Finite Group Databasebeta

← #13149809

Subgroups — #13149809

Order 1728.

971 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 971 conjugacy classes of subgroups — too many to draw. Only its 193 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
1#11722491C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3#64110⋊C3C212yesmaximal, complemented
4#11721065C211yesmaximal, characteristic, complemented
5#11721055C212yesmaximal, complemented
6#11720841C214yesmaximal, complemented
7#11720835C214yesmaximal, complemented
8#10594668—32nomaximal
9C2×C4×#272C2211yescharacteristic
10#3828⋊C3C2212yes—
11#88677C412yes—
12#88672C412yes—
13#88655C411yescharacteristic
14C4×((C3⋊D3)⋊C3)⋊C2C2212yes—
15#88651C411yescharacteristic
16#88650C2212yes—
17#88649C412yes—
18C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
19C2×#6180C2211yescharacteristic, complemented
20#88527C2212yes—
21#88527C2212yescomplemented
22#3748⋊C3C2212yescomplemented
23#3743⋊C3C2214yescomplemented
24C8×(C3⋊D3)⋊C3C2214yescomplemented
25#88435C2212yes—
26#88434C2212yescomplemented
27#88430C2214yescomplemented
28((C3⋊D3)⋊C3)⋊C8C2214yescomplemented
29D3×#235—32no—
30#64110D312yescomplemented
31C3×#901—32no—
32C2×C3×ΓL(1,𝔽25)—32no—
33C2×C3×ΓL(1,𝔽25)—61no—
34#63817—32no—
35#63814—32no—
36D3×#208—32no—
37#63805—34no—
38#63804—34no—
39#63791—34no—
40D3×ΓL(1,𝔽25)—34no—
41C22×#272C2×C411yescharacteristic
42C2×((C3⋊D3)⋊C3)⋊C2C2×C412yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084C2×C411yescharacteristic
45C4×#272C2×C411yescharacteristic
46C4×#272C2×C411yescharacteristic
47C4×#272C2311yescharacteristic
48C4×#272C2311yescharacteristic
49#638⋊C3C2×C414yes—
50C4⋊((C3⋊D3)⋊C3)C2×C412yes—
51C4×(C3⋊D3)⋊C3C2314yes—
52#634⋊C3C2×C412yes—
53#6200C2×C414yes—
54#6199C2×C412yes—
55#6198C2×C412yes—
56((C3⋊D3)⋊C3)⋊C4C2×C414yes—
57#6196C2×C412yes—
58C8×H(3,𝔽3)C2312yes—
59#6180C2312yes—
60(C3⋊(C3⋊C8))⋊C3C2314yes—
61#5750—92no—
62#5745—91no—
63#3828C2×D312yes—
64C3×#235—32no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67D3×#67—34no—
68#3804—34no—
69D3×D12—32no—
70C4×D32—34no—
71#3799—32no—
72#3798—34no—
73#3795—32no—
74D3×Dic6—32no—
75C2×C32×C8—21no—
76C2×C32×C8C2×D312yescomplemented
77#3748C2×D312yescomplemented
78#3743C2×D314yescomplemented
79C8×C3⋊D3C2×D314yescomplemented
80C3×#208—32no—
81C2×C3×C3⋊C8—32no—
82C2×C3×C3⋊C8—61no—
83C3×ΓL(1,𝔽25)—32no—
84C3×ΓL(1,𝔽25)—32no—
85C3×ΓL(1,𝔽25)—32no—
86C3×ΓL(1,𝔽25)—32no—
87C3×ΓL(1,𝔽25)—34no—
88C3×ΓL(1,𝔽25)—62no—
89C3×ΓL(1,𝔽25)—62no—
90C3×D3×C8—34no—
91#3712—34no—
92#3711—34no—
93#3711—34no—
94#3710—34no—
95D3×C3⋊C8—34no—
96D3×C3⋊C8—34no—
97C22×H(3,𝔽3)C22×C411yescharacteristic
98C2×#272—41no—
99C2×#272C22×C411yescharacteristic
100C2×#272C22×C411yescharacteristic
101C2×(C3⋊D3)⋊C3C22×C414yes—
102((C3⋊D3)⋊C3)⋊C2—42no—
103C4×H(3,𝔽3)C22×C411yescharacteristic
104C4×H(3,𝔽3)C2411yescharacteristic
105#1084C22×C411yescharacteristic
106#1084C22×C411yescharacteristic
107(C3⋊Dic3)⋊C3C22×C414yes—
108D3×#41—92no—
109C2×#235—91no—
110C3×#131—92no—
111C2×C3×M16—91no—
112#948—92no—
113C2×#208—91no—
114#907—94no—
115#907—94no—
116#907—92no—
117D3×M16—94no—
118D3×M16—94no—
119D3×M16—92no—
120#901—32no—
121#901—92no—
122#901—94no—
123#901—92no—
124C2×ΓL(1,𝔽25)—32no—
