Finite Group Databasebeta

← #13149876

Subgroups — #13149876

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
2M16×#272C211yesmaximal, characteristic, complemented
3#64116⋊C3C212yesmaximal, complemented
4C2×#88435C211yesmaximal, characteristic, complemented
5#11721054C212yesmaximal, complemented
6#11720842C212yesmaximal, complemented
7#11720840C212yesmaximal, complemented
8#11720835C214yesmaximal, complemented
9#10594706—32nomaximal
10C2×C4×#272C2211yescharacteristic
11#3828⋊C3C2212yes—
12#88677C412yes—
13#88672C412yes—
14#88655C411yescharacteristic
15C4×((C3⋊D3)⋊C3)⋊C2C2212yes—
16#88651C411yescharacteristic
17#88650C2212yes—
18#88649C412yes—
19M16×H(3,𝔽3)C2211yescharacteristic, complemented
20#88538C2211yescharacteristic, complemented
21#88527C2212yes—
22C8×#272C2212yescomplemented
23C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
24#3743⋊C3C2214yescomplemented
25C8×(C3⋊D3)⋊C3C2214yescomplemented
26#88435C2212yes—
27#88435C2211yescharacteristic, complemented
28#88435C2211yescharacteristic, complemented
29#88430C2214yescomplemented
30((C3⋊D3)⋊C3)⋊C8C2214yescomplemented
31D3×#235—32no—
32#64116D312yescomplemented
33C3×#907—32no—
34C3×D3×M16—32no—
35C3×D3×M16—61no—
36C2×#3711—32no—
37#63817—32no—
38#63815—32no—
39#63804—34no—
40#63803—34no—
41#63791—34no—
42D3×ΓL(1,𝔽25)—34no—
43C22×#272C2×C411yescharacteristic
44C2×((C3⋊D3)⋊C3)⋊C2C2×C412yes—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×#1084C2×C411yescharacteristic
47C4×#272C2×C411yescharacteristic
48C4×#272C2×C411yescharacteristic
49C4×#272C2311yescharacteristic
50C4×#272C2311yescharacteristic
51#638⋊C3C2×C414yes—
52C4⋊((C3⋊D3)⋊C3)C2×C412yes—
53C4×(C3⋊D3)⋊C3C2314yes—
54#634⋊C3C2×C412yes—
55#6200C2×C414yes—
56#6199C2×C412yes—
57#6198C2×C412yes—
58((C3⋊D3)⋊C3)⋊C4C2×C414yes—
59#6196C2×C412yes—
60C8×H(3,𝔽3)C2312yes—
61#6180C2312yes—
62(C3⋊(C3⋊C8))⋊C3C2312yes—
63(C3⋊(C3⋊C8))⋊C3C2312yes—
64#5750—92no—
65#5745—91no—
66#3828C2×D312yes—
67C3×#235—32no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—61no—
70D3×#67—34no—
71#3804—34no—
72D3×D12—32no—
73C4×D32—34no—
74#3799—32no—
75#3798—34no—
76#3795—32no—
77D3×Dic6—32no—
78C32×M16—21no—
79C32×M16C2×D312yescomplemented
80C2×C3⋊(C3⋊C8)C2×D312yescomplemented
81#3743C2×D314yescomplemented
82C8×C3⋊D3C2×D314yescomplemented
83C3×#208—32no—
84C3×#208—61no—
85C2×C3×C3⋊C8—32no—
86C3×ΓL(1,𝔽25)—34no—
87C3×ΓL(1,𝔽25)—34no—
88C3×ΓL(1,𝔽25)—62no—
89C3×D3×C8—34no—
90C3×D3×C8—34no—
91C3×D3×C8—62no—
92#3712—34no—
93#3711—32no—
94#3711—32no—
95#3711—34no—
96#3711—32no—
97#3711—32no—
98#3710—34no—
99D3×C3⋊C8—34no—
100C22×H(3,𝔽3)C22×C411yescharacteristic
101C2×#272—41no—
102C2×#272C22×C411yescharacteristic
103C2×#272C22×C411yescharacteristic
104C2×(C3⋊D3)⋊C3C22×C414yes—
105((C3⋊D3)⋊C3)⋊C2—42no—
106C4×H(3,𝔽3)C22×C411yescharacteristic
107C4×H(3,𝔽3)C2411yescharacteristic
108#1084C22×C411yescharacteristic
109#1084C22×C411yescharacteristic
110(C3⋊Dic3)⋊C3C22×C414yes—
111D3×#41—92no—
112C2×#235—91no—
113C3×#131—92no—
114C2×C3×M16—91no—
115#948—92no—
116C2×#208—91no—
117#907—32no—
118#907—92no—
119#907—94no—
120#907—92no—
121D3×M16—32no—
122D3×M16—61no—
123D3×M16—94no—
124D3×M16—94no—
125#901—92no—
126#901—94no—
127#901—92no—
128#901—92no—
129C2×ΓL(1,𝔽25)—94no—
130C2×ΓL(1,𝔽25)—92no—
131C22×C3×D3—32no—
