Finite Group Databasebeta

← #13149808

Subgroups — #13149808

Order 1728.

1007 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722491C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3#64110⋊C3C212yesmaximal, complemented
4#11721065C211yesmaximal, characteristic, complemented
5#11721055C212yesmaximal, complemented
6#11720842C212yesmaximal, complemented
7#11720840C212yesmaximal, complemented
8#11720836C212yesmaximal, complemented
9C8×((C3⋊D3)⋊C3)⋊C2C212yesmaximal, complemented
10D3×#901—32nomaximal
11C2×C4×#272C2211yescharacteristic
12#3828⋊C3C2212yes—
13#88677C412yes—
14#88672C412yes—
15#88655C411yescharacteristic
16C4×((C3⋊D3)⋊C3)⋊C2C2212yes—
17#88651C411yescharacteristic
18#88650C2212yes—
19#88649C412yes—
20C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
21C2×#6180C2211yescharacteristic, complemented
22C8×#272C2211yescharacteristic
23C8×#272C2211yescharacteristic
24C8×#272C2211yescharacteristic, complemented
25C8×#272C2211yescharacteristic, complemented
26#3748⋊C3C2212yescomplemented
27#3743⋊C3C2214yescomplemented
28C8×(C3⋊D3)⋊C3C2214yescomplemented
29#88435C2212yescomplemented
30#88434C2212yes—
31#88430C2214yescomplemented
32((C3⋊D3)⋊C3)⋊C8C2214yescomplemented
33D3×#235—32no—
34#64110D312yescomplemented
35C3×#901—32no—
36C2×C3×D3×C8—32no—
37C2×C3×D3×C8—61no—
38#63817—32no—
39#63814—32no—
40D3×#208—32no—
41#63804—34no—
42#63803—34no—
43D3×ΓL(1,𝔽25)—34no—
44D32×C8—34no—
45C22×#272C2×C411yescharacteristic
46C2×((C3⋊D3)⋊C3)⋊C2C2×C412yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084C2×C411yescharacteristic
49C4×#272C2×C411yescharacteristic
50C4×#272C2×C411yescharacteristic
51C4×#272C2311yescharacteristic
52C4×#272C2311yescharacteristic
53#638⋊C3C2×C414yes—
54C4⋊((C3⋊D3)⋊C3)C2×C412yes—
55C4×(C3⋊D3)⋊C3C2314yes—
56#634⋊C3C2×C412yes—
57#6200C2×C414yes—
58#6199C2×C412yes—
59#6198C2×C412yes—
60((C3⋊D3)⋊C3)⋊C4C2×C414yes—
61#6196C2×C412yes—
62C8×H(3,𝔽3)C2311yescharacteristic
63C8×H(3,𝔽3)C2311yescharacteristic
64#6180C2311yescharacteristic
65#6180C2311yescharacteristic
66(C3⋊(C3⋊C8))⋊C3C2314yes—
67D3×#131—92no—
68C2×#901—91no—
69#3828C2×D312yes—
70C3×#235—32no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—61no—
73D3×#67—34no—
74#3804—34no—
75D3×D12—32no—
76C4×D32—34no—
77#3799—32no—
78#3798—34no—
79#3795—32no—
80D3×Dic6—32no—
81C2×C32×C8—21no—
82C2×C32×C8C2×D312yescomplemented
83#3748C2×D312yescomplemented
84#3743C2×D314yescomplemented
85C8×C3⋊D3C2×D314yescomplemented
86C3×#208—32no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—61no—
89C3×ΓL(1,𝔽25)—34no—
90C3×D3×C8—32no—
91C3×D3×C8—32no—
92C3×D3×C8—32no—
93C3×D3×C8—32no—
94C3×D3×C8—34no—
95C3×D3×C8—61no—
96C3×D3×C8—61no—
97C3×D3×C8—61no—
98C3×D3×C8—61no—
99#3712—34no—
100#3711—34no—
101#3711—34no—
102#3710—34no—
103D3×C3⋊C8—34no—
104D3×C3⋊C8—34no—
105C22×H(3,𝔽3)C22×C411yescharacteristic
106C2×#272—22no—
107C2×#272C22×C411yescharacteristic
108C2×#272C22×C411yescharacteristic
109C2×(C3⋊D3)⋊C3C22×C414yes—
110((C3⋊D3)⋊C3)⋊C2—24no—
111C4×H(3,𝔽3)C22×C411yescharacteristic
112C4×H(3,𝔽3)C2411yescharacteristic
113#1084C22×C411yescharacteristic
114#1084C22×C411yescharacteristic
115(C3⋊Dic3)⋊C3C22×C414yes—
116D3×#41—92no—
117C2×#235—91no—
118C3×#131—92no—
119C22×C3×C8—91no—
120#948—92no—
121C2×#208—91no—
122#907—94no—
123#907—92no—
124D3×M16—94no—
125D3×M16—92no—
126#901—32no—
127#901—92no—
128#901—92no—
129#901—94no—
130#901—92no—
131#901—92no—
132C2×ΓL(1,𝔽25)—92no—
