Finite Group Databasebeta

← #13151385

Subgroups — #13151385

Order 1728.

1606 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#88676C214yesmaximal, complemented
2#11721850C211yesmaximal, characteristic, complemented
3#64146⋊C3C212yesmaximal, complemented
4#11721312C214yesmaximal, complemented
5#11721163C212yesmaximal, complemented
6#11721138C212yesmaximal, complemented
7#10595295—32nomaximal
8C23×(C3⋊D3)⋊C3C2212yescomplemented
9C22×((C3⋊D3)⋊C3)⋊C2C2212yescomplemented
10C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×#6235C2214yescomplemented
12C2×C4⋊#272C2211yescharacteristic, complemented
13C2×#638⋊C3C2214yescomplemented
14#88676—216no—
15C2×#6200C2214yescomplemented
16C2×#6198C2212yes—
17C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
18C4×#1084C2211yescharacteristic, complemented
19#3757⋊C3C2212yescomplemented
20#3752⋊C3C2214yescomplemented
21#88451C2212yescomplemented
22#88449C2212yescomplemented
23C2×#3811—34no—
24C2×D3×#67—34no—
25#64146D312yescomplemented
26C3×#940—32no—
27C3×#940—61no—
28C3×#937—32no—
29#63970—34no—
30#63958—34no—
31#63922—34no—
32#63880—34no—
33C23×H(3,𝔽3)C2312yes—
34C22×#272C2312yes—
35C22×(C3⋊D3)⋊C3—28no—
36C22×(C3⋊D3)⋊C3—42no—
37C22×(C3⋊D3)⋊C3C2314yes—
38C2×((C3⋊D3)⋊C3)⋊C2—28no—
39C2×((C3⋊D3)⋊C3)⋊C2—42no—
40D4×H(3,𝔽3)—41no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#6235—28no—
43C2×#1084C2312yes—
44C4⋊#272—41no—
45#638⋊C3—28no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47#6200—28no—
48#6198D418yescomplemented
49((C3⋊D3)⋊C3)⋊C4—28no—
50#5717—91no—
51#5590—92no—
52C23×C3⋊D3C2×D312yescomplemented
53C23×C3×D3—32no—
54C22×D32—34no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D312yescomplemented
57C2×#638C2×D314yescomplemented
58C2×C3×#67—34no—
59C2×C3×#67—34no—
60C2×C3×#67—34no—
61C2×C3×#67—64no—
62C2×C3×D12—32no—
63C2×C3×D12—61no—
64#3811—616no—
65D3×#67—616no—
66C2×#626—34no—
67C2×#626—34no—
68C2×#625—34no—
69C2×#624—34no—
70C2×D3×Dic3—34no—
71#3757C2×D312yescomplemented
72#3752C2×D314yescomplemented
73C3×#217—32no—
74C3×(C2×D3)⋊C4—34no—
75C3×C4×Dic3—32no—
76C3×C4×Dic3—61no—
77Dic3⋊Dic3—34no—
78(C2×D3)⋊Dic3—34no—
79C22×H(3,𝔽3)—24no—
80C22×H(3,𝔽3)—22no—
81C22×H(3,𝔽3)C2411yescharacteristic
82C2×#272—24no—
83C2×#272—22no—
84C2×(C3⋊D3)⋊C3—24no—
85C2×(C3⋊D3)⋊C3—24no—
86C2×(C3⋊D3)⋊C3—48no—
87C2×(C3⋊D3)⋊C3C2×D418yescomplemented
88((C3⋊D3)⋊C3)⋊C2—48no—
89C4×H(3,𝔽3)—21no—
90#1084C2×D414yescomplemented
91(C3⋊Dic3)⋊C3—24no—
92C22×#67—94no—
93C2×D3×D4—94no—
94C2×D3×D4—94no—
95C3×#127—91no—
96C3×#120—92no—
97#940—32no—
98#940—61no—
99#939—94no—
100#938—94no—
101#937—32no—
102#937—92no—
103#937—92no—
104#884—94no—
105#882—94no—
106#67⋊C4—94no—
107C4×D12—92no—
108C23×C32—22no—
109C23×C32C22×D312yes—
110C23×C32C22×D312yes—
111C22×C3⋊D3—28no—
112C22×C3⋊D3—42no—
113C22×C3⋊D3C22×D314yes—
114C22×C3×D3—34no—
115C22×C3×D3—34no—
116C22×C3×D3—62no—
117C22×C3×D3—68no—
118C22×C3×D3—122no—
119C2×D32—68no—
120C2×D32—68no—
121C2×D32—124no—
122C32×D4—42no—
123C32×D4—42no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D312yes—