132C22×C3×D3—61no—
133C2×D32—34no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D312yes—
136#638C4×D314yes—
137C4⋊(C3⋊D3)C4×D312yes—
138C4×C3⋊D3C22×D314yes—
139#634C4×D312yes—
140C3×#67—34no—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—61no—
143C3×D12—32no—
144C3×C4×D3—32no—
145C3×C4×D3—32no—
146C3×C4×D3—34no—
147C3×C4×D3—32no—
148C3×C4×D3—32no—
149C3×C4×D3—61no—
150C3×C4×D3—61no—
151C3×C4×D3—61no—
152C3×C4×D3—61no—
153C3×Dic6—32no—
154#627—34no—
155#626—34no—
156#626—34no—
157#625—34no—
158#624—34no—
159D3×Dic3—34no—
160D3×Dic3—34no—
161C32×C8—22no—
162C32×C8C22×D314yes—
163C3⋊(C3⋊C8)C22×D312yes—
164C3⋊(C3⋊C8)C22×D312yes—
165C3×C3⋊C8—32no—
166C3×C3⋊C8—32no—
167C3×C3⋊C8—34no—
168C3×C3⋊C8—62no—
169#568—271noSylow
170C2×H(3,𝔽3)—21no—
171C2×H(3,𝔽3)C23×C411yescharacteristic
172#272—21no—
173#272—21no—
174(C3⋊D3)⋊C3—24no—
175C3×#41—92no—
176C22×C3×C4—91no—
177C2×#67—92no—
178#239—92no—
179D3×Q8—92no—
180#237—92no—
181#237—94no—
182D3×D4—92no—
183D3×D4—94no—
184#235—32no—
185#235—92no—
186#235—92no—
187#235—92no—
188#235—94no—
189#235—92no—
190C2×D12—91no—
191C2×C4×D3—32no—
192C2×C4×D3—61no—
193C2×C4×D3—92no—
194C2×C4×D3—94no—
195C2×Dic6—91no—
196C3×M16—32no—
197C3×M16—61no—
198C3×M16—94no—
199C3×M16—91no—
200C3×M16—92no—
201C3×M16D3211yescharacteristic
202C2×C3×C8—94no—
203C2×C3×C8—92no—
204C2×C3×C8—92no—
205#208—32no—
206#208—61no—
207#208—91no—
208#208—91no—
209#208—92no—
210#208—94no—
211C2×C3⋊C8—32no—
212C2×C3⋊C8—92no—
213C2×C3⋊C8—94no—
214ΓL(1,𝔽25)—34no—
215ΓL(1,𝔽25)—34no—
216ΓL(1,𝔽25)—62no—
217ΓL(1,𝔽25)—94no—
218ΓL(1,𝔽25)—94no—
219ΓL(1,𝔽25)—94no—
220ΓL(1,𝔽25)—92no—
221ΓL(1,𝔽25)—92no—
222ΓL(1,𝔽25)—92no—
223ΓL(1,𝔽25)—92no—
224D3×C8—34no—
225D3×C8—34no—
226D3×C8—62no—
227D3×C8—94no—
228D3×C8—94no—
229D3×C8—94no—
230C22×C32—21no—
231C22×C32C2×C4×D312yes—
232C2×C3⋊D3C2×C4×D314yes—
233C2×C3×D3—34no—
234C2×C3×D3—32no—
235C2×C3×D3—32no—
236C2×C3×D3—61no—
237C2×C3×D3—61no—
238C2×C3×D3—122no—
239C2×C3×D3—122no—
240D32—124no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4C2×C4×D312yes—
244C32×C4C23×D312yes—
245C3⋊Dic3C2×C4×D314yes—
246C3×Dic3—34no—
247C3×Dic3—32no—
248C3×Dic3—32no—
249C3×Dic3—61no—
250C3×Dic3—61no—
251C2×#41—271no—
252#131—92no—
253#131—272no—
254#131—272no—
255#131—272no—
256C2×M16—91no—
257C2×M16—271no—
258C2×M16—274no—
259H(3,𝔽3)#56811yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C22×D3—32no—
262C22×D3—61no—
263C22×D3—92no—
264C22×D3—94no—
265C3×Q8—92no—
266C3×D4—92no—
267C3×D4—94no—
268C2×C3×C4—32no—
269C2×C3×C4—61no—
270C2×C3×C4—94no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—91no—
276C2×C3×C4C2×D3211yescharacteristic
277#67—34no—
278#67—94no—
279#67—94no—
280#67—94no—
281#67—94no—
282C2×Dic3—32no—
283C2×Dic3—61no—
284C2×Dic3—92no—
285C2×Dic3—94no—
286D12—32no—
287D12—92no—
288D12—92no—
289D12—94no—
290C4×D3—32no—
291C4×D3—32no—
292C4×D3—32no—
293C4×D3—32no—
294C4×D3—34no—
295C4×D3—61no—
296C4×D3—61no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—94no—
300C4×D3—94no—
301C4×D3—94no—
302C4×D3—94no—
303Dic6—32no—
304Dic6—92no—
305Dic6—92no—
306Dic6—94no—
307C3×C8—34no—
308C3×C8—62no—
309C3×C8—92no—
310C3×C8—92no—
311C3×C8—92no—
312C3×C8C2×D3212yes—
313C3⋊C8—34no—
314C3⋊C8—32no—
315C3⋊C8—32no—
316C3⋊C8—62no—
317C3⋊C8—92no—
