Finite Group Databasebeta

← #13150897

Subgroups — #13150897

Order 1728.

1343 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1343 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88677C212yesmaximal, complemented
2C2×#88542C211yesmaximal, characteristic, complemented
3#64139⋊C3C212yesmaximal, complemented
4#11721318C212yesmaximal, complemented
5#11721162C214yesmaximal, complemented
6#11721102C214yesmaximal, complemented
7#10595072—32nomaximal
8C23×#272C2211yescharacteristic
9C22×((C3⋊D3)⋊C3)⋊C2C2212yescomplemented
10C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
11C22×#1084C2211yescharacteristic, complemented
12C2×#638⋊C3C2214yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
14#88677—28no—
15C2×#6200C2214yes—
16C2×#6199C2212yescomplemented
17C2×#6198C2212yescomplemented
18#88542C2214yescomplemented
19#3752⋊C3C2214yescomplemented
20C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
21#88450C2214yescomplemented
22#88449C2212yescomplemented
23C2×D3×#67—34no—
24#64139D312yescomplemented
25C3×#930—32no—
26C2×C3×(C2×D3)⋊C4—32no—
27C2×C3×(C2×D3)⋊C4—61no—
28#63978—34no—
29#63923—34no—
30#63922—34no—
31#63891—34no—
32#63890—34no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×#272—41no—
35C22×#272—41no—
36C22×#272D414yescomplemented
37C22×#272C2312yes—
38C22×(C3⋊D3)⋊C3C2314yes—
39C2×((C3⋊D3)⋊C3)⋊C2—24no—
40C2×((C3⋊D3)⋊C3)⋊C2—42no—
41C2×((C3⋊D3)⋊C3)⋊C2—42no—
42C2×C4×H(3,𝔽3)—41no—
43C2×C4×H(3,𝔽3)C2312yes—
44C2×#1084—41no—
45C2×#1084C2312yes—
46#638⋊C3—28no—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C4⋊((C3⋊D3)⋊C3)—42no—
49#6200—28no—
50#6199—24no—
51#6198—42no—
52#5605—92no—
53#5493—91no—
54C23×C3×D3—32no—
55C23×C3×D3—61no—
56C22×D32—34no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D312yescomplemented
59C2×#638C2×D314yescomplemented
60C2×C4⋊(C3⋊D3)C2×D312yescomplemented
61C2×C3×#67—34no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C2×C3×D12—32no—
65D3×#67—616no—
66C2×#626—34no—
67C2×#626—34no—
68C2×#625—34no—
69C2×D3×Dic3—34no—
70#3752C2×D314yescomplemented
71C4⋊(C3⋊Dic3)C2×D312yescomplemented
72C3×(C2×D3)⋊C4—34no—
73C3×(C2×D3)⋊C4—34no—
74C3×(C2×D3)⋊C4—34no—
75C3×(C2×D3)⋊C4—64no—
76C3×C4⋊Dic3—32no—
77Dic3⋊Dic3—34no—
78#3722—34no—
79(C2×D3)⋊Dic3—34no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)—21no—
82C22×H(3,𝔽3)C2×D412yes—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×#272—22no—
85C2×#272—22no—
86C2×#272—44no—
87C2×#272C2×D414yes—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3—24no—
91((C3⋊D3)⋊C3)⋊C2—48no—
92C4×H(3,𝔽3)—22no—
93#1084—22no—
94(C3⋊Dic3)⋊C3—24no—
95C22×#67—92no—
96C2×D3×D4—94no—
97C3×#121—92no—
98C2×C3×C22⋊C4—91no—
99#939—94no—
100#937—94no—
101#930—32no—
102#930—92no—
103#930—92no—
104C2×(C2×D3)⋊C4—32no—
105C2×(C2×D3)⋊C4—61no—
106#894—94no—
107#886—94no—
108#883—94no—
109#882—94no—
110#882—94no—
111C23×C32—21no—
112C23×C32C22×D312yes—
113C22×C3⋊D3C22×D314yes—
114C22×C3×D3—34no—
115C22×C3×D3—38no—
116C22×C3×D3—34no—
117C22×C3×D3—64no—
118C22×C3×D3—62no—
119C22×C3×D3—122no—
120C22×C3×D3—122no—
121C22×C3×D3—122no—
122C22×C3×D3—122no—
123C2×D32—68no—
124C2×D32—124no—
125C2×D32—124no—
