C32C2×#919semidirect
313. C2×C3#63810semidirect
314. C2×C3#64128semidirect
331. C3C2×#63810semidirect
332. C3C2×#64128semidirect
335. C2#11723345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890172
C2×C8×C3⋊(C3⋊D3)2
C2×C3×#37452
C2×C32×Dic122
C2×#887932
#117233452
C2×#641283
C2×#638103