Finite Group Databasebeta

← C22×#88964 #13181343

Subgroups — C22×#88964 #13181343

Order 1728.

4322 conjugacy classes of subgroups; only the 716 normal subgroups are shown.

normal subgroups only

This group has 4322 conjugacy classes of subgroups — too many to draw. Only its 716 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
1C22×#6295C211yesmaximal, characteristic, complemented
2C22×Dic3×C3⋊D3C212yesmaximal, complemented
3C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C22×C3×#627C211yesmaximal, characteristic, complemented
5C22×C3×#626C212yesmaximal, complemented
6C2×#88964C2124yesmaximal, direct product factor
7C22×#3805—31nomaximal
8C22×#3796—32nomaximal
9C23×C3×C3⋊D3C2212yes—
10C23×C32×D3C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×#6295C22112yescomplemented
15C2×Dic3×C3⋊D3C22124yescomplemented
16C2×D3×C3⋊Dic3C22112yescomplemented
17C2×C3×#627C22112yescomplemented
18C2×C3×#626C22124yescomplemented
19#88964C22164yesdirect product factor
20C23×C3⋊Dic3—31no—
21C22×C4×C3⋊D3—32no—
22C22×C3×#67—32no—
23C22×C3×D12—32no—
24C22×C3×Dic6—32no—
25C22×#627D311yescharacteristic, complemented
26C22×#626—32no—
27C22×#626D312yescomplemented
28C22×#625—31no—
29C2×#3805—324no—
30C22×D3×Dic3—32no—
31C22×D3×Dic3—32no—
32C22×D3×Dic3—32no—
33C22×D3×Dic3—61no—
34C22×D3×Dic3—62no—
35C2×#3796—348no—
36C23×C33C2311yescharacteristic
37C22×C3×C3⋊D3—28no—
38C22×C3×C3⋊D3C23112yes—
39C22×C32×D3—24no—
40C22×C32×D3C2316yes—
41C2×C3⋊(C3⋊Dic3)C2316yes—
42C2×C3×C3⋊Dic3C2316yes—
43C2×C32×Dic3C23112yes—
44#6295C23116yescomplemented
45Dic3×C3⋊D3C23132yescomplemented
46D3×C3⋊Dic3C23116yescomplemented
47C3×#627C23116yescomplemented
48C3×#626C23132yescomplemented
49C22×#239—91no—
50C22×#237—92no—
51C24×C32—31no—
52C23×C3⋊D3C2×D312yes—
53C23×C3×D3—21no—
54C23×C3×D3—32no—
55C23×C3×D3—32no—
56C23×C3×D3—62no—
57C23×C3×D3C2×D312yes—
58C22×C32×C4—32no—
59C22×C3⋊Dic3—32no—
60C22×C3⋊Dic3—31no—
61C22×C3⋊Dic3—312no—
62C22×C3⋊Dic3—61no—
63C22×C3⋊Dic3—62no—
64C22×C3⋊Dic3—121no—
65C22×C3⋊Dic3C2×D311yescharacteristic
66C2×C4×C3⋊D3—324no—
67C2×C3×#67—324no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—61no—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C2×C3×D12—324no—
74C2×C3×Dic6—324no—
75C2×#627C2×D3112yescomplemented
76C2×#626—324no—
77C2×#626C2×D3124yescomplemented
78C2×#625—312no—
79#3805—364no—
80C2×D3×Dic3—324no—
81C2×D3×Dic3—324no—
82C2×D3×Dic3—324no—
83C2×D3×Dic3—624no—
84C2×D3×Dic3—612no—
85#3796—3128no—
