Finite Group Databasebeta

← #13141060

Subgroups — #13141060

Order 1728.

433 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721869C211yesmaximal, characteristic, complemented
2#64159⋊C3C211yesmaximal, characteristic
3#11721055C211yesmaximal, characteristic, complemented
4#63664⋊C3C211yesmaximal, characteristic, complemented
5#11720322C211yesmaximal, characteristic, complemented
6#11720319C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10591673—31nomaximal
9#10591662—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12#88538—21no—
13#88538C2211yescharacteristic
14C2×#6180—21no—
15C2×#6180C2211yescharacteristic
16#3755⋊C3C411yescharacteristic, complemented
17#3756⋊C3C411yescharacteristic, complemented
18#3753⋊C3C411yescharacteristic, complemented
19#3754⋊C3C411yescharacteristic, complemented
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21#3748⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88430—21no—
24((C3⋊D3)⋊C3)⋊C8—21no—
25#64159D311yescharacteristic
26C3×#950—31no—
27C3×#948—31no—
28C3×#948—31no—
29C3×#948—61no—
30#63817—31no—
31#63814—31no—
32#63664D311yescharacteristic, complemented
33C3×#837—31no—
34#63578—31no—
35#63577—31no—
36#63570—31no—
37#63568—31no—
38#63553—31no—
39Dic3×C3⋊C8—31no—
40Q8×H(3,𝔽3)C2×C411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3C2×C411yescharacteristic
50#6180—21no—
51#6180D411yescharacteristic
52#6180D411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
56#5196—91no—
57#5144—91no—
58#4824—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D311yescharacteristic
62#3828C2×D311yescharacteristic
63C3×#235—31no—
64#3756Dic311yescharacteristic, complemented
65#3755Dic311yescharacteristic, complemented
66#3754Dic311yescharacteristic, complemented
67#3753Dic311yescharacteristic, complemented
68C4×C3⋊Dic3C2×D311yescharacteristic
69#3748C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#216—31no—
72C3×#215—31no—
73C3×#214—31no—
74C3×#213—31no—
75C3×C4×Dic3—31no—
76C3×#208—31no—
77C3×#208—31no—
78C3×#208—31no—
79C3×#208—61no—
80C3×#208—61no—
81C3×#208—61no—
82C3×#208—62no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—61no—
87C2×C3×C3⋊C8—61no—
88C2×C3×C3⋊C8—61no—
89C2×C3×C3⋊C8—62no—
90#3712—61no—
91#3711—61no—
92#3710—61no—
93D3×C3⋊C8—61no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)#4111yescharacteristic
96C2×(C3⋊D3)⋊C3—21no—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)C22×C411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100(C3⋊Dic3)⋊C3—22no—
101(C3⋊Dic3)⋊C3—21no—
102C3×#135—91no—
103C3×#131—91no—
104#950—31no—
105#950—91no—
106#948—31no—
107#948—31no—
108#948—61no—
109#948—91no—
110#907—91no—
111#901—91no—
112C3×#104—91no—
113#837—31no—
114#837—91no—
115#837—91no—
116#825—91no—
117#822—91no—
118#822—91no—
119#819—91no—
120C8×Dic3—91no—
121C8×Dic3—91no—
122#805—91no—
123C4×C3⋊C8—91no—
124C32×Q8—21no—
125C32×Q8C4×D311yescharacteristic
126C32×Q8C2×Dic311yescharacteristic
127C32×D4—22no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4C4×D311yescharacteristic
132C32×D4C2×Dic311yescharacteristic
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139#638—21no—
140C2×C3⋊Dic3—21no—
141C4⋊(C3⋊D3)C2×Dic311yescharacteristic
142C4×C3⋊D3—21no—
143#634C2×Dic311yescharacteristic
144C3×#67—61no—
145C2×C3×Dic3—61no—
146C3×D12—31no—
147C3×C4×D3—61no—
148C3×Dic6—31no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)C2×Dic311yescharacteristic
151C3⋊(C3⋊C8)C2×Dic311yescharacteristic
152C3×C3⋊C8—31no—
153C3×C3⋊C8—31no—
154C3×C3⋊C8—31no—
155C3×C3⋊C8—31no—
156C3×C3⋊C8—31no—
157C3×C3⋊C8—31no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#443—271noSylow
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)—41no—
167C2×H(3,𝔽3)C4×D411yescharacteristic
168(C3⋊D3)⋊C3—41no—
169C3×#41—31no—
170C3×#41—31no—
171C3×#41—61no—
172C3×#41—91no—
173C3×#41D3211yescharacteristic
174#235—31no—
175#235—91no—
176C3×Q16—91no—
177C3×SD16—91no—
178C3×SD16—91no—
179C3×D8—91no—
180C3×M16—91no—
181C3×M16—91no—
182C3×M16—181no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C2×C3×C8—181no—
186C3×C42—91no—
187#216—31no—
188#216—91no—
189#215—31no—
190#215—91no—
191#214—31no—
192#214—91no—
193#213—31no—
194#213—91no—
195C4×Dic3—31no—
196C4×Dic3—91no—
197#208—31no—
198#208—31no—
199#208—31no—
200#208—61no—
201#208—61no—
202#208—61no—
203#208—62no—
204#208—91no—