126C2×C32×C4—22no—
127C2×C32×C4—42no—
128C2×C32×C4—42no—
129C2×C32×C4C22×D312yes—
130C2×C32×C4C22×D312yes—
131#638—28no—
132C2×C3⋊Dic3C22×D314yes—
133C4⋊(C3⋊D3)—42no—
134C3×#67—68no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—122no—
139C2×C3×Dic3—122no—
140C3×D12—122no—
141#626—68no—
142#626—124no—
143#625—68no—
144D3×Dic3—68no—
145#534—271noSylow
146C2×H(3,𝔽3)—22no—
147C2×H(3,𝔽3)C22×D411yescharacteristic
148C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
149C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
150#272—24no—
151(C3⋊D3)⋊C3—44no—
152C24×C3—91no—
153C23×D3—32no—
154C23×D3—61no—
155C23×D3—92no—
156C23×D3—94no—
157C2×C3×D4—92no—
158C2×C3×D4—94no—
159C22×C3×C4—32no—
160C22×C3×C4—61no—
161C22×C3×C4—91no—
162C22×C3×C4D3211yescharacteristic
163C2×#67—34no—
164C2×#67—94no—
165C2×#67—94no—
166C2×#67—94no—
167C2×#67—94no—
168C2×#67—188no—
169C22×Dic3—32no—
170C22×Dic3—61no—
171C22×Dic3—92no—
172D3×D4—1816no—
173C2×D12—32no—
174C2×D12—92no—
175C2×D12—94no—
176C2×D12—92no—
177C2×C4×D3—94no—
178C3×C4⋊C4—92no—
179C3×C22⋊C4—94no—
180C3×C22⋊C4—94no—
181#217—94no—
182(C2×D3)⋊C4—34no—
183(C2×D3)⋊C4—34no—
184(C2×D3)⋊C4—34no—
185(C2×D3)⋊C4—64no—
186(C2×D3)⋊C4—94no—
187(C2×D3)⋊C4—94no—
188(C2×D3)⋊C4—94no—
189C4⋊Dic3—32no—
190C4⋊Dic3—92no—
191C4⋊Dic3—92no—
192Dic3⋊C4—94no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—21no—
198C22×C32—22no—
199C22×C32D3×D414yes—
200C22×C32C23×D312yes—
201C2×C3⋊D3—24no—
202C2×C3⋊D3—24no—
203C2×C3⋊D3—24no—
204C2×C3×D3—38no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213C2×C3×D3—128no—
214C2×C3×D3—128no—
215D32—1216no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4—42no—
219C3⋊Dic3—24no—
220C3×Dic3—64no—
221C3×Dic3—64no—
222C3×Dic3—122no—
223C22×D4—272no—
224#123—274no—
225#121—92no—
226#121—272no—
227#120—274no—
228C2×C22⋊C4—91no—
229H(3,𝔽3)#53411yescharacteristic, complemented, Sylow
230C23×C3—32no—
231C23×C3—61no—
232C23×C3—94no—
233C23×C3—92no—
234C23×C3—94no—
235C23×C3—361no—
236C23×C3—361no—
237C23×C3C2×D3211yescharacteristic
238C22×D3—34no—
239C22×D3—38no—
240C22×D3—34no—
241C22×D3—62no—
242C22×D3—64no—
243C22×D3—94no—
244C22×D3—122no—
245C22×D3—122no—
246C22×D3—122no—
247C22×D3—122no—
248C22×D3—188no—
249C22×D3—184no—
250C22×D3—362no—
251C22×D3—362no—
252C22×D3—364no—
253C22×D3—364no—
254C3×D4—188no—
255C3×D4—362no—
256C2×C3×C4—32no—
257C2×C3×C4—32no—
258C2×C3×C4—41no—
259C2×C3×C4—62no—
260C2×C3×C4—94no—
261C2×C3×C4—92no—
262C2×C3×C4—122no—
263C2×C3×C4—122no—
264C2×C3×C4—361no—
265C2×C3×C4C2×D3212yes—
266#67—68no—
267#67—188no—
268#67—188no—
269#67—188no—
270#67—364no—
271C2×Dic3—34no—
272C2×Dic3—34no—
273C2×Dic3—62no—
274C2×Dic3—94no—
275C2×Dic3—122no—
276C2×Dic3—122no—
277C2×Dic3—184no—
278D12—122no—
279D12—188no—
280D12—362no—
281D12—362no—
282C4×D3—188no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32C2×D3×D412yes—
289C2×C32#100412yes—
290C2×C32#100912yes—
291C3⋊D3—44no—
292C3×D3—68no—
293C3×D3—124no—
294C3×D3—124no—
295C24—91no—
296C24—272no—
297C2×D4—94no—
298C2×D4—92no—
299C2×D4—274no—
300C2×D4—272no—
301C2×D4—548no—
302C22×C4—91no—
303C22×C4—272no—