125C2×ΓL(1,𝔽25)—61no—
126C2×ΓL(1,𝔽25)—94no—
127C22×C3×D3—32no—
128C22×C3×D3—61no—
129C2×D32—34no—
130C2×C32×C4—21no—
131C2×C32×C4C22×D312yes—
132#638C4×D314yes—
133C4⋊(C3⋊D3)C4×D312yes—
134C4×C3⋊D3C22×D314yes—
135#634C4×D312yes—
136C3×#67—34no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—61no—
139C3×D12—32no—
140C3×C4×D3—32no—
141C3×C4×D3—32no—
142C3×C4×D3—34no—
143C3×C4×D3—32no—
144C3×C4×D3—32no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C3×Dic6—32no—
150#627—34no—
151#626—34no—
152#626—34no—
153#625—34no—
154#624—34no—
155D3×Dic3—34no—
156D3×Dic3—34no—
157C32×C8—22no—
158C32×C8C22×D312yes—
159C32×C8C22×D312yes—
160C3⋊(C3⋊C8)C22×D314yes—
161C3×C3⋊C8—34no—
162C3×C3⋊C8—32no—
163C3×C3⋊C8—32no—
164C3×C3⋊C8—62no—
165#568—271noSylow
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)C23×C411yescharacteristic
168#272—21no—
169#272—21no—
170(C3⋊D3)⋊C3—24no—
171C3×#41—92no—
172C22×C3×C4—91no—
173C2×#67—92no—
174#239—92no—
175D3×Q8—92no—
176#237—92no—
177#237—94no—
178D3×D4—92no—
179D3×D4—94no—
180#235—32no—
181#235—92no—
182#235—92no—
183#235—92no—
184#235—94no—
185#235—92no—
186C2×D12—91no—
187C2×C4×D3—32no—
188C2×C4×D3—61no—
189C2×C4×D3—92no—
190C2×C4×D3—94no—
191C2×Dic6—91no—
192C3×M16—94no—
193C3×M16—92no—
194C3×M16—92no—
195C3×M16—92no—
196C2×C3×C8—32no—
197C2×C3×C8—61no—
198C2×C3×C8—94no—
199C2×C3×C8—91no—
200C2×C3×C8D3211yescharacteristic
201#208—32no—
202#208—92no—
203#208—92no—
204#208—94no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—61no—
207C2×C3⋊C8—92no—
208C2×C3⋊C8—94no—
209ΓL(1,𝔽25)—32no—
210ΓL(1,𝔽25)—32no—
211ΓL(1,𝔽25)—32no—
212ΓL(1,𝔽25)—32no—
213ΓL(1,𝔽25)—34no—
214ΓL(1,𝔽25)—62no—
215ΓL(1,𝔽25)—62no—
216ΓL(1,𝔽25)—94no—
217ΓL(1,𝔽25)—94no—
218ΓL(1,𝔽25)—94no—
219ΓL(1,𝔽25)—94no—
220D3×C8—34no—
221D3×C8—94no—
222D3×C8—94no—
223D3×C8—94no—
224D3×C8—94no—
225C22×C32—21no—
226C22×C32C2×C4×D312yes—
227C2×C3⋊D3C2×C4×D314yes—
228C2×C3×D3—34no—
229C2×C3×D3—32no—
230C2×C3×D3—32no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—122no—
234C2×C3×D3—122no—
235D32—124no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4C2×C4×D312yes—
239C32×C4C23×D312yes—
240C3⋊Dic3C2×C4×D314yes—
241C3×Dic3—34no—
242C3×Dic3—32no—
243C3×Dic3—32no—
244C3×Dic3—61no—
245C3×Dic3—61no—
246C2×#41—271no—
247#131—92no—
248#131—274no—
249#131—272no—
250C2×M16—91no—
251C2×M16—274no—
252C2×M16—271no—
253H(3,𝔽3)#56811yescharacteristic, complemented, Sylow
254C23×C3—91no—
255C22×D3—32no—
256C22×D3—61no—
257C22×D3—92no—
258C22×D3—94no—
259C3×Q8—92no—
260C3×D4—92no—
261C3×D4—94no—
262C2×C3×C4—32no—
263C2×C3×C4—61no—
264C2×C3×C4—94no—
265C2×C3×C4—91no—
266C2×C3×C4—91no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4C2×D3211yescharacteristic
271#67—34no—
272#67—94no—
273#67—94no—
274#67—94no—
275#67—94no—
276C2×Dic3—32no—
277C2×Dic3—61no—
278C2×Dic3—92no—
279C2×Dic3—94no—
280D12—32no—
281D12—92no—
282D12—92no—
283D12—94no—
284C4×D3—32no—
285C4×D3—32no—
286C4×D3—32no—
287C4×D3—32no—
288C4×D3—34no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—94no—
294C4×D3—94no—
295C4×D3—94no—
296C4×D3—94no—
297Dic6—32no—
298Dic6—92no—
299Dic6—92no—
300Dic6—94no—
301C3×C8—32no—
302C3×C8—32no—
303C3×C8—62no—
304C3×C8—94no—
305C3×C8—92no—