205#208—181no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—61no—
210C2×C3⋊C8—61no—
211C2×C3⋊C8—61no—
212C2×C3⋊C8—62no—
213C2×C3⋊C8—91no—
214C2×C3⋊C8—181no—
215ΓL(1,𝔽25)—181no—
216ΓL(1,𝔽25)—181no—
217D3×C8—181no—
218D3×C8—181no—
219C22×C32—22no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32#23711yescharacteristic
224C22×C32#23711yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3×D3—61no—
227C32×C4—22no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4C2×C4×D311yescharacteristic
233C32×C4D3×D411yescharacteristic
234C32×C4D3×D411yescharacteristic
235C32×C4C22×Dic311yescharacteristic
236C3⋊Dic3—22no—
237C3⋊Dic3—21no—
238C3×Dic3—62no—
239C3×Dic3—61no—
240#135—91no—
241#131—91no—
242#131—271no—
243#108—271no—
244#104—91no—
245#104—271no—
246C4×C8—271no—
247H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
248C3×Q8—31no—
249C3×Q8—31no—
250C3×Q8—61no—
251C3×Q8—91no—
252C3×Q8D3×Dic311yescharacteristic
253C3×D4—21no—
254C3×D4—31no—
255C3×D4—31no—
256C3×D4—61no—
257C3×D4—61no—
258C3×D4—62no—
259C3×D4—61no—
260C3×D4—91no—
261C3×D4—181no—
262C3×D4D3×Dic311yescharacteristic
263C2×C3×C4—21no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—62no—
269C2×C3×C4—61no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3211yescharacteristic
273#67—61no—
274#67—181no—
275C2×Dic3—61no—
276C2×Dic3—181no—
277D12—31no—
278D12—91no—
279C4×D3—61no—
280C4×D3—181no—
281Dic6—31no—
282Dic6—91no—
283C3×C8—91no—
284C3×C8—91no—
285C3×C8—91no—
286C3×C8—91no—
287C3×C8—181no—
288C3×C8—181no—
289C3⋊C8—31no—
290C3⋊C8—31no—
291C3⋊C8—31no—
292C3⋊C8—31no—
293C3⋊C8—31no—
294C3⋊C8—31no—
295C3⋊C8—61no—
296C3⋊C8—61no—
297C3⋊C8—61no—
298C3⋊C8—61no—
299C3⋊C8—62no—
300C3⋊C8—61no—
301C3⋊C8—91no—
302C3⋊C8—91no—
303C3⋊C8—181no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32#67⋊C411yescharacteristic
312C2×C32D4×Dic311yescharacteristic
313C3⋊D3—41no—
314C3×D3—121no—
315#41—91no—
316#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
317Q16—91no—
318SD16—91no—
319SD16—91no—
320D8—91no—
321M16—91no—
322M16—91no—
323M16—181no—
324M16—541no—
325C2×C8—91no—
326C2×C8—91no—
327C2×C8—181no—
328C2×C8—541no—
329C42—91no—
330C22×C3—21no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—62no—
336C22×C3—61no—
337C22×C3—181no—
338C22×C3#380511yescharacteristic
339C2×D3—61no—
340C2×D3—181no—
341C3×C4—21no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—62no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—181no—
352C3×C4—182no—
353C3×C4C2×D3×Dic311yescharacteristic
354C3×C4#381111yescharacteristic
355Dic3—62no—
356Dic3—61no—
357Dic3—181no—
358Dic3—182no—
359C32—21no—
360C32#482411yescharacteristic, complemented
361C32#519611yescharacteristic, complemented
362Q8—91no—
363Q8((C3⋊D3)⋊C3)⋊C411yescharacteristic
364D4—21no—
365D4—91no—
366D4—181no—
367D4((C3⋊D3)⋊C3)⋊C411yescharacteristic
368C2×C4—21no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C8—91no—
373C8—91no—
374C8—91no—
375C8—91no—
376C8—181no—
377C8—181no—
378C2×C3—21no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—41no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—121no—
386C2×C3—121no—
387C2×C3—121no—
388C2×C3—122no—
389C2×C3—361no—
390C2×C3#67⋊Dic311yescharacteristic
391D3—121no—
392D3—361no—
393C22—21no—
394C22—181no—
395C22#8867211yescharacteristic
396C4—21no—
397C4—182no—
398C4—181no—
399C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
400C4#8867711yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3#1059167511yescharacteristic
405C2—21no—
406C2—41no—
407C2—361no—
408C2#1172131511yescharacteristic
subgroupG/H
centerC4#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#67⋊Dic3
prime radical#64159⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11721869C2semidirect
3. #11721055C2semidirect
4. #63664⋊C3C2semidirect
5. #11720322C2semidirect
6. #11720319C2semidirect
7. #11720310C2semidirect
16. #3755⋊C3C4semidirect
17. #3756⋊C3C4semidirect
18. #3753⋊C3C4semidirect
19. #3754⋊C3C4semidirect
32. #63664D3semidirect
64. #3756Dic3semidirect
65. #3755Dic3semidirect
66. #3754Dic3semidirect
67. #3753Dic3semidirect
247. H(3,𝔽3)#443semidirect
360. C32#4824semidirect
361. C32#5196semidirect

Maximal subgroups

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

subgroupindex
#117218692
#64159⋊C32
#117210552
#63664⋊C32
#117203222
#117203192
#117203102
#105916733
#105916623