Finite Group Databasebeta

← C22×#88965 #13181344

Subgroups — C22×#88965 #13181344

Order 1728.

4402 conjugacy classes of subgroups; only the 716 normal subgroups are shown.

normal subgroups only

This group has 4402 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×#6292C212yesmaximal, complemented
3C22×#6289C211yesmaximal, characteristic, complemented
4C22×C3×#627C211yesmaximal, characteristic, complemented
5C22×C3×#624C212yesmaximal, complemented
6C2×#88965C2124yesmaximal, direct product factor
7C22×#3799—32nomaximal
8C22×#3798—31nomaximal
9C23×C3⋊(C3⋊D3)C2211yescharacteristic
10C23×C3×C3⋊D3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×#6295C22112yescomplemented
15C2×#6292C22124yescomplemented
16C2×#6289C22112yescomplemented
17C2×C3×#627C22112yescomplemented
18C2×C3×#624C22124yescomplemented
19#88965C22164yesdirect product factor
20C22×C4⋊(C3⋊D3)—32no—
21C22×C4×C3⋊D3—31no—
22C22×C3×C4×D3—32no—
23C22×C3×C4×D3—32no—
24C22×C3×Dic6—32no—
25C22×#627D311yescharacteristic, complemented
26C22×#626—32no—
27C22×#626—32no—
28C22×#626—32no—
29C22×#626—62no—
30C22×#625—31no—
31C22×#624—32no—
32C22×#624—61no—
33C22×#624D312yescomplemented
34C2×#3799—348no—
35C2×#3798—324no—
36C23×C33C2311yescharacteristic
37C22×C3⋊(C3⋊D3)—24no—
38C22×C3⋊(C3⋊D3)C2316yes—
39C22×C3×C3⋊D3—28no—
40C22×C3×C3⋊D3C23112yes—
41C2×C3×C3⋊Dic3C2316yes—
42C2×C32×Dic3C2316yes—
43C2×C32×Dic3C23112yes—
44#6295C23116yescomplemented
45#6292C23132yescomplemented
46#6289C23116yescomplemented
47C3×#627C23116yescomplemented
48C3×#624C23132yescomplemented
49C22×#239—91no—
50C22×#235—92no—
51C23×C3⋊D3—32no—
52C23×C3⋊D3—31no—
53C23×C3⋊D3—61no—
54C23×C3⋊D3—62no—
55C23×C3⋊D3—121no—
56C23×C3⋊D3C2×D312yes—
57C23×C3×D3—32no—
58C23×C3×D3—32no—
59C23×C3×D3—62no—
60C22×C32×C4—32no—
61C22×C32×C4—31no—
62C22×C3⋊Dic3C2×D311yescharacteristic
63C2×C4⋊(C3⋊D3)—324no—
64C2×C4×C3⋊D3—312no—
65C22×C3×Dic3—21no—
66C22×C3×Dic3—22no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—61no—
69C22×C3×Dic3C2×D312yes—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C2×D312yes—
72C2×C3×C4×D3—324no—
73C2×C3×C4×D3—324no—
74C2×C3×Dic6—324no—
75C2×#627C2×D3112yescomplemented
76C2×#626—324no—
77C2×#626—324no—
78C2×#626—324no—
79C2×#626—624no—
80C2×#625—312no—
81C2×#624—324no—
82C2×#624—612no—
83C2×#624C2×D3124yescomplemented
84#3799—3128no—
85#3798—364no—
86C22×C33#4114yescomplemented
87C22×C33C2413yes—
88C2×C3⋊(C3⋊D3)—212no—
89C2×C3⋊(C3⋊D3)C2414yes—
90C2×C3×C3⋊D3—224no—
91C2×C3×C3⋊D3C2418yes—
92C3×C3⋊Dic3C2414yes—
93C32×Dic3C2414yes—
94C32×Dic3C2418yes—
95C22×C3×Q8—91no—
96C23×C3×C4—92no—
97C22×#67—92no—
98C22×#67—92no—
99C2×#239—924no—
100C2×#235—948no—
101C22×D12—92no—
102C22×D12—92no—
103C22×D12—91no—
104C22×D12—182no—
105C22×C4×D3—32no—
106C22×C4×D3—32no—
107C22×C4×D3—92no—
108C22×C4×D3—91no—
109C22×C4×D3—181no—
110C22×Dic6—32no—
111C23×C32—22no—
112C23×C32—21no—
113C23×C32—41no—
114C23×C32C22×D312yes—
115C23×C32C22×D311yescharacteristic
116C22×C3⋊D3—28no—
117C22×C3⋊D3—312no—
118C22×C3⋊D3—36no—
119C22×C3⋊D3—66no—
120C22×C3⋊D3—612no—
121C22×C3⋊D3—64no—
122C22×C3⋊D3—68no—
123C22×C3⋊D3—68no—
124C22×C3⋊D3—126no—
125C22×C3⋊D3—128no—
126C22×C3⋊D3—128no—
127C22×C3⋊D3C22×D3112yes—
128C22×C3×D3—312no—
129C22×C3×D3—312no—
130C22×C3×D3—612no—
