Finite Group Databasebeta

← #13148584

Subgroups — #13148584

Order 1728.

1259 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 203 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×#6198C211yesmaximal, characteristic
2#11721733C211yesmaximal, characteristic, complemented
3C22⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
4#11721310C212yesmaximal, complemented
5#11721298C211yesmaximal, characteristic, complemented
6#11721138C212yesmaximal, complemented
7C4×#6198C212yesmaximal, complemented
8#11721122C214yesmaximal, complemented
9#10594182—32nomaximal
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×#1084C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
13C2×#6198C418yescomplemented
14C2×#6198C2212yes—
15C2×#6198C2211yescharacteristic
16C2×#6198C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
18C22⋊C4×H(3,𝔽3)C2211yescharacteristic
19#88552C2211yescharacteristic
20#88540C2212yescomplemented
21C4×#1084C2212yescomplemented
22#3757⋊C3C2212yes—
23#3752⋊C3C2214yescomplemented
24#88450C2214yescomplemented
25#88449C2212yescomplemented
26#88447C2212yescomplemented
27C22×#626—32no—
28C22⋊C4×C3⋊D3D312yescomplemented
29C3×D3×C22⋊C4—32no—
30C3×#877—32no—
31C3×#877—61no—
32#63974—32no—
33D3×#217—32no—
34#63955—32no—
35#63904—34no—
36C4×#626—34no—
37#63881—34no—
38#63880—34no—
39C23×H(3,𝔽3)C2311yescharacteristic
40C22×(C3⋊D3)⋊C3—44no—
41C22×(C3⋊D3)⋊C3C2×C418yescomplemented
42C22×(C3⋊D3)⋊C3C2312yes—
43C22×(C3⋊D3)⋊C3C2312yes—
44C2×C4×H(3,𝔽3)C2312yes—
45C2×#1084C2×C414yescomplemented
46C2×#1084C2312yes—
47C2×#1084C2311yescharacteristic
48C2×#1084C2311yescharacteristic
49C2×(C3⋊Dic3)⋊C3C2314yes—
50C4×(C3⋊D3)⋊C3—44no—
51#6198C2×C418yescomplemented
52#6198C2×C414yescomplemented
53#6198C2×C414yescomplemented
54((C3⋊D3)⋊C3)⋊C4—44no—
55#5521—91no—
56#5486—92no—
57C23×C3⋊D3C2×D312yes—
58C23×C3×D3—32no—
59C2×C4×C3⋊D3C2×D314yescomplemented
60C22×C3×Dic3—32no—
61C22×C3×Dic3—61no—
62C2×C3×C4×D3—34no—
63C2×#626—316no—
64C2×#626—32no—
65C2×#626—32no—
66C2×#626—32no—
67C2×#626—32no—
68C2×#624—34no—
69C2×D3×Dic3—34no—
70C32×C22⋊C4—21no—
71C32×C22⋊C4C2×D312yes—
72#3757C2×D312yes—
73#3752C2×D314yescomplemented
74C3×#217—32no—
75C3×#217—32no—
76C3×#217—61no—
77C3×(C2×D3)⋊C4—34no—
78C3×Dic3⋊C4—34no—
79C3×Dic3⋊C4—62no—
80C3×C4×Dic3—34no—
81C3×C4×Dic3—62no—
82Dic3⋊Dic3—34no—
83#3722—34no—
84(C2×D3)⋊Dic3—34no—
85Dic32—34no—
86C22×H(3,𝔽3)—22no—
87C22×H(3,𝔽3)C22×C412yes—
88C22×H(3,𝔽3)C2411yescharacteristic
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3—24no—
91C2×(C3⋊D3)⋊C3—48no—
92C2×(C3⋊D3)⋊C3C22×C414yes—
93C2×(C3⋊D3)⋊C3C22×C414yes—
94C4×H(3,𝔽3)—22no—
95#1084—22no—
96#1084C22×C412yes—
97#1084C22×C412yes—
98(C3⋊Dic3)⋊C3—24no—
99C22×#67—92no—
100C22×D12—91no—
101C3×#117—91no—
102C2×C3×C22⋊C4—92no—
103C2×#217—92no—
104C4×#67—94no—
105C4×#67—94no—
106#926—91no—
107#893—94no—
108#67⋊C4—94no—
109#67⋊C4—94no—
110D3×C22⋊C4—32no—
111D3×C22⋊C4—92no—
112D3×C22⋊C4—92no—
113D3×C22⋊C4—92no—
114#877—32no—
115#877—61no—
116#877—92no—
117C4×D12—92no—
118C4×D12—92no—
119C23×C32—21no—
120C23×C32C22×D312yes—
121C22×C3⋊D3—44no—
