Finite Group Databasebeta

← C22×#88959 #13181338

Subgroups — C22×#88959 #13181338

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×#6296C211yesmaximal, characteristic, complemented
2C22×#6291C212yesmaximal, complemented
3C22×#6289C211yesmaximal, characteristic, complemented
4C22×C3×#625C211yesmaximal, characteristic, complemented
5C22×C3×D3×Dic3C212yesmaximal, complemented
6C2×#88959C2124yesmaximal, direct product factor
7C22×#3804—32nomaximal
8C22×#3798—31nomaximal
9C23×C3⋊(C3⋊D3)C2211yescharacteristic
10C23×C32×D3C2212yes—
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C32×Dic3C2211yescharacteristic
14C2×#6296C22112yescomplemented
15C2×#6291C22124yescomplemented
16C2×#6289C22112yescomplemented
17C2×C3×#625C22112yescomplemented
18C2×C3×D3×Dic3C22124yescomplemented
19#88959C22164yesdirect product factor
20C22×#638—32no—
21C22×C4×C3⋊D3—31no—
22C22×C3×#67—32no—
23C23×C3×Dic3—32no—
24C22×C3×C4×D3—32no—
25C22×#627—31no—
26C22×#627—32no—
27C22×#626—32no—
28C22×#626—32no—
29C22×#626—62no—
30C22×#625D311yescharacteristic, complemented
31C22×#624—32no—
32C22×#624—61no—
33C2×#3804—348no—
34C22×D3×Dic3D312yescomplemented
35C2×#3798—324no—
36C23×C33C2311yescharacteristic
37C22×C3⋊(C3⋊D3)—24no—
38C22×C3⋊(C3⋊D3)C2316yes—
39C22×C32×D3—28no—
40C22×C32×D3C23112yes—
41C2×C3×C3⋊Dic3C23112yes—
42C2×C3×C3⋊Dic3C2316yes—
43C2×C32×Dic3C2316yes—
44#6296C23116yescomplemented
45#6291C23132yescomplemented
46#6289C23116yescomplemented
47C3×#625C23116yescomplemented
48C3×D3×Dic3C23132yescomplemented
49C22×#237—91no—
50C22×#235—92no—
51C24×C32—32no—
52C23×C3⋊D3—32no—
53C23×C3⋊D3—31no—
54C23×C3⋊D3—61no—
55C23×C3⋊D3—62no—
56C23×C3⋊D3—121no—
57C23×C3×D3—22no—
58C23×C3×D3C2×D312yes—
59C23×C3×D3C2×D312yes—
60C22×C32×C4—31no—
61C2×#638—324no—
62C22×C3⋊Dic3C2×D312yes—
63C22×C3⋊Dic3C2×D311yescharacteristic
64C2×C4×C3⋊D3—312no—
65C2×C3×#67—324no—
66C22×C3×Dic3—21no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—324no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3—62no—
73C22×C3×Dic3C2×D312yes—
74C2×C3×C4×D3—324no—
75C2×#627—312no—
76C2×#627—324no—
77C2×#626—324no—
78C2×#626—324no—
79C2×#626—624no—
80C2×#625C2×D3112yescomplemented
81C2×#624—324no—
82C2×#624—612no—
83#3804—3128no—
84C2×D3×Dic3C2×D3124yescomplemented
85#3798—364no—
86C22×C33#4114yescomplemented
87C22×C33C2413yes—
88C2×C3⋊(C3⋊D3)—212no—
89C2×C3⋊(C3⋊D3)C2414yes—
90C2×C32×D3—224no—
91C2×C32×D3C2418yes—
92C3×C3⋊Dic3C2418yes—
93C3×C3⋊Dic3C2414yes—
94C32×Dic3C2414yes—
95C22×C3×D4—91no—
96C23×C3×C4—92no—
97C22×#67—32no—
98C22×#67—92no—
99C22×#67—92no—
100C22×#67—182no—
101C23×Dic3—32no—
102C2×#237—924no—
103C2×#235—948no—
104C22×D12—92no—
105C22×C4×D3—32no—
106C22×C4×D3—92no—
107C22×C4×D3—91no—
108C22×C4×D3—181no—
109C22×Dic6—92no—
110C22×Dic6—91no—
111C23×C32—22no—
112C23×C32—21no—
113C23×C32—312no—
114C23×C32—41no—
115C23×C32—68no—
116C23×C32C22×D312yes—
117C23×C32C22×D311yescharacteristic
118C22×C3⋊D3—312no—
119C22×C3⋊D3—36no—
120C22×C3⋊D3—66no—
121C22×C3⋊D3—612no—
122C22×C3⋊D3—64no—
123C22×C3⋊D3—68no—
124C22×C3⋊D3—68no—
125C22×C3⋊D3—126no—
126C22×C3⋊D3—128no—
127C22×C3⋊D3—128no—
128C22×C3×D3—28no—
129C22×C3×D3—28no—
130C22×C3×D3—212no—
131C22×C3×D3—48no—
132C22×C3×D3C22×D3112yes—
133C22×C3×D3C22×D3112yes—