131C22×C3×D3—68no—
132C22×C3×D3—68no—
133C22×C3×D3—128no—
134C2×C32×C4—312no—
135C2×C32×C4—36no—
136C2×C3⋊Dic3C22×D316yes—
137C4⋊(C3⋊D3)—332no—
138C4×C3⋊D3—316no—
139C2×C3×Dic3—26no—
140C2×C3×Dic3—212no—
141C2×C3×Dic3—312no—
142C2×C3×Dic3—66no—
143C2×C3×Dic3C22×D3112yes—
144C2×C3×Dic3C22×D3112yes—
145C2×C3×Dic3C22×D3112yes—
146C3×C4×D3—332no—
147C3×C4×D3—332no—
148C3×Dic6—332no—
149#627C22×D3116yescomplemented
150#626—332no—
151#626—332no—
152#626—332no—
153#626—632no—
154#625—316no—
155#624—332no—
156#624—616no—
157#624C22×D3132yescomplemented
158C22×#41—271noSylow
159C2×C33C2×#4116yescomplemented
160C2×C33C2511yescharacteristic
161C3⋊(C3⋊D3)—24no—
162C3×C3⋊D3—28no—
163C24×C3—92no—
164C23×D3—32no—
165C23×D3—32no—
166C23×D3—62no—
167C23×D3—91no—
168C23×D3—92no—
169C23×D3—181no—
170C23×D3—182no—
171C23×D3—361no—
172C2×C3×Q8—912no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4—32no—
176C22×C3×C4—61no—
177C22×C3×C4—62no—
178C22×C3×C4—924no—
179C22×C3×C4—91no—
180C2×#67—924no—
181C2×#67—924no—
182C22×Dic3—32no—
183C22×Dic3—61no—
184C22×Dic3D3212yes—
185C22×Dic3D3211yescharacteristic
186#239—964no—
187#235—9128no—
188C2×D12—924no—
189C2×D12—924no—
190C2×D12—912no—
191C2×D12—1824no—
192C2×C4×D3—324no—
193C2×C4×D3—324no—
194C2×C4×D3—924no—
195C2×C4×D3—912no—
196C2×C4×D3—1812no—
197C2×Dic6—324no—
198C22×C32—23no—
199C22×C32—26no—
200C22×C32—24no—
201C22×C32—28no—
202C22×C32—44no—
203C22×C32—43no—
204C22×C32#23518yescomplemented
205C22×C32#23914yescomplemented
206C22×C32C23×D316yes—
207C22×C32C23×D313yes—
208C2×C3⋊D3—224no—
209C2×C3⋊D3—38no—
210C2×C3⋊D3—34no—
211C2×C3⋊D3—64no—
212C2×C3⋊D3—68no—
213C2×C3⋊D3—612no—
214C2×C3⋊D3—624no—
215C2×C3⋊D3—624no—
216C2×C3⋊D3—124no—
217C2×C3⋊D3—1224no—
218C2×C3⋊D3—1224no—
219C2×C3⋊D3C23×D318yes—
220C2×C3×D3—38no—
221C2×C3×D3—38no—
222C2×C3×D3—68no—
223C2×C3×D3—624no—
224C2×C3×D3—624no—
225C2×C3×D3—1224no—
226C32×C4—38no—
227C32×C4—34no—
228C3⋊Dic3C23×D314yes—
229C3×Dic3—24no—
230C3×Dic3—28no—
231C3×Dic3—38no—
232C3×Dic3—64no—
233C3×Dic3C23×D318yes—
234C3×Dic3C23×D318yes—
235C3×Dic3C23×D318yes—
236C2×#41—2724no—
237C22×Q8—91no—
238C22×D4—272no—
239C22×D4—271no—
240C23×C4—92no—
241C23×C4—271no—
242C33C22×#4111yescharacteristic, complemented, Sylow
243C23×C3—21no—
244C23×C3—22no—
245C23×C3—41no—
246C23×C3—912no—
247C23×C3—188no—
248C23×C3C2×D3211yescharacteristic
249C23×C3C2×D3212yes—
250C22×D3—312no—
251C22×D3—312no—
252C22×D3—612no—
253C22×D3—68no—
254C22×D3—68no—
255C22×D3—96no—
256C22×D3—912no—
257C22×D3—128no—
258C22×D3—186no—
259C22×D3—1812no—
260C22×D3—188no—
261C22×D3—184no—
262C22×D3—188no—
263C22×D3—366no—
264C22×D3—368no—
265C22×D3—368no—
266C3×Q8—916no—
267C2×C3×C4—312no—
268C2×C3×C4—312no—
269C2×C3×C4—312no—
270C2×C3×C4—66no—
271C2×C3×C4—612no—
272C2×C3×C4—932no—
273C2×C3×C4—96no—
274#67—932no—
275#67—932no—
276C2×Dic3—312no—
277C2×Dic3—66no—
278C2×Dic3C2×D32112yes—
279C2×Dic3C2×D3216yes—
280D12—932no—
281D12—932no—
282D12—916no—
283D12—1832no—
284C4×D3—332no—
285C4×D3—332no—
286C4×D3—932no—
287C4×D3—916no—
288C4×D3—1816no—
289Dic6—332no—
290C2×C32—26no—
291C2×C32—212no—
292C2×C32—21no—
293C2×C32—22no—
294C2×C32—46no—
295C2×C32—41no—
296C2×C32C2×#235112yescomplemented
297C2×C32C2×#23916yescomplemented
298C2×C32C24×D312yes—