133C2×D3×C8—32no—
134C2×D3×C8—61no—
135C2×D3×C8—92no—
136C2×D3×C8—94no—
137C22×C3×D3—32no—
138C22×C3×D3—61no—
139C2×D32—34no—
140C2×C32×C4—21no—
141C2×C32×C4C22×D312yes—
142#638C4×D314yes—
143C4⋊(C3⋊D3)C4×D312yes—
144C4×C3⋊D3C22×D314yes—
145#634C4×D312yes—
146C3×#67—34no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—61no—
149C3×D12—32no—
150C3×C4×D3—32no—
151C3×C4×D3—32no—
152C3×C4×D3—34no—
153C3×C4×D3—32no—
154C3×C4×D3—32no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157C3×C4×D3—61no—
158C3×C4×D3—61no—
159C3×Dic6—32no—
160#627—34no—
161#626—34no—
162#626—34no—
163#625—34no—
164#624—34no—
165D3×Dic3—34no—
166D3×Dic3—34no—
167C32×C8—21no—
168C32×C8—21no—
169C32×C8C22×D312yes—
170C32×C8C22×D312yes—
171C3⋊(C3⋊C8)C22×D314yes—
172C3×C3⋊C8—34no—
173C3×C3⋊C8—32no—
174C3×C3⋊C8—32no—
175C3×C3⋊C8—61no—
176C3×C3⋊C8—61no—
177C2×#131—271noSylow
178C2×H(3,𝔽3)—21no—
179C2×H(3,𝔽3)C23×C411yescharacteristic
180#272—21no—
181#272#13112yescomplemented
182(C3⋊D3)⋊C3—24no—
183C3×#41—92no—
184C22×C3×C4—91no—
185C2×#67—92no—
186#239—92no—
187D3×Q8—92no—
188#237—92no—
189#237—94no—
190D3×D4—92no—
191D3×D4—94no—
192#235—32no—
193#235—92no—
194#235—92no—
195#235—92no—
196#235—94no—
197#235—92no—
198C2×D12—91no—
199C2×C4×D3—32no—
200C2×C4×D3—61no—
201C2×C4×D3—92no—
202C2×C4×D3—94no—
203C2×Dic6—91no—
204C3×M16—94no—
205C3×M16—92no—
206C2×C3×C8—32no—
207C2×C3×C8—61no—
208C2×C3×C8—94no—
209C2×C3×C8—91no—
210C2×C3×C8—91no—
211C2×C3×C8—91no—
212C2×C3×C8—91no—
213C2×C3×C8—91no—
214C2×C3×C8D3211yescharacteristic
215#208—32no—
216#208—92no—
217#208—92no—
218#208—94no—
219C2×C3⋊C8—32no—
220C2×C3⋊C8—61no—
221C2×C3⋊C8—92no—
222C2×C3⋊C8—94no—
223ΓL(1,𝔽25)—34no—
224ΓL(1,𝔽25)—94no—
225ΓL(1,𝔽25)—94no—
226ΓL(1,𝔽25)—94no—
227ΓL(1,𝔽25)—94no—
228D3×C8—32no—
229D3×C8—32no—
230D3×C8—32no—
231D3×C8—32no—
232D3×C8—34no—
233D3×C8—61no—
234D3×C8—61no—
235D3×C8—61no—
236D3×C8—61no—
237D3×C8—94no—
238D3×C8—94no—
239D3×C8—94no—
240D3×C8—94no—
241C22×C32—21no—
242C22×C32C2×C4×D312yes—
243C2×C3⋊D3C2×C4×D314yes—
244C2×C3×D3—32no—
245C2×C3×D3—34no—
246C2×C3×D3—32no—
247C2×C3×D3—61no—
248C2×C3×D3—61no—
249C2×C3×D3—64no—
250C2×C3×D3—64no—
251D32—68no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4C2×C4×D312yes—
255C32×C4C23×D312yes—
256C3⋊Dic3C2×C4×D314yes—
257C3×Dic3—34no—
258C3×Dic3—32no—
259C3×Dic3—32no—
260C3×Dic3—61no—
261C3×Dic3—61no—
262C2×#41—271no—
263#131—92no—
264#131—272no—
265#131—272no—
266#131—272no—
267C2×M16—272no—
268C2×M16—271no—
269C22×C8—91no—
270C22×C8—272no—
271H(3,𝔽3)C2×#13111yescharacteristic, complemented, Sylow
272C23×C3—91no—
273C22×D3—32no—
274C22×D3—61no—
275C22×D3—92no—
276C22×D3—94no—
277C3×Q8—92no—
278C3×D4—92no—
279C3×D4—94no—
280C2×C3×C4—32no—
281C2×C3×C4—61no—
282C2×C3×C4—94no—
283C2×C3×C4—91no—
284C2×C3×C4—91no—
285C2×C3×C4—91no—
286C2×C3×C4—91no—
287C2×C3×C4—91no—
288C2×C3×C4C2×D3211yescharacteristic
289#67—34no—
290#67—94no—
291#67—94no—
292#67—94no—
293#67—94no—
294C2×Dic3—32no—
295C2×Dic3—61no—
296C2×Dic3—92no—
297C2×Dic3—94no—
298D12—32no—
299D12—92no—
300D12—92no—
301D12—94no—
302C4×D3—32no—
303C4×D3—32no—
304C4×D3—32no—
305C4×D3—32no—
306C4×D3—34no—
307C4×D3—61no—
308C4×D3—61no—
309C4×D3—61no—
310C4×D3—61no—
311C4×D3—94no—
312C4×D3—94no—
313C4×D3—94no—
314C4×D3—94no—
315Dic6—32no—
316Dic6—92no—
317Dic6—92no—
318Dic6—94no—
319C3×C8—32no—
320C3×C8—32no—
321C3×C8—61no—
322C3×C8—61no—
323C3×C8—94no—
324C3×C8—91no—
325C3×C8—91no—
326C3×C8C2×D3211yescharacteristic
