Finite Group Databasebeta

← C22×#88677 #13164819

Subgroups — C22×#88677 #13164819

Order 1728.

5515 conjugacy classes of subgroups; only the 635 normal subgroups are shown.

normal subgroups only

This group has 5515 conjugacy classes of subgroups — too many to draw. Only its 635 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
1C24×#272C211yesmaximal, characteristic, complemented
2C23×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
3C22×#638⋊C3C212yesmaximal, complemented
4C2×#88677C2124yesmaximal, direct product factor
5C22×#6200C212yesmaximal, complemented
6C22×#6199C211yesmaximal, characteristic, complemented
7C22×D3×#67—32nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic, complemented
9C23×#272—28no—
10C23×#272C2211yescharacteristic
11C23×#272C22112yescomplemented
12C23×#272C2211yescharacteristic, complemented
13C23×(C3⋊D3)⋊C3C2212yescomplemented
14C22×((C3⋊D3)⋊C3)⋊C2—28no—
15C22×((C3⋊D3)⋊C3)⋊C2C22112yescomplemented
16C2×#638⋊C3C22124yescomplemented
17C22×(C3⋊Dic3)⋊C3C2212yescomplemented
18#88677C22164yesdirect product factor
19C2×#6200C22124yescomplemented
20C2×#6199C22112yescomplemented
21C24×C3×D3—32no—
22C24×C3×D3—61no—
23C23×D32—32no—
24C22×#638D312yescomplemented
25C22×C3×#67—32no—
26C2×D3×#67—348no—
27C22×#626—32no—
28C22×#625—32no—
29C22×D3×Dic3—32no—
30C23×H(3,𝔽3)—24no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C23×H(3,𝔽3)C2316yescomplemented
33C22×#272—24no—
34C22×#272—248no—
35C22×#272D418yescomplemented
36C22×#272C2316yes—
37C22×#272C23116yescomplemented
38C22×#272C2316yescomplemented
39C22×(C3⋊D3)⋊C3—28no—
40C22×(C3⋊D3)⋊C3C23112yescomplemented
41C2×((C3⋊D3)⋊C3)⋊C2—248no—
42C2×((C3⋊D3)⋊C3)⋊C2C23116yescomplemented
43#638⋊C3C23132yescomplemented
44C2×(C3⋊Dic3)⋊C3C23112yescomplemented
45#6200C23132yescomplemented
46#6199C23116yescomplemented
47C23×#67—91no—
48C22×D3×D4—92no—
49C24×C32—21no—
50C24×C32C2×D312yescomplemented
51C23×C3⋊D3C2×D312yescomplemented
52C23×C3×D3—32no—
53C23×C3×D3—32no—
54C23×C3×D3—32no—
55C23×C3×D3—324no—
56C23×C3×D3—616no—
57C23×C3×D3—61no—
58C23×C3×D3—612no—
59C23×C3×D3—61no—
60C23×C3×D3—616no—
61C22×D32—324no—
62C22×D32—616no—
63C2×#638C2×D3124yescomplemented
64C22×C3⋊Dic3C2×D312yescomplemented
65C2×C3×#67—324no—
66C22×C3×Dic3—32no—
67D3×#67—3128no—
68C2×#626—324no—
69C2×#625—324no—
70C2×D3×Dic3—324no—
71C22×H(3,𝔽3)—212no—
72C22×H(3,𝔽3)C2×D414yescomplemented
73C22×H(3,𝔽3)C2413yes—
74C22×H(3,𝔽3)C2414yescomplemented
75C2×#272—212no—
76C2×#272—232no—
77C2×#272C2×D4124yescomplemented
78C2×#272C2414yes—
79C2×#272C2414yescomplemented
80C2×(C3⋊D3)⋊C3—224no—
81C2×(C3⋊D3)⋊C3C2418yescomplemented
82((C3⋊D3)⋊C3)⋊C2—232no—
83(C3⋊Dic3)⋊C3C2418yescomplemented
84C25×C3—91no—
85C24×D3—32no—
86C24×D3—61no—
87C24×D3—91no—
88C24×D3—92no—
89C22×C3×D4—92no—
90C22×#67—32no—
91C22×#67—92no—
92C22×#67—92no—
93C22×#67—924no—
94C22×#67—92no—
95C23×Dic3—91no—
96C2×D3×D4—948no—
97C22×D12—92no—
98C22×C4×D3—92no—
99C23×C32—28no—
100C23×C32—28no—
101C23×C32—21no—
102C23×C32—26no—
103C23×C32C22×D312yes—
104C23×C32C22×D3112yescomplemented
105C22×C3⋊D3—28no—
106C22×C3⋊D3C22×D3112yescomplemented
107C22×C3×D3—316no—
108C22×C3×D3—312no—
109C22×C3×D3—312no—
110C22×C3×D3—312no—
111C22×C3×D3—332no—
112C22×C3×D3—68no—
113C22×C3×D3—66no—
114C22×C3×D3—616no—
115C22×C3×D3—66no—
116C22×C3×D3—696no—