126#638—28no—
127C2×C3⋊Dic3C22×D314yes—
128C3×#67—68no—
129C3×#67—68no—
130C3×#67—68no—
131C3×#67—128no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—62no—
135C3×D12—122no—
136C3×D12—122no—
137#626—316no—
138#626—68no—
139#625—68no—
140#624—68no—
141D3×Dic3—68no—
142D42—271noSylow
143C2×H(3,𝔽3)—42no—
144C2×H(3,𝔽3)C22×D412yes—
145C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
146#272—42no—
147(C3⋊D3)⋊C3—28no—
148C24×C3—92no—
149C23×D3—32no—
150C23×D3—92no—
151C23×D3—92no—
152C23×D3—94no—
153C2×C3×D4—32no—
154C2×C3×D4—61no—
155C2×C3×D4—94no—
156C2×C3×D4—94no—
157C2×C3×D4—91no—
158C2×C3×D4D3211yescharacteristic
159C2×#67—34no—
160C2×#67—34no—
161C2×#67—34no—
162C2×#67—64no—
163C2×#67—94no—
164C2×#67—94no—
165C2×#67—94no—
166C2×#67—94no—
167C2×#67—1816no—
168C22×Dic3—94no—
169D3×D4—1816no—
170D3×D4—1816no—
171C2×D12—32no—
172C2×D12—61no—
173C2×D12—92no—
174C2×D12—94no—
175C2×C4×D3—94no—
176C2×C4×D3—94no—
177C3×C22⋊C4—94no—
178C3×C22⋊C4—92no—
179C3×C42—91no—
180#217—32no—
181#217—92no—
182#217—92no—
183(C2×D3)⋊C4—34no—
184(C2×D3)⋊C4—94no—
185(C2×D3)⋊C4—94no—
186C4⋊Dic3—92no—
187Dic3⋊C4—94no—
188C4×Dic3—32no—
189C4×Dic3—61no—
190C22×C32—24no—
191C22×C32—24no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—44no—
196C22×C32—42no—
197C22×C32C23×D312yes—
198C2×C3⋊D3—24no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—48no—
201C2×C3⋊D3D3×D418yescomplemented
202C2×C3×D3—38no—
203C2×C3×D3—64no—
204C2×C3×D3—64no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C2×C3×D3—122no—
209C2×C3×D3—124no—
210C2×C3×D3—128no—
211D32—1216no—
212C32×C4—22no—
213C32×C4—22no—
214C3⋊Dic3—24no—
215C3×Dic3—38no—
216C3×Dic3—64no—
217C3×Dic3—64no—
218C22×D4—274no—
219#127—91no—
220#121—274no—
221#120—92no—
222#120—272no—
223C4×D4—272no—
224H(3,𝔽3)D4211yescharacteristic, complemented, Sylow
225C23×C3—32no—
226C23×C3—32no—
227C23×C3—62no—
228C23×C3—94no—
229C23×C3—92no—
230C23×C3—188no—
231C23×C3—362no—
232C23×C3C2×D3212yes—
233C22×D3—34no—
234C22×D3—34no—
235C22×D3—62no—
236C22×D3—68no—
237C22×D3—94no—
238C22×D3—122no—
239C22×D3—188no—
240C22×D3—188no—
241C22×D3—188no—
242C22×D3—188no—
243C22×D3—362no—
244C22×D3—362no—
245C22×D3—364no—
246C3×D4—41no—
247C3×D4—122no—
248C3×D4—122no—
249C3×D4—188no—
250C3×D4—188no—
251C3×D4—361no—
252C2×C3×C4—32no—
253C2×C3×C4—61no—
254C2×C3×C4—92no—
255C2×C3×C4—94no—
256C2×C3×C4C2×D3211yescharacteristic
257#67—68no—
258#67—68no—
259#67—68no—
260#67—916no—
261#67—128no—
262#67—188no—
263#67—188no—
264#67—188no—
265C2×Dic3—34no—
266C2×Dic3—34no—
267C2×Dic3—62no—
268C2×Dic3—94no—
269C2×Dic3—188no—
270D12—98no—
271D12—122no—
272D12—122no—
273D12—188no—
274C4×D3—188no—
275C4×D3—188no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32—42no—
279C2×C32—42no—
280C2×C32—82no—
281C2×C32C2×D3×D414yes—
282C2×C32#100412yes—
283C3⋊D3—28no—
284C3×D3—68no—
285C3×D3—124no—
286C3×D3—242no—
287C24—92no—
288C24—272no—
289C2×D4—94no—
290C2×D4—94no—
291C2×D4—91no—
292C2×D4—272no—
293C2×D4—274no—