86C22×C33#4114yescomplemented
87C22×C33C2413yes—
88C2×C3×C3⋊D3—224no—
89C2×C3×C3⋊D3C2418yes—
90C2×C32×D3—212no—
91C2×C32×D3C2414yes—
92C3⋊(C3⋊Dic3)C2414yes—
93C3×C3⋊Dic3C2414yes—
94C32×Dic3C2418yes—
95C22×C3×Q8—91no—
96C22×C3×D4—92no—
97C22×#67—32no—
98C22×#67—92no—
99C23×Dic3—92no—
100C23×Dic3—92no—
101C23×Dic3—181no—
102C2×#239—924no—
103C2×#237—948no—
104C22×D12—32no—
105C22×D12—91no—
106C22×C4×D3—92no—
107C22×C4×D3—92no—
108C22×C4×D3—91no—
109C22×C4×D3—182no—
110C22×Dic6—32no—
111C23×C32—21no—
112C23×C32—22no—
113C23×C32—36no—
114C23×C32—41no—
115C23×C32—64no—
116C23×C32C22×D312yes—
117C23×C32C22×D311yescharacteristic
118C22×C3⋊D3—28no—
119C22×C3⋊D3C22×D3112yes—
120C22×C3×D3—28no—
121C22×C3×D3—28no—
122C22×C3×D3—26no—
123C22×C3×D3—312no—
124C22×C3×D3—312no—
125C22×C3×D3—612no—
126C22×C3×D3—616no—
127C22×C3×D3—68no—
128C22×C3×D3—68no—
129C22×C3×D3C22×D3112yes—
130C2×C32×C4—312no—
131C2×C3⋊Dic3—312no—
132C2×C3⋊Dic3—36no—
133C2×C3⋊Dic3—316no—
134C2×C3⋊Dic3—66no—
135C2×C3⋊Dic3—612no—
136C2×C3⋊Dic3—126no—
137C2×C3⋊Dic3C22×D316yes—
138C4×C3⋊D3—332no—
139C3×#67—332no—
140C2×C3×Dic3—212no—
141C2×C3×Dic3—312no—
142C2×C3×Dic3—66no—
143C2×C3×Dic3C22×D3112yes—
144C2×C3×Dic3C22×D3112yes—
145C3×D12—332no—
146C3×Dic6—332no—
147#627C22×D3116yescomplemented
148#626—332no—
149#626C22×D3132yescomplemented
150#625—316no—
151D3×Dic3—332no—
152D3×Dic3—332no—
153D3×Dic3—332no—
154D3×Dic3—616no—
155D3×Dic3—632no—
156C22×#41—271noSylow
157C2×C33C2×#4116yescomplemented
158C2×C33C2511yescharacteristic
159C3×C3⋊D3—28no—
160C32×D3—24no—
161C24×C3—32no—
162C24×C3—61no—
163C24×C3—92no—
164C23×D3—32no—
165C23×D3—32no—
166C23×D3—62no—
167C23×D3D3211yescharacteristic
168C2×C3×Q8—912no—
169C2×C3×D4—924no—
170C22×C3×C4—32no—
171C22×C3×C4—32no—
172C22×C3×C4—62no—
173C22×C3×C4—91no—
174C2×#67—324no—
175C2×#67—924no—
176C22×Dic3—32no—
177C22×Dic3—61no—
178C22×Dic3—924no—
179C22×Dic3—91no—
180C22×Dic3—92no—
181C22×Dic3—924no—
182C22×Dic3—1812no—
183C22×Dic3—182no—
184C22×Dic3—181no—
185C22×Dic3—361no—
186C22×Dic3D3212yes—
187#239—964no—
188#237—9128no—
189C2×D12—324no—
190C2×D12—912no—
191C2×C4×D3—924no—
192C2×C4×D3—924no—
193C2×C4×D3—912no—
194C2×C4×D3—1824no—
195C2×Dic6—324no—
196C22×C32—26no—
197C22×C32—23no—
198C22×C32—28no—
199C22×C32—24no—
200C22×C32—34no—
201C22×C32—44no—
202C22×C32—43no—
203C22×C32—612no—
204C22×C32#23718yescomplemented
205C22×C32#23914yescomplemented
206C22×C32C23×D316yes—
207C22×C32C23×D313yes—
208C2×C3⋊D3—224no—