134C2×C32×C4—36no—
135#638—332no—
136C2×C3⋊Dic3C22×D3112yes—
137C2×C3⋊Dic3C22×D316yes—
138C4×C3⋊D3—316no—
139C3×#67—332no—
140C2×C3×Dic3—26no—
141C2×C3×Dic3—312no—
142C2×C3×Dic3—312no—
143C2×C3×Dic3—312no—
144C2×C3×Dic3—332no—
145C2×C3×Dic3—66no—
146C2×C3×Dic3—612no—
147C2×C3×Dic3C22×D3112yes—
148C3×C4×D3—332no—
149#627—316no—
150#627—332no—
151#626—332no—
152#626—332no—
153#626—632no—
154#625C22×D3116yescomplemented
155#624—332no—
156#624—616no—
157D3×Dic3C22×D3132yescomplemented
158C22×#41—271noSylow
159C2×C33C2×#4116yescomplemented
160C2×C33C2511yescharacteristic
161C3⋊(C3⋊D3)—24no—
162C32×D3—28no—
163C24×C3—32no—
164C24×C3—32no—
165C24×C3—62no—
166C23×D3—92no—
167C23×D3—91no—
168C23×D3—181no—
169C23×D3—182no—
170C23×D3—361no—
171C23×D3D3212yes—
172C2×C3×D4—912no—
173C22×C3×C4—32no—
174C22×C3×C4—61no—
175C22×C3×C4—924no—
176C22×C3×C4—91no—
177C22×C3×C4—92no—
178C2×#67—324no—
179C2×#67—924no—
180C2×#67—924no—
181C2×#67—1824no—
182C22×Dic3—32no—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—324no—
186C22×Dic3—61no—
187C22×Dic3—62no—
188C22×Dic3D3211yescharacteristic
189#237—964no—
190#235—9128no—
191C2×D12—924no—
192C2×C4×D3—324no—
193C2×C4×D3—924no—
194C2×C4×D3—912no—
195C2×C4×D3—1812no—
196C2×Dic6—924no—
197C2×Dic6—912no—
198C22×C32—23no—
199C22×C32—26no—
200C22×C32—24no—
201C22×C32—28no—
202C22×C32—38no—
203C22×C32—44no—
204C22×C32—43no—
205C22×C32—624no—
206C22×C32#23518yescomplemented
207C22×C32#23714yescomplemented
208C22×C32C23×D316yes—
209C22×C32C23×D313yes—
210C2×C3⋊D3—38no—
211C2×C3⋊D3—34no—
212C2×C3⋊D3—64no—
213C2×C3⋊D3—68no—
214C2×C3⋊D3—612no—
215C2×C3⋊D3—624no—
216C2×C3⋊D3—624no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—1224no—
219C2×C3⋊D3—1224no—
220C2×C3×D3—224no—
221C2×C3×D3—224no—
222C2×C3×D3—28no—
223C2×C3×D3—424no—
224C2×C3×D3C23×D318yes—
225C2×C3×D3C23×D318yes—
226C32×C4—34no—
227C3⋊Dic3C23×D318yes—
228C3⋊Dic3C23×D314yes—
229C3×Dic3—24no—
230C3×Dic3—38no—
231C3×Dic3—38no—
232C3×Dic3—38no—
233C3×Dic3—64no—
234C3×Dic3—68no—
235C3×Dic3C23×D318yes—
236C2×#41—2724no—
237C22×Q8—271no—
238C22×D4—91no—
239C22×D4—272no—
240C23×C4—92no—
241C23×C4—271no—
242C33C22×#4111yescharacteristic, complemented, Sylow
243C23×C3—21no—
244C23×C3—22no—
245C23×C3—312no—
246C23×C3—312no—
247C23×C3—41no—
248C23×C3—612no—
249C23×C3—68no—
250C23×C3—68no—
251C23×C3—128no—
252C23×C3C2×D3211yescharacteristic
253C23×C3C2×D3212yes—
254C22×D3—28no—
255C22×D3—912no—
256C22×D3—96no—
257C22×D3—186no—
258C22×D3—1812no—
259C22×D3—188no—
260C22×D3—184no—
261C22×D3—188no—
262C22×D3—366no—
263C22×D3—368no—
264C22×D3—368no—
265C22×D3C2×D32112yes—
266C3×D4—916no—
267C2×C3×C4—312no—
268C2×C3×C4—66no—
269C2×C3×C4—932no—
270C2×C3×C4—96no—
271C2×C3×C4—912no—
272#67—332no—
273#67—932no—
274#67—932no—
275#67—1832no—
276C2×Dic3—312no—
277C2×Dic3—312no—
278C2×Dic3—312no—
279C2×Dic3—332no—
280C2×Dic3—66no—
281C2×Dic3—612no—
282C2×Dic3C2×D3216yes—
283D12—932no—
284C4×D3—332no—
285C4×D3—932no—
286C4×D3—916no—
287C4×D3—1816no—
288Dic6—932no—
289Dic6—916no—
290C2×C32—26no—
291C2×C32—212no—
292C2×C32—21no—
293C2×C32—22no—
294C2×C32—46no—
295C2×C32—41no—
296C2×C32—68no—
297C2×C32C2×#235112yescomplemented
298C2×C32C2×#23716yescomplemented
299C2×C32C24×D312yes—
300C2×C32C24×D311yescharacteristic