299C2×C32C24×D311yescharacteristic
300C3⋊D3—28no—
301C3⋊D3—64no—
302C3⋊D3—68no—
303C3⋊D3—68no—
304C3⋊D3—128no—
305C3⋊D3—128no—
306C3×D3—68no—
307C3×D3—68no—
308C3×D3—128no—
309C24—92no—
310C24—271no—
311#41—2764no—
312C2×Q8—912no—
313C2×D4—2724no—
314C2×D4—2712no—
315C22×C4—31no—
316C22×C4—32no—
317C22×C4—924no—
318C22×C4—91no—
319C22×C4—2712no—
320C22×C3—23no—
321C22×C3—26no—
322C22×C3—28no—
323C22×C3—24no—
324C22×C3—43no—
325C22×C3—44no—
326C22×C3—98no—
327C22×C3—1824no—
328C22×C3#379814yescomplemented
329C22×C3#379918yescomplemented
330C22×C3C22×D3213yes—
331C22×C3C22×D3216yes—
332C2×D3—38no—
333C2×D3—38no—
334C2×D3—68no—
335C2×D3—624no—
336C2×D3—624no—
337C2×D3—94no—
338C2×D3—98no—
339C2×D3—1224no—
340C2×D3—184no—
341C2×D3—188no—
342C2×D3—1824no—
343C2×D3—1812no—
344C2×D3—1824no—
345C2×D3—364no—
346C2×D3—3624no—
347C2×D3—3624no—
348C3×C4—38no—
349C3×C4—38no—
350C3×C4—38no—
351C3×C4—64no—
352C3×C4—68no—
353C3×C4—94no—
354Dic3—38no—
355Dic3—64no—
356Dic3C22×D3218yes—
357Dic3C22×D3214yes—
358C32—21no—
359C32—22no—
360C32—41no—
361C32C22×#23512yescomplemented
362C32C22×#23911yescharacteristic, complemented
363C23—912no—
364C23—188no—
365C23—276no—
366C23—544no—
367C23D3311yescharacteristic
368Q8—916no—
369D4—2732no—
370D4—2716no—
371C2×C4—36no—
372C2×C4—312no—
373C2×C4—932no—
374C2×C4—96no—
375C2×C4—2716no—
376C2×C3—212no—
377C2×C3—26no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—46no—
381C2×C3—41no—
382C2×C3—188no—
383C2×C3C2×#379816yescomplemented
384C2×C3C2×#3799112yescomplemented
385C2×C3C23×D3211yescharacteristic
386C2×C3C23×D3212yes—
387D3—68no—
388D3—68no—
389D3—128no—
390D3—188no—
391D3—184no—
392D3—188no—
393D3—368no—
394D3—368no—
395C22—98no—
396C22—1824no—
397C22—274no—
398C22—5412no—
399C22#8896514yesdirect product factor
400C22C2×D3313yes—
401C4—34no—
402C4—38no—
403C4—94no—
404C3—22no—
405C3—21no—
406C3—41no—
407C3C22×#379811yescharacteristic, complemented
408C3C22×#379912yescomplemented
409C2—188no—
410C2—544no—
411C2C2×#8896516yesdirect product factor
412C2C22×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×#6292C2semidirect
3. C22×#6289C2semidirect
4. C22×C3×#627C2semidirect
5. C22×C3×#624C2semidirect
6. C2×#88965C2direct (N × H)
14. C2×#6295C22semidirect
15. C2×#6292C22semidirect
16. C2×#6289C22semidirect
17. C2×C3×#627C22semidirect
18. C2×C3×#624C22semidirect
19. #88965C22direct (N × H)
25. C22×#627D3semidirect
33. C22×#624D3semidirect
44. #6295C23semidirect
45. #6292C23semidirect
46. #6289C23semidirect
47. C3×#627C23semidirect
48. C3×#624C23semidirect
75. C2×#627C2×D3semidirect
83. C2×#624C2×D3semidirect
86. C22×C33#41semidirect
149. #627C22×D3semidirect
157. #624C22×D3semidirect
159. C2×C33C2×#41semidirect
204. C22×C32#235semidirect
205. C22×C32#239semidirect
242. C33C22×#41semidirect
296. C2×C32C2×#235semidirect
297. C2×C32C2×#239semidirect
328. C22×C3#3798semidirect
329. C22×C3#3799semidirect
361. C32C22×#235semidirect
362. C32C22×#239semidirect
383. C2×C3C2×#3798semidirect
384. C2×C3C2×#3799semidirect
399. C22#88965direct (N × H)
407. C3C22×#3798semidirect
408. C3C22×#3799semidirect
411. C2C2×#88965direct (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×#62922
C22×#62892
C22×C3×#6272
C22×C3×#6242
C2×#889652
C22×#37993
C22×#37983