117C22×C3×D3—696no—
118C22×C3×D3—68no—
119C22×C3×D3—68no—
120C22×C3×D3—68no—
121C2×D32—332no—
122C2×D32—696no—
123#638C22×D3132yescomplemented
124C2×C3⋊Dic3C22×D3112yescomplemented
125C3×#67—332no—
126C2×C3×Dic3—312no—
127#626—332no—
128#625—332no—
129D3×Dic3—332no—
130C23×D4—271noSylow
131C2×H(3,𝔽3)—24no—
132C2×H(3,𝔽3)C22×D416yescomplemented
133C2×H(3,𝔽3)C2511yescharacteristic
134#272—24no—
135#272C22×D418yescomplemented
136(C3⋊D3)⋊C3—28no—
137C24×C3—32no—
138C24×C3—61no—
139C24×C3—92no—
140C24×C3—91no—
141C24×C3—91no—
142C24×C3—912no—
143C24×C3—188no—
144C24×C3D3211yescharacteristic
145C23×D3—32no—
146C23×D3—32no—
147C23×D3—324no—
148C23×D3—32no—
149C23×D3—61no—
150C23×D3—61no—
151C23×D3—612no—
152C23×D3—616no—
153C23×D3—616no—
154C23×D3—92no—
155C23×D3—912no—
156C23×D3—924no—
157C23×D3—1816no—
158C23×D3—188no—
159C2×C3×D4—924no—
160C22×C3×C4—92no—
161C2×#67—324no—
162C2×#67—924no—
163C2×#67—924no—
164C2×#67—964no—
165C2×#67—924no—
166C22×Dic3—32no—
167C22×Dic3—92no—
168C22×Dic3—912no—
169D3×D4—9128no—
170C2×D12—924no—
171C2×C4×D3—924no—
172C22×C32—224no—
173C22×C32—224no—
174C22×C32—23no—
175C22×C32—24no—
176C22×C32—24no—
177C22×C32D3×D418yescomplemented
178C22×C32C23×D316yes—
179C22×C32C23×D318yescomplemented
180C2×C3⋊D3—224no—
181C2×C3⋊D3C23×D318yescomplemented
182C2×C3×D3—348no—
183C2×C3×D3—38no—
184C2×C3×D3—38no—
185C2×C3×D3—38no—
186C2×C3×D3—624no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C2×C3×D3—664no—
190C2×C3×D3—664no—
191C2×C3×D3—624no—
192C2×C3×D3—624no—
193C2×C3×D3—624no—
194D32—664no—
195C3⋊Dic3C23×D318yescomplemented
196C3×Dic3—38no—
197C25—91no—
198C25—271no—
199C22×D4—92no—
200C22×D4—272no—
201C22×D4—2724no—
202C23×C4—271no—
203H(3,𝔽3)C23×D411yescharacteristic, complemented, Sylow
204C23×C3—24no—
205C23×C3—32no—
206C23×C3—312no—
207C23×C3—68no—
208C23×C3—68no—
209C23×C3—61no—
210C23×C3—66no—
211C23×C3—912no—
212C23×C3—96no—
213C23×C3—98no—
214C23×C3—96no—
215C23×C3—916no—
216C23×C3—1848no—
217C23×C3—188no—
218C23×C3—184no—
219C23×C3C2×D3211yescharacteristic
220C23×C3C2×D3216yes—
221C22×D3—312no—
222C22×D3—316no—
223C22×D3—312no—
224C22×D3—332no—
225C22×D3—312no—
226C22×D3—66no—
227C22×D3—68no—
228C22×D3—66no—
229C22×D3—616no—
230C22×D3—696no—
231C22×D3—68no—
232C22×D3—696no—
233C22×D3—68no—
234C22×D3—68no—
235C22×D3—912no—
236C22×D3—916no—
237C22×D3—932no—
238C22×D3—188no—
239C22×D3—1896no—
240C22×D3—1848no—
241C3×D4—932no—
242C2×C3×C4—912no—
243#67—332no—
244#67—932no—
245#67—932no—
246#67—932no—
247C2×Dic3—312no—
248C2×Dic3—912no—
249C2×Dic3—916no—
250D12—932no—
251C4×D3—932no—
252C2×C32—28no—
253C2×C32—28no—
254C2×C32—26no—
255C2×C32—21no—
256C2×C32C2×D3×D4112yescomplemented
257C2×C32C24×D312yes—
258C3⋊D3—28no—
259C3×D3—316no—
260C3×D3—68no—
261C3×D3—68no—
262C3×D3—68no—
263C3×D3—68no—
264C24—91no—
265C24—92no—
266C24—91no—
267C24—912no—
268C24—188no—
269C24—2712no—
270C24—548no—
271C24((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
272C2×D4—924no—
273C2×D4—2724no—
274C2×D4—2764no—
275C22×C4—92no—
276C22×C4—2712no—
277C22×C3—212no—
278C22×C3—36no—
279C22×C3—38no—
280C22×C3—38no—
281C22×C3—624no—
282C22×C3—624no—
283C22×C3—63no—
284C22×C3—64no—
285C22×C3—64no—
286C22×C3—98no—
287C22×C3—924no—
288C22×C3—94no—
289C22×C3—94no—
290C22×C3—1832no—
291C22×C3—1824no—
292C22×C3—1812no—