327C3×C8C2×D3211yescharacteristic
328C3⋊C8—32no—
329C3⋊C8—32no—
330C3⋊C8—34no—
331C3⋊C8—61no—
332C3⋊C8—61no—
333C3⋊C8—94no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32C22×C4×D312yes—
338C3⋊D3—24no—
339C3×D3—34no—
340C3×D3—62no—
341C3×D3—62no—
342C3×D3—64no—
343C3×D3—62no—
344#41—92no—
345#41—272no—
346#41—272no—
347#41—272no—
348C2×Q8—271no—
349C2×D4—271no—
350C2×D4—272no—
351C22×C4—91no—
352C22×C4—272no—
353M16—92no—
354M16—94no—
355M16—274no—
356M16—272no—
357C2×C8—91no—
358C2×C8—91no—
359C2×C8—91no—
360C2×C8—91no—
361C2×C8—91no—
362C2×C8—94no—
363C2×C8—272no—
364C2×C8—274no—
365C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
366C22×C3—32no—
367C22×C3—61no—
368C22×C3—94no—
369C22×C3—91no—
370C22×C3—91no—
371C22×C3—182no—
372C22×C3C4×D3211yescharacteristic
373C2×D3—32no—
374C2×D3—32no—
375C2×D3—34no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—64no—
379C2×D3—64no—
380C2×D3—94no—
381C2×D3—188no—
382C2×D3—184no—
383C3×C4—32no—
384C3×C4—32no—
385C3×C4—61no—
386C3×C4—61no—
387C3×C4—94no—
388C3×C4—91no—
389C3×C4—91no—
390C3×C4C4×D3211yescharacteristic
391C3×C4C22×D3211yescharacteristic
392Dic3—32no—
393Dic3—32no—
394Dic3—34no—
395Dic3—61no—
396Dic3—61no—
397Dic3—94no—
398C32—21no—
399C32D3×#13112yescomplemented
400C23—91no—
401C23—272no—
402Q8—92no—
403Q8—272no—
404D4—94no—
405D4—92no—
406D4—272no—
407D4—274no—
408C2×C4—91no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—94no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—272no—
415C2×C4—274no—
416C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C8—91no—
418C8—91no—
419C8—94no—
420C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
422C2×C3—21no—
423C2×C3—32no—
424C2×C3—62no—
425C2×C3—62no—
426C2×C3—61no—
427C2×C3—92no—
428C2×C3—184no—
429C2×C3—181no—
430C2×C3C2×C4×D3211yescharacteristic
431D3—34no—
432D3—62no—
433D3—62no—
434D3—62no—
435D3—64no—
436D3—184no—
437C22—91no—
438C22—91no—
439C22—94no—
440C22—182no—
441C22—544no—
442C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
443C4—91no—
444C4—91no—
445C4—94no—
446C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
447C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
448C3—32no—
449C3—61no—
450C3#1059466911yescharacteristic
451C2—21no—
452C2—92no—
453C2—181no—
454C2—184no—
455C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC8C2×((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)C2×#131
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×#272C2semidirect
3. #64110⋊C3C2semidirect
4. #11721065C2semidirect
5. #11721055C2semidirect
6. #11720842C2semidirect
7. #11720840C2semidirect
8. #11720836C2semidirect
9. C8×((C3⋊D3)⋊C3)⋊C2C2semidirect
20. C2×C8×H(3,𝔽3)C22semidirect
21. C2×#6180C22semidirect
24. C8×#272C22semidirect
25. C8×#272C22semidirect
26. #3748⋊C3C22semidirect
27. #3743⋊C3C22semidirect
28. C8×(C3⋊D3)⋊C3C22semidirect
29. #88435C22semidirect
31. #88430C22semidirect
32. ((C3⋊D3)⋊C3)⋊C8C22semidirect
34. #64110D3semidirect
82. C2×C32×C8C2×D3semidirect
83. #3748C2×D3semidirect
84. #3743C2×D3semidirect
85. C8×C3⋊D3C2×D3semidirect
181. #272#131semidirect
271. H(3,𝔽3)C2×#131semidirect
365. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
399. C32D3×#131semidirect

Maximal subgroups

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

subgroupindex
#117224912
C2×C8×#2722
#64110⋊C32
#117210652
#117210552
#117208422
#117208402
#117208362
C8×((C3⋊D3)⋊C3)⋊C22
D3×#9013