122C22×C3⋊D3C4×D318yescomplemented
123C22×C3⋊D3C22×D312yes—
124C22×C3⋊D3C22×D312yes—
125C22×C3×D3—32no—
126C22×C3×D3—32no—
127C22×C3×D3—38no—
128C22×C3×D3—124no—
129C2×C32×C4—22no—
130C2×C32×C4C22×D314yes—
131C2×C3⋊Dic3C22×D314yes—
132C4×C3⋊D3—44no—
133C2×C3×Dic3—38no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—62no—
142C3×C4×D3—124no—
143#626—38no—
144#626—38no—
145#626—38no—
146#626—38no—
147#624—124no—
148D3×Dic3—124no—
149#518—271noSylow
150C2×H(3,𝔽3)—22no—
151C2×H(3,𝔽3)C23×C411yescharacteristic
152C2×H(3,𝔽3)H(5,𝔽2)12yes—
153(C3⋊D3)⋊C3—24no—
154(C3⋊D3)⋊C3—24no—
155C24×C3—92no—
156C23×D3—32no—
157C23×D3—92no—
158C22×C3×C4—94no—
159C22×C3×C4—91no—
160C2×#67—92no—
161C2×#67—92no—
162C2×#67—92no—
163C2×#67—92no—
164C2×#67—916no—
165C22×Dic3—32no—
166C22×Dic3—61no—
167C22×Dic3—94no—
168C2×D12—91no—
169C2×D12—91no—
170C2×D12—92no—
171C2×D12—98no—
172C2×C4×D3—34no—
173C2×C4×D3—94no—
174C2×C4×D3—94no—
175C2×C4×D3—94no—
176C3×C4⋊C4—92no—
177C3×C22⋊C4—32no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—94no—
180C3×C22⋊C4—92no—
181C3×C22⋊C4—91no—
182C3×C22⋊C4D3211yescharacteristic
183C3×C42—92no—
184#217—32no—
185#217—32no—
186#217—61no—
187#217—92no—
188#217—94no—
189(C2×D3)⋊C4—34no—
190(C2×D3)⋊C4—94no—
191(C2×D3)⋊C4—94no—
192(C2×D3)⋊C4—94no—
193C4⋊Dic3—92no—
194Dic3⋊C4—34no—
195Dic3⋊C4—62no—
196Dic3⋊C4—94no—
197C4×Dic3—34no—
198C4×Dic3—62no—
199C4×Dic3—92no—
200C4×Dic3—94no—
201C22×C32—24no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32—42no—
205C22×C32C2×C4×D314yes—
206C22×C32C23×D312yes—
207C2×C3⋊D3—24no—
208C2×C3⋊D3—24no—
209C2×C3⋊D3—48no—
210C2×C3⋊D3C2×C4×D314yes—
211C2×C3⋊D3C2×C4×D314yes—
212C2×C3×D3—34no—
213C2×C3×D3—34no—
214C2×C3×D3—64no—
215C2×C3×D3—64no—
216C2×C3×D3—128no—
217C32×C4—24no—
218C32×C4—42no—
219C3⋊Dic3—24no—
220C3×Dic3—34no—
221C3×Dic3—34no—
222C3×Dic3—62no—
223C3×Dic3—62no—
224C3×Dic3—64no—
225C3×Dic3—64no—
226C3×Dic3—122no—
227C22×D4—271no—
228C4×D4—272no—
229C4×D4—272no—
230C4×D4—274no—
231#117—91no—
232#117—271no—
233C2×C22⋊C4—92no—
234C2×C22⋊C4—272no—
235H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
236C23×C3—32no—
237C23×C3—61no—
238C23×C3—92no—
239C23×C3—92no—
240C23×C3—98no—
241C23×C3—364no—
242C23×C3C2×D3211yescharacteristic
243C22×D3—32no—
244C22×D3—32no—
245C22×D3—38no—
246C22×D3—92no—
247C22×D3—92no—
248C22×D3—98no—
249C22×D3—124no—
250C22×D3—364no—
251C2×C3×C4—34no—
252C2×C3×C4—62no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—94no—
256C2×C3×C4—94no—
257C2×C3×C4—92no—
258C2×C3×C4—364no—
259C2×C3×C4C2×D3212yes—
260#67—98no—
261#67—98no—
262#67—98no—
263#67—98no—
264C2×Dic3—32no—
265C2×Dic3—32no—
266C2×Dic3—38no—
267C2×Dic3—34no—
268C2×Dic3—34no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—64no—
272C2×Dic3—62no—
273C2×Dic3—94no—
274C2×Dic3—364no—
275D12—94no—
276D12—94no—
277D12—98no—
278C4×D3—124no—
279C4×D3—364no—
280C4×D3—364no—
281C4×D3—364no—
282C2×C32—24no—
283C2×C32—21no—
284C2×C32—22no—
285C2×C32—42no—
286C2×C32C22×C4×D312yes—
287C2×C32#100414yes—
288C3⋊D3—24no—
289C3⋊D3—24no—
290C3×D3—64no—
291C3×D3—64no—