293C22×C3#381114yes—
294C22×C3C22×D3213yes—
295C22×C3C22×D3214yes—
296C2×D3—38no—
297C2×D3—348no—
298C2×D3—38no—
299C2×D3—38no—
300C2×D3—64no—
301C2×D3—624no—
302C2×D3—64no—
303C2×D3—664no—
304C2×D3—624no—
305C2×D3—664no—
306C2×D3—624no—
307C2×D3—624no—
308C2×D3—98no—
309C2×D3—1824no—
310C2×D3—1864no—
311C2×D3—1832no—
312C3×C4—98no—
313Dic3—38no—
314Dic3—98no—
315C32—21no—
316C32C22×D3×D412yescomplemented
317C23—24no—
318C23—96no—
319C23—98no—
320C23—912no—
321C23—96no—
322C23—916no—
323C23—1848no—
324C23—184no—
325C23—188no—
326C23—2716no—
327C23—5448no—
328C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C23C2×((C3⋊D3)⋊C3)⋊C216yescomplemented
330D4—932no—
331D4—2732no—
332C2×C4—912no—
333C2×C4—2716no—
334C2×C3—24no—
335C2×C3—32no—
336C2×C3—312no—
337C2×C3—68no—
338C2×C3—68no—
339C2×C3—61no—
340C2×C3—66no—
341C2×C3—98no—
342C2×C3—188no—
343C2×C3—184no—
344C2×C3C2×#381116yes—
345C2×C3C23×D3211yescharacteristic
346D3—316no—
347D3—68no—
348D3—68no—
349D3—68no—
350D3—68no—
351D3—188no—
352C22—212no—
353C22—94no—
354C22—924no—
355C22—98no—
356C22—94no—
357C22—1832no—
358C22—1812no—
359C22—1824no—
360C22—5432no—
361C22#8867714yesdirect product factor
362C22C22×((C3⋊D3)⋊C3)⋊C213yes—
363C22C22×((C3⋊D3)⋊C3)⋊C214yescomplemented
364C4—98no—
365C3—32no—
366C3—61no—
367C3C22×#381111yescharacteristic
368C2—24no—
369C2—98no—
370C2—184no—
371C2—188no—
372C2C2×#8867716yesdirect product factor
373C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C23×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×#272C2semidirect
2. C23×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. C22×#638⋊C3C2semidirect
4. C2×#88677C2direct (N × H)
5. C22×#6200C2semidirect
6. C22×#6199C2semidirect
8. C24×H(3,𝔽3)C22semidirect
11. C23×#272C22semidirect
12. C23×#272C22semidirect
13. C23×(C3⋊D3)⋊C3C22semidirect
15. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
16. C2×#638⋊C3C22semidirect
17. C22×(C3⋊Dic3)⋊C3C22semidirect
18. #88677C22direct (N × H)
19. C2×#6200C22semidirect
20. C2×#6199C22semidirect
24. C22×#638D3semidirect
32. C23×H(3,𝔽3)C23semidirect
35. C22×#272D4semidirect
37. C22×#272C23semidirect
38. C22×#272C23semidirect
40. C22×(C3⋊D3)⋊C3C23semidirect
42. C2×((C3⋊D3)⋊C3)⋊C2C23semidirect
43. #638⋊C3C23semidirect
44. C2×(C3⋊Dic3)⋊C3C23semidirect
45. #6200C23semidirect
46. #6199C23semidirect
50. C24×C32C2×D3semidirect
51. C23×C3⋊D3C2×D3semidirect
63. C2×#638C2×D3semidirect
64. C22×C3⋊Dic3C2×D3semidirect
72. C22×H(3,𝔽3)C2×D4semidirect
74. C22×H(3,𝔽3)C24semidirect
77. C2×#272C2×D4semidirect
79. C2×#272C24semidirect
81. C2×(C3⋊D3)⋊C3C24semidirect
83. (C3⋊Dic3)⋊C3C24semidirect
104. C23×C32C22×D3semidirect
106. C22×C3⋊D3C22×D3semidirect
123. #638C22×D3semidirect
124. C2×C3⋊Dic3C22×D3semidirect
132. C2×H(3,𝔽3)C22×D4semidirect
135. #272C22×D4semidirect
177. C22×C32D3×D4semidirect
179. C22×C32C23×D3semidirect
181. C2×C3⋊D3C23×D3semidirect
195. C3⋊Dic3C23×D3semidirect
203. H(3,𝔽3)C23×D4semidirect
256. C2×C32C2×D3×D4semidirect
271. C24((C3⋊D3)⋊C3)⋊C2semidirect
316. C32C22×D3×D4semidirect
329. C23C2×((C3⋊D3)⋊C3)⋊C2semidirect
361. C22#88677direct (N × H)
363. C22C22×((C3⋊D3)⋊C3)⋊C2semidirect
372. C2C2×#88677direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#2722
C23×((C3⋊D3)⋊C3)⋊C22
C22×#638⋊C32
C2×#886772
C22×#62002
C22×#61992
C22×D3×#673