209C2×C3⋊D3C23×D318yes—
210C2×C3×D3—224no—
211C2×C3×D3—224no—
212C2×C3×D3—24no—
213C2×C3×D3—38no—
214C2×C3×D3—38no—
215C2×C3×D3—68no—
216C2×C3×D3—648no—
217C2×C3×D3—624no—
218C2×C3×D3—624no—
219C2×C3×D3C23×D318yes—
220C32×C4—38no—
221C3⋊Dic3—38no—
222C3⋊Dic3—34no—
223C3⋊Dic3—64no—
224C3⋊Dic3—68no—
225C3⋊Dic3—124no—
226C3⋊Dic3C23×D314yes—
227C3×Dic3—28no—
228C3×Dic3—38no—
229C3×Dic3—64no—
230C3×Dic3C23×D318yes—
231C3×Dic3C23×D318yes—
232C2×#41—2724no—
233C22×Q8—91no—
234C22×D4—92no—
235C22×D4—271no—
236C23×C4—272no—
237C23×C4—271no—
238C33C22×#4111yescharacteristic, complemented, Sylow
239C23×C3—21no—
240C23×C3—22no—
241C23×C3—312no—
242C23×C3—41no—
243C23×C3—66no—
244C23×C3—68no—
245C23×C3—68no—
246C23×C3—912no—
247C23×C3—188no—
248C23×C3C2×D3212yes—
249C23×C3C2×D3211yescharacteristic
250C22×D3—24no—
251C22×D3—312no—
252C22×D3—312no—
253C22×D3—612no—
254C22×D3—616no—
255C22×D3—68no—
256C22×D3—68no—
257C22×D3C2×D3216yes—
258C3×Q8—916no—
259C3×D4—932no—
260C2×C3×C4—312no—
261C2×C3×C4—312no—
262C2×C3×C4—612no—
263C2×C3×C4—96no—
264#67—332no—
265#67—932no—
266C2×Dic3—312no—
267C2×Dic3—66no—
268C2×Dic3—932no—
269C2×Dic3—96no—
270C2×Dic3—912no—
271C2×Dic3—932no—
272C2×Dic3—1816no—
273C2×Dic3—1812no—
274C2×Dic3—186no—
275C2×Dic3—366no—
276C2×Dic3C2×D32112yes—
277D12—332no—
278D12—916no—
279C4×D3—932no—
280C4×D3—932no—
281C4×D3—916no—
282C4×D3—1832no—
283Dic6—332no—
284C2×C32—212no—
285C2×C32—26no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—46no—
289C2×C32—41no—
290C2×C32—64no—
291C2×C32C2×#237112yescomplemented
292C2×C32C2×#23916yescomplemented
293C2×C32C24×D312yes—
294C2×C32C24×D311yescharacteristic
295C3⋊D3—28no—
296C3×D3—28no—
297C3×D3—28no—
298C3×D3—616no—
299C3×D3—68no—
300C3×D3—68no—
301C24—31no—
302C24—92no—
303#41—2764no—
304C2×Q8—912no—
305C2×D4—924no—
306C2×D4—2712no—
307C22×C4—32no—
308C22×C4—91no—
309C22×C4—2724no—
310C22×C4—2712no—
311C22×C4—271no—
312C22×C3—23no—
313C22×C3—26no—
314C22×C3—28no—
315C22×C3—24no—
316C22×C3—38no—
317C22×C3—43no—
318C22×C3—44no—
319C22×C3—64no—
320C22×C3—624no—
321C22×C3—624no—
322C22×C3—98no—
323C22×C3—1824no—
324C22×C3#379618yescomplemented
325C22×C3#380514yescomplemented
326C22×C3C22×D3216yes—
327C22×C3C22×D3213yes—
328C2×D3—212no—
329C2×D3—38no—
330C2×D3—38no—
331C2×D3—68no—
332C2×D3—648no—
333C2×D3—624no—
334C2×D3—624no—
335C2×D3C22×D3214yes—
336C3×C4—38no—
337C3×C4—38no—
338C3×C4—68no—
339C3×C4—94no—
340Dic3—38no—
341Dic3—64no—
342Dic3—94no—
343Dic3—98no—