292C24—92no—
293C2×D4—271no—
294C2×D4—271no—
295C2×D4—272no—
296C2×D4—278no—
297C22×C4—91no—
298C22×C4—94no—
299C22×C4—274no—
300C4⋊C4—92no—
301C4⋊C4—272no—
302C22⋊C4—92no—
303C22⋊C4—91no—
304C22⋊C4—94no—
305C22⋊C4—274no—
306C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C42—92no—
308C42—272no—
309C22×C3—22no—
310C22×C3—32no—
311C22×C3—34no—
312C22×C3—64no—
313C22×C3—61no—
314C22×C3—62no—
315C22×C3—94no—
316C22×C3—94no—
317C22×C3—122no—
318C22×C3—184no—
319C22×C3—184no—
320C22×C3—368no—
321C22×C3C4×D3212yes—
322C22×C3C22×D3211yescharacteristic
323C2×D3—34no—
324C2×D3—34no—
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—94no—
328C2×D3—94no—
329C2×D3—128no—
330C2×D3—184no—
331C2×D3—184no—
332C2×D3—368no—
333C3×C4—22no—
334C3×C4—64no—
335C3×C4—92no—
336C3×C4—92no—
337C3×C4—122no—
338C3×C4—184no—
339C3×C4—182no—
340Dic3—34no—
341Dic3—34no—
342Dic3—62no—
343Dic3—62no—
344Dic3—64no—
345Dic3—64no—
346Dic3—122no—
347Dic3—184no—
348C32—21no—
349C32#548612yescomplemented
350C23—92no—
351C23—92no—
352C23—98no—
353C23—364no—
354C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
355D4—274no—
356D4—274no—
357D4—278no—
358C2×C4—91no—
359C2×C4—91no—
360C2×C4—94no—
361C2×C4—92no—
362C2×C4—94no—
363C2×C4—364no—
364C2×C4—1084no—
365C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
366C2×C3—22no—
367C2×C3—34no—
368C2×C3—32no—
369C2×C3—64no—
370C2×C3—62no—
371C2×C3—61no—
372C2×C3—122no—
373C2×C3—184no—
374C2×C3—184no—
375C2×C3C2×C4×D3211yescharacteristic
376C2×C3#6431312yes—
377D3—64no—
378D3—64no—
379D3—184no—
380D3—184no—
381C22—22no—
382C22—94no—
383C22—94no—
384C22—184no—
385C22—184no—
386C22—368no—
387C22C4×((C3⋊D3)⋊C3)⋊C212yes—
388C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
389C4—22no—
390C4—92no—
391C4—92no—
392C4—184no—
393C4—182no—
394C3—32no—
395C3—61no—
396C3#1059418111yescharacteristic
397C2—22no—
398C2—184no—
399C2—184no—
400C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C2#1172250612yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC22×#6198C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#518
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721733C2semidirect
3. C22⋊C4×(C3⋊D3)⋊C3C2semidirect
4. #11721310C2semidirect
5. #11721298C2semidirect
6. #11721138C2semidirect
7. C4×#6198C2semidirect
8. #11721122C2semidirect
12. C2×C4×(C3⋊D3)⋊C3C22semidirect
13. C2×#6198C4semidirect
17. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
20. #88540C22semidirect
21. C4×#1084C22semidirect
23. #3752⋊C3C22semidirect
24. #88450C22semidirect
25. #88449C22semidirect
26. #88447C22semidirect
28. C22⋊C4×C3⋊D3D3semidirect
41. C22×(C3⋊D3)⋊C3C2×C4semidirect
45. C2×#1084C2×C4semidirect
51. #6198C2×C4semidirect
52. #6198C2×C4semidirect
53. #6198C2×C4semidirect
59. C2×C4×C3⋊D3C2×D3semidirect
73. #3752C2×D3semidirect
122. C22×C3⋊D3C4×D3semidirect
235. H(3,𝔽3)#518semidirect
349. C32#5486semidirect

Maximal subgroups

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

subgroupindex
C22×#61982
#117217332
C22⋊C4×(C3⋊D3)⋊C32
#117213102
#117212982
#117211382
C4×#61982
#117211222
#105941823