301C3⋊D3—64no—
302C3⋊D3—68no—
303C3⋊D3—68no—
304C3⋊D3—128no—
305C3⋊D3—128no—
306C3×D3—28no—
307C3×D3—28no—
308C3×D3—48no—
309C24—32no—
310C24—271no—
311#41—2764no—
312C2×Q8—2712no—
313C2×D4—912no—
314C2×D4—2724no—
315C22×C4—31no—
316C22×C4—924no—
317C22×C4—92no—
318C22×C4—91no—
319C22×C4—2712no—
320C22×C3—23no—
321C22×C3—26no—
322C22×C3—24no—
323C22×C3—28no—
324C22×C3—38no—
325C22×C3—38no—
326C22×C3—43no—
327C22×C3—44no—
328C22×C3—68no—
329C22×C3—624no—
330C22×C3—624no—
331C22×C3—1224no—
332C22×C3#379814yescomplemented
333C22×C3#380418yescomplemented
334C22×C3C22×D3213yes—
335C22×C3C22×D3216yes—
336C2×D3—224no—
337C2×D3—98no—
338C2×D3—94no—
339C2×D3—184no—
340C2×D3—188no—
341C2×D3—1824no—
342C2×D3—1812no—
343C2×D3—1824no—
344C2×D3—364no—
345C2×D3—3624no—
346C2×D3—3624no—
347C2×D3C22×D3218yes—
348C3×C4—38no—
349C3×C4—64no—
350C3×C4—94no—
351C3×C4—98no—
352Dic3—38no—
353Dic3—38no—
354Dic3—38no—
355Dic3—64no—
356Dic3—68no—
357Dic3C22×D3214yes—
358C32—21no—
359C32—22no—
360C32—41no—
361C32C22×#23512yescomplemented
362C32C22×#23711yescharacteristic, complemented
363C23—312no—
364C23—68no—
365C23—276no—
366C23—544no—
367C23D3311yescharacteristic
368Q8—2716no—
369D4—916no—
370D4—2732no—
371C2×C4—36no—
372C2×C4—932no—
373C2×C4—912no—
374C2×C4—96no—
375C2×C4—2716no—
376C2×C3—26no—
377C2×C3—212no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—46no—
381C2×C3—41no—
382C2×C3—68no—
383C2×C3—68no—
384C2×C3—128no—
385C2×C3C2×#379816yescomplemented
386C2×C3C2×#3804112yescomplemented
387C2×C3C23×D3211yescharacteristic
388C2×C3C23×D3212yes—
389D3—28no—
390D3—188no—
391D3—184no—
392D3—188no—
393D3—368no—
394D3—368no—
395C22—38no—
396C22—624no—
397C22—274no—
398C22—5412no—
399C22#8895914yesdirect product factor
400C22C2×D3313yes—
401C4—34no—
402C4—98no—
403C4—94no—
404C3—21no—
405C3—22no—
406C3—41no—
407C3C22×#379811yescharacteristic, complemented
408C3C22×#380412yescomplemented
409C2—68no—
410C2—544no—
411C2C2×#8895916yesdirect 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×#6296C2semidirect
2. C22×#6291C2semidirect
3. C22×#6289C2semidirect
4. C22×C3×#625C2semidirect
5. C22×C3×D3×Dic3C2semidirect
6. C2×#88959C2direct (N × H)
14. C2×#6296C22semidirect
15. C2×#6291C22semidirect
16. C2×#6289C22semidirect
17. C2×C3×#625C22semidirect
18. C2×C3×D3×Dic3C22semidirect
19. #88959C22direct (N × H)
30. C22×#625D3semidirect
34. C22×D3×Dic3D3semidirect
44. #6296C23semidirect
45. #6291C23semidirect
46. #6289C23semidirect
47. C3×#625C23semidirect
48. C3×D3×Dic3C23semidirect
80. C2×#625C2×D3semidirect
84. C2×D3×Dic3C2×D3semidirect
86. C22×C33#41semidirect
154. #625C22×D3semidirect
157. D3×Dic3C22×D3semidirect
159. C2×C33C2×#41semidirect
206. C22×C32#235semidirect
207. C22×C32#237semidirect
242. C33C22×#41semidirect
297. C2×C32C2×#235semidirect
298. C2×C32C2×#237semidirect
332. C22×C3#3798semidirect
333. C22×C3#3804semidirect
361. C32C22×#235semidirect
362. C32C22×#237semidirect
385. C2×C3C2×#3798semidirect
386. C2×C3C2×#3804semidirect
399. C22#88959direct (N × H)
407. C3C22×#3798semidirect
408. C3C22×#3804semidirect
411. C2C2×#88959direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62962
C22×#62912
C22×#62892
C22×C3×#6252
C22×C3×D3×Dic32
C2×#889592
C22×#38043
C22×#37983