304C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
305C4⋊C4—92no—
306C4⋊C4—272no—
307C22⋊C4—94no—
308C22⋊C4—94no—
309C22⋊C4—274no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—32no—
313C22×C3—34no—
314C22×C3—62no—
315C22×C3—62no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—61no—
319C22×C3—62no—
320C22×C3—94no—
321C22×C3—184no—
322C22×C3—184no—
323C22×C3—184no—
324C22×C3—182no—
325C22×C3—182no—
326C22×C3—364no—
327C22×C3#380212yes—
328C22×C3C22×D3211yescharacteristic
329C2×D3—38no—
330C2×D3—64no—
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—64no—
334C2×D3—64no—
335C2×D3—64no—
336C2×D3—122no—
337C2×D3—122no—
338C2×D3—128no—
339C2×D3—128no—
340C2×D3—184no—
341C2×D3—184no—
342C2×D3—184no—
343C2×D3—3616no—
344C2×D3—368no—
345C3×C4—22no—
346C3×C4—62no—
347C3×C4—62no—
348C3×C4—122no—
349C3×C4—184no—
350C3×C4—182no—
351Dic3—64no—
352Dic3—64no—
353Dic3—122no—
354Dic3—184no—
355C32—21no—
356C32#560512yescomplemented
357C23—92no—
358C23—94no—
359C23—94no—
360C23—361no—
361C23—361no—
362C23—544no—
363C23—1082no—
364C23—1082no—
365C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366D4—188no—
367D4—362no—
368D4—548no—
369D4—1082no—
370C2×C4—41no—
371C2×C4—92no—
372C2×C4—94no—
373C2×C4—361no—
374C2×C4—544no—
375C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
376C2×C3—22no—
377C2×C3—32no—
378C2×C3—32no—
379C2×C3—32no—
380C2×C3—64no—
381C2×C3—64no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—184no—
386C2×C3—364no—
387C2×C3C2×#380211yescharacteristic
388C2×C3#6430811yescharacteristic
389C2×C3#6433011yescharacteristic
390D3—68no—
391D3—124no—
392D3—124no—
393D3—364no—
394C22—21no—
395C22—21no—
396C22—94no—
397C22—182no—
398C22—182no—
399C22—184no—
400C22—184no—
401C22—184no—
402C22—364no—
403C22—1088no—
404C22#8865512yes—
405C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C4—22no—
407C4—184no—
408C4—182no—
409C3—32no—
410C3—61no—
411C3#1059508411yescharacteristic
412C2—22no—
413C2—184no—
414C2—364no—
415C2C2×#8865511yescharacteristic
416C2#1172249311yescharacteristic
417C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#534
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88677C2semidirect
2. C2×#88542C2semidirect
3. #64139⋊C3C2semidirect
4. #11721318C2semidirect
5. #11721162C2semidirect
6. #11721102C2semidirect
9. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
10. C22×C4×H(3,𝔽3)C22semidirect
11. C22×#1084C22semidirect
12. C2×#638⋊C3C22semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
16. C2×#6199C22semidirect
17. C2×#6198C22semidirect
18. #88542C22semidirect
19. #3752⋊C3C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. #88450C22semidirect
22. #88449C22semidirect
24. #64139D3semidirect
36. C22×#272D4semidirect
58. C22×C32×C4C2×D3semidirect
59. C2×#638C2×D3semidirect
60. C2×C4⋊(C3⋊D3)C2×D3semidirect
70. #3752C2×D3semidirect
71. C4⋊(C3⋊Dic3)C2×D3semidirect
229. H(3,𝔽3)#534semidirect
304. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
356. C32#5605semidirect

Maximal subgroups

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

subgroupindex
C2×#886772
C2×#885422
#64139⋊C32
#117213182
#117211622
#117211022
#105950723