344Dic3—188no—
345Dic3—184no—
346Dic3—364no—
347Dic3C22×D3218yes—
348C32—22no—
349C32—21no—
350C32—41no—
351C32C22×#23712yescomplemented
352C32C22×#23911yescharacteristic, complemented
353C23—36no—
354C23—64no—
355C23—912no—
356C23—188no—
357C23D3311yescharacteristic
358Q8—916no—
359D4—932no—
360D4—2716no—
361C2×C4—312no—
362C2×C4—96no—
363C2×C4—2732no—
364C2×C4—2716no—
365C2×C4—276no—
366C2×C3—212no—
367C2×C3—26no—
368C2×C3—21no—
369C2×C3—22no—
370C2×C3—46no—
371C2×C3—41no—
372C2×C3—68no—
373C2×C3—68no—
374C2×C3—188no—
375C2×C3C2×#3796112yescomplemented
376C2×C3C2×#380516yescomplemented
377C2×C3C23×D3212yes—
378C2×C3C23×D3211yescharacteristic
379D3—24no—
380D3—616no—
381D3—68no—
382D3—68no—
383C22—34no—
384C22—612no—
385C22—98no—
386C22—1824no—
387C22#8896414yesdirect product factor
388C22C2×D3313yes—
389C4—38no—
390C4—94no—
391C4—274no—
392C3—22no—
393C3—21no—
394C3—41no—
395C3C22×#379612yescomplemented
396C3C22×#380511yescharacteristic, complemented
397C2—64no—
398C2—188no—
399C2C2×#8896416yesdirect product factor
400C2C22×D3311yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC2×C33C25
FrattiniC2C22×D33
Fitting, socleC23×C33C23
hypocenterC33C22×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6295C2semidirect
2. C22×Dic3×C3⋊D3C2semidirect
3. C22×D3×C3⋊Dic3C2semidirect
4. C22×C3×#627C2semidirect
5. C22×C3×#626C2semidirect
6. C2×#88964C2direct (N × H)
14. C2×#6295C22semidirect
15. C2×Dic3×C3⋊D3C22semidirect
16. C2×D3×C3⋊Dic3C22semidirect
17. C2×C3×#627C22semidirect
18. C2×C3×#626C22semidirect
19. #88964C22direct (N × H)
25. C22×#627D3semidirect
27. C22×#626D3semidirect
44. #6295C23semidirect
45. Dic3×C3⋊D3C23semidirect
46. D3×C3⋊Dic3C23semidirect
47. C3×#627C23semidirect
48. C3×#626C23semidirect
75. C2×#627C2×D3semidirect
77. C2×#626C2×D3semidirect
86. C22×C33#41semidirect
147. #627C22×D3semidirect
149. #626C22×D3semidirect
157. C2×C33C2×#41semidirect
204. C22×C32#237semidirect
205. C22×C32#239semidirect
238. C33C22×#41semidirect
291. C2×C32C2×#237semidirect
292. C2×C32C2×#239semidirect
324. C22×C3#3796semidirect
325. C22×C3#3805semidirect
351. C32C22×#237semidirect
352. C32C22×#239semidirect
375. C2×C3C2×#3796semidirect
376. C2×C3C2×#3805semidirect
387. C22#88964direct (N × H)
395. C3C22×#3796semidirect
396. C3C22×#3805semidirect
399. C2C2×#88964direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62952
C22×Dic3×C3⋊D32
C22×D3×C3⋊Dic32
C22×C3×#6272
C22×C3×#6262
C2×#889642
C22×#38053
C22×#37963