294C2×D4—5416no—
295C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
296C22×C4—274no—
297C4⋊C4—272no—
298C22⋊C4—92no—
299C22⋊C4—94no—
300C22⋊C4—272no—
301C42—91no—
302C22×C3—24no—
303C22×C3—22no—
304C22×C3—32no—
305C22×C3—64no—
306C22×C3—64no—
307C22×C3—62no—
308C22×C3—62no—
309C22×C3—61no—
310C22×C3—98no—
311C22×C3—124no—
312C22×C3—122no—
313C22×C3—184no—
314C22×C3—184no—
315C22×C3—184no—
316C22×C3—182no—
317C22×C3—368no—
318C22×C3C22×D3211yescharacteristic
319C2×D3—38no—
320C2×D3—64no—
321C2×D3—64no—
322C2×D3—64no—
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—98no—
326C2×D3—122no—
327C2×D3—124no—
328C2×D3—128no—
329C2×D3—184no—
330C2×D3—184no—
331C2×D3—368no—
332C2×D3—368no—
333C2×D3—3616no—
334C3×C4—21no—
335C3×C4—62no—
336C3×C4—62no—
337C3×C4—94no—
338C3×C4—184no—
339Dic3—38no—
340Dic3—64no—
341Dic3—64no—
342Dic3—184no—
343C32—21no—
344C32#559012yescomplemented
345C23—94no—
346C23—92no—
347C23—188no—
348C23—362no—
349C23—548no—
350C23—1082no—
351C23C2×((C3⋊D3)⋊C3)⋊C212yes—
352D4—41no—
353D4—188no—
354D4—188no—
355D4—278no—
356D4—361no—
357D4—548no—
358C2×C4—92no—
359C2×C4—94no—
360C2×C4—548no—
361C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C2×C3—34no—
363C2×C3—32no—
364C2×C3—42no—
365C2×C3—62no—
366C2×C3—61no—
367C2×C3—122no—
368C2×C3—122no—
369C2×C3—188no—
370C2×C3—242no—
371C2×C3—362no—
372C2×C3D32×D412yes—
373C2×C3#6433011yescharacteristic
374D3—68no—
375D3—124no—
376D3—188no—
377D3—242no—
378C22—24no—
379C22—22no—
380C22—98no—
381C22—184no—
382C22—184no—
383C22—184no—
384C22—182no—
385C22—368no—
386C22—1088no—
387C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
388C4—21no—
389C4—94no—
390C4—184no—
391C3—32no—
392C3—61no—
393C3#1059528911yescharacteristic
394C2—42no—
395C2—188no—
396C2—362no—
397C2D4×((C3⋊D3)⋊C3)⋊C212yes—
398C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)D42
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88676C2semidirect
2. #11721850C2semidirect
3. #64146⋊C3C2semidirect
4. #11721312C2semidirect
5. #11721163C2semidirect
6. #11721138C2semidirect
8. C23×(C3⋊D3)⋊C3C22semidirect
9. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
10. C2×D4×H(3,𝔽3)C22semidirect
11. C2×#6235C22semidirect
12. C2×C4⋊#272C22semidirect
13. C2×#638⋊C3C22semidirect
15. C2×#6200C22semidirect
17. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
18. C4×#1084C22semidirect
19. #3757⋊C3C22semidirect
20. #3752⋊C3C22semidirect
21. #88451C22semidirect
22. #88449C22semidirect
25. #64146D3semidirect
48. #6198D4semidirect
52. C23×C3⋊D3C2×D3semidirect
56. C2×C32×D4C2×D3semidirect
57. C2×#638C2×D3semidirect
71. #3757C2×D3semidirect
72. #3752C2×D3semidirect
87. C2×(C3⋊D3)⋊C3C2×D4semidirect
90. #1084C2×D4semidirect
201. C2×C3⋊D3D3×D4semidirect
224. H(3,𝔽3)D42semidirect
295. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
344. C32#5590semidirect

Maximal subgroups

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

subgroupindex
C2×#886762
#117218502
#64146⋊C32
#117213122
#117211632
#117211382
#105952953