306C3×C8C2×D3212yes—
307C3⋊C8—32no—
308C3⋊C8—32no—
309C3⋊C8—34no—
310C3⋊C8—62no—
311C3⋊C8—94no—
312C2×C32—22no—
313C2×C32—22no—
314C2×C32—21no—
315C2×C32C22×C4×D312yes—
316C3⋊D3—24no—
317C3×D3—64no—
318C3×D3—62no—
319C3×D3—62no—
320C3×D3—121no—
321C3×D3—121no—
322#41—92no—
323#41—272no—
324#41—272no—
325#41—272no—
326C2×Q8—271no—
327C2×D4—271no—
328C2×D4—272no—
329C22×C4—91no—
330C22×C4—272no—
331M16—92no—
332M16—92no—
333M16—92no—
334M16—94no—
335M16—272no—
336M16—274no—
337C2×C8—91no—
338C2×C8—94no—
339C2×C8—274no—
340C2×C8—272no—
341C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
342C22×C3—32no—
343C22×C3—61no—
344C22×C3—94no—
345C22×C3—91no—
346C22×C3—91no—
347C22×C3—361no—
348C22×C3C4×D3211yescharacteristic
349C2×D3—32no—
350C2×D3—32no—
351C2×D3—34no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—94no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—362no—
358C2×D3—364no—
359C3×C4—32no—
360C3×C4—32no—
361C3×C4—61no—
362C3×C4—61no—
363C3×C4—94no—
364C3×C4—91no—
365C3×C4—91no—
366C3×C4C4×D3211yescharacteristic
367C3×C4C22×D3211yescharacteristic
368Dic3—32no—
369Dic3—32no—
370Dic3—34no—
371Dic3—61no—
372Dic3—61no—
373Dic3—94no—
374C32—21no—
375C32#575012yescomplemented
376C23—91no—
377C23—272no—
378Q8—92no—
379Q8—272no—
380D4—94no—
381D4—92no—
382D4—272no—
383D4—274no—
384C2×C4—91no—
385C2×C4—94no—
386C2×C4—91no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—91no—
390C2×C4—272no—
391C2×C4—274no—
392C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C8—92no—
394C8—94no—
395C8C2×((C3⋊D3)⋊C3)⋊C212yes—
396C2×C3—21no—
397C2×C3—32no—
398C2×C3—62no—
399C2×C3—62no—
400C2×C3—61no—
401C2×C3—184no—
402C2×C3—181no—
403C2×C3—181no—
404C2×C3C2×C4×D3211yescharacteristic
405D3—62no—
406D3—62no—
407D3—64no—
408D3—121no—
409D3—121no—
410D3—184no—
411C22—91no—
412C22—91no—
413C22—94no—
414C22—361no—
415C22—1082no—
416C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C4—91no—
418C4—91no—
419C4—94no—
420C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
422C3—32no—
423C3—61no—
424C3#1059467011yescharacteristic
425C2—21no—
426C2—181no—
427C2—181no—
428C2—184no—
429C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC4C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×C4×D32
prime radical#11722491C2
cosocleC4×H(3,𝔽3)C24
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#568
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88527C2semidirect
3. #64110⋊C3C2semidirect
4. #11721065C2semidirect
5. #11721055C2semidirect
6. #11720841C2semidirect
7. #11720835C2semidirect
18. C2×C8×H(3,𝔽3)C22semidirect
19. C2×#6180C22semidirect
21. #88527C22semidirect
22. #3748⋊C3C22semidirect
23. #3743⋊C3C22semidirect
24. C8×(C3⋊D3)⋊C3C22semidirect
26. #88434C22semidirect
27. #88430C22semidirect
28. ((C3⋊D3)⋊C3)⋊C8C22semidirect
30. #64110D3semidirect
76. C2×C32×C8C2×D3semidirect
77. #3748C2×D3semidirect
78. #3743C2×D3semidirect
79. C8×C3⋊D3C2×D3semidirect
253. H(3,𝔽3)#568semidirect
341. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
375. C32#5750semidirect

Maximal subgroups

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

subgroupindex
#117224912
C2×#885272
#64110⋊C32
#117210652
#117210552
#117208412
#117208352
#105946683