318C3⋊C8—92no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—41no—
322C2×C32C22×C4×D312yes—
323C3⋊D3—24no—
324C3×D3—62no—
325C3×D3—64no—
326C3×D3—62no—
327C3×D3—62no—
328C3×D3—121no—
329#41—92no—
330#41—272no—
331#41—272no—
332#41—272no—
333C2×Q8—271no—
334C2×D4—271no—
335C2×D4—272no—
336C22×C4—91no—
337C22×C4—272no—
338M16—91no—
339M16—92no—
340M16—94no—
341M16—272no—
342M16—271no—
343M16—274no—
344M16—271no—
345M16((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
346C2×C8—92no—
347C2×C8—92no—
348C2×C8—94no—
349C2×C8—274no—
350C22×C3—32no—
351C22×C3—61no—
352C22×C3—94no—
353C22×C3—91no—
354C22×C3—91no—
355C22×C3—361no—
356C22×C3C4×D3211yescharacteristic
357C2×D3—32no—
358C2×D3—32no—
359C2×D3—34no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—94no—
363C2×D3—122no—
364C2×D3—122no—
365C2×D3—362no—
366C2×D3—364no—
367C3×C4—32no—
368C3×C4—32no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—94no—
372C3×C4—91no—
373C3×C4—91no—
374C3×C4C4×D3211yescharacteristic
375C3×C4C22×D3211yescharacteristic
376Dic3—32no—
377Dic3—32no—
378Dic3—34no—
379Dic3—61no—
380Dic3—61no—
381Dic3—94no—
382C32—21no—
383C32#575012yescomplemented
384C23—91no—
385C23—272no—
386Q8—92no—
387Q8—272no—
388D4—94no—
389D4—92no—
390D4—272no—
391D4—274no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—94no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—272no—
399C2×C4—274no—
400C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C8—92no—
402C8—92no—
403C8—92no—
404C8C2×((C3⋊D3)⋊C3)⋊C212yes—
405C2×C3—21no—
406C2×C3—32no—
407C2×C3—62no—
408C2×C3—61no—
409C2×C3—121no—
410C2×C3—184no—
411C2×C3—181no—
412C2×C3—181no—
413C2×C3C2×C4×D3211yescharacteristic
414D3—62no—
415D3—62no—
416D3—62no—
417D3—64no—
418D3—121no—
419D3—184no—
420C22—91no—
421C22—91no—
422C22—94no—
423C22—361no—
424C22—1082no—
425C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
426C4—91no—
427C4—91no—
428C4—94no—
429C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
430C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
431C3—32no—
432C3—61no—
433C3#1059470811yescharacteristic
434C2—21no—
435C2—181no—
436C2—181no—
437C2—184no—
438C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC4C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3C2×C4×D32
prime radical#11722491C2
cosocleC4×H(3,𝔽3)C24
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#568
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×#272C2semidirect
3. #64116⋊C3C2semidirect
4. C2×#88435C2semidirect
5. #11721054C2semidirect
6. #11720842C2semidirect
7. #11720840C2semidirect
8. #11720835C2semidirect
19. M16×H(3,𝔽3)C22semidirect
20. #88538C22semidirect
22. C8×#272C22semidirect
23. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
24. #3743⋊C3C22semidirect
25. C8×(C3⋊D3)⋊C3C22semidirect
27. #88435C22semidirect
28. #88435C22semidirect
29. #88430C22semidirect
30. ((C3⋊D3)⋊C3)⋊C8C22semidirect
32. #64116D3semidirect
79. C32×M16C2×D3semidirect
80. C2×C3⋊(C3⋊C8)C2×D3semidirect
81. #3743C2×D3semidirect
82. C8×C3⋊D3C2×D3semidirect
259. H(3,𝔽3)#568semidirect
345. M16((C3⋊D3)⋊C3)⋊C2semidirect
383. C32#5750semidirect

Maximal subgroups

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

subgroupindex
#117224912
M16×#2722
#64116⋊C32
C2×#884352
#117210542
#117208422
#117208402
#117208352
#105947063