Finite Group Databasebeta

← #13139021

Subgroups — #13139021

Order 1728.

426 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721776C211yesmaximal, characteristic, complemented
2#64110⋊C3C211yesmaximal, characteristic, complemented
3#11721073C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic, complemented
5#11720328C211yesmaximal, characteristic, complemented
6#11720322C211yesmaximal, characteristic, complemented
7#11720319C211yesmaximal, characteristic, complemented
8#10590924—31nomaximal
9#10590922—31nomaximal
10#88740C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13#88538C2211yescharacteristic
14#88527—21no—
15C8×#272—21no—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3748⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#3743⋊C3—21no—
20C8×(C3⋊D3)⋊C3—21no—
21#88439C411yescharacteristic, complemented
22#88438C411yescharacteristic, complemented
23#88437C411yescharacteristic, complemented
24#88436C411yescharacteristic, complemented
25#64110D311yescharacteristic, complemented
26C3×#901—31no—
27C3×#901—31no—
28C3×#901—31no—
29C3×#901—61no—
30#63829—31no—
31#63822—31no—
32#63642D311yescharacteristic, complemented
33C3×#815—31no—
34#63578—31no—
35#63577—31no—
36#63570—31no—
37#63569—31no—
38#63568—31no—
39#63568—31no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—21no—
42C4⋊#272C2×C411yescharacteristic
43C4×#272—21no—
44#6229C2×C411yescharacteristic
45#638⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3C2×C411yescharacteristic
50C8×H(3,𝔽3)D411yescharacteristic
51C8×H(3,𝔽3)D411yescharacteristic
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
56#5145—91no—
57#5145—91no—
58#4717—91no—
59#3828C2×D311yescharacteristic
60C3×#235—31no—
61C3×#235—31no—
62C3×#235—31no—
63C3×#235—61no—
64C2×C32×C8—21no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68#3748C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70#3743—21no—
71C8×C3⋊D3—21no—
72C3×C4×Dic3—31no—
73C3×#208—31no—
74C3×#208—31no—
75C3×#208—31no—
76C3×#208—61no—
77C2×C3×C3⋊C8—31no—
78C3×ΓL(1,𝔽25)—62no—
79C3×ΓL(1,𝔽25)—61no—
80C3×ΓL(1,𝔽25)—61no—
81C3×ΓL(1,𝔽25)—61no—
82C3×D3×C8—62no—
83C3×D3×C8—61no—
84C3×D3×C8—61no—
85C3×D3×C8—61no—
86#3719—31no—
87#3718—31no—
88#3717—31no—
89#3716—31no—
90#3715—31no—
91#3715—31no—
92#3714—31no—
93#3713—31no—
94C22×H(3,𝔽3)#4111yescharacteristic
95C2×#272—21no—
96C2×(C3⋊D3)⋊C3—21no—
97C4×H(3,𝔽3)C22×C411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99#1084—21no—
100(C3⋊Dic3)⋊C3—21no—
101(C3⋊Dic3)⋊C3—41no—
102C3×#131—91no—
103C3×#131—91no—
104#950—91no—
105#950—91no—
106#904—91no—
107#901—31no—
108#901—31no—
109#901—31no—
110#901—61no—
111#901—91no—
112C3×#97—91no—
113#837—91no—
114#837—91no—
115#825—91no—
116#825—91no—
117#822—91no—
118#822—91no—
119#819—91no—
120#815—31no—
121#815—91no—
122#805—91no—
123#805—91no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127#638—21no—
128C2×C3⋊Dic3—21no—
129C4⋊(C3⋊D3)C4×D311yescharacteristic
130C4×C3⋊D3—21no—
131#634C4×D311yescharacteristic
132C3×#67—62no—
133C3×#67—61no—
134C3×#67—61no—
135C3×#67—61no—
136C2×C3×Dic3—61no—
137C3×D12—31no—
138C3×D12—31no—
139C3×D12—31no—
140C3×D12—61no—
141C3×C4×D3—62no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×Dic6—31no—
146C3×Dic6—31no—
147C3×Dic6—31no—
148C3×Dic6—61no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8D1211yescharacteristic
152C32×C8D1211yescharacteristic
153C32×C8#6711yescharacteristic
154C32×C8#6711yescharacteristic
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)C4×D311yescharacteristic
157C3⋊(C3⋊C8)C4×D311yescharacteristic
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#444—271noSylow
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)C4×D411yescharacteristic
167#272—41no—
168(C3⋊D3)⋊C3—41no—
169C3×#41—91no—
170C3×#41—91no—
171#235—31no—
172#235—31no—
173#235—31no—
174#235—61no—
175#235—91no—
176C3×M16—91no—
177C3×M16—91no—
178C3×M16—181no—
179C3×M16—181no—
180C2×C3×C8—31no—
181C2×C3×C8—31no—
182C2×C3×C8—61no—
183C2×C3×C8—91no—
184C2×C3×C8—181no—
185C2×C3×C8—181no—
186C2×C3×C8D3211yescharacteristic
187C3×C42—91no—
188#216—91no—
189#216—91no—
190#215—91no—
191#215—91no—
192#214—91no—
193#214—91no—
194#213—91no—
195#213—91no—
196C4×Dic3—31no—
197C4×Dic3—91no—
198#208—31no—
199#208—31no—
200#208—31no—
201#208—61no—
202#208—91no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205Dic12—91no—
206D24—91no—
207#204—91no—
208#204—91no—
209ΓL(1,𝔽25)—61no—
210ΓL(1,𝔽25)—61no—
211ΓL(1,𝔽25)—62no—
212ΓL(1,𝔽25)—61no—
213ΓL(1,𝔽25)—181no—
214D3×C8—61no—
215D3×C8—61no—
216D3×C8—62no—
217D3×C8—61no—
218D3×C8—181no—
219C22×C32—21no—
220C22×C32#23511yescharacteristic
221C22×C32#23511yescharacteristic
222C2×C3⋊D3—21no—
223C2×C3×D3—62no—
224C2×C3×D3—61no—
225C2×C3×D3—61no—
226C2×C3×D3—61no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4C2×C4×D311yescharacteristic
230C32×C4C2×C4×D311yescharacteristic
231C32×C4C2×D1211yescharacteristic
232C32×C4C2×#6711yescharacteristic
233C3⋊Dic3—21no—
234C3⋊Dic3—41no—
235C3×Dic3—62no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—121no—
240#135—271no—
241#131—91no—
242#131—91no—
243#108—271no—
244#104—271no—
245#104—271no—
246#97—91no—
247H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
248C3×Q8—91no—
249C3×Q8—91no—
250C3×D4—91no—
251C3×D4—91no—
252C3×D4—181no—
253C3×D4—181no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3211yescharacteristic
261#67—61no—
262#67—61no—
263#67—62no—
264#67—61no—
265#67—181no—
266C2×Dic3—61no—
267C2×Dic3—181no—
268D12—31no—
269D12—31no—
270D12—31no—
271D12—61no—
272D12—91no—
273C4×D3—61no—
274C4×D3—61no—
275C4×D3—62no—
276C4×D3—61no—
277C4×D3—181no—
278Dic6—31no—
279Dic6—31no—
280Dic6—31no—
281Dic6—61no—
282Dic6—91no—
283C3×C8—31no—
284C3×C8—31no—
285C3×C8—31no—
286C3×C8—31no—
287C3×C8—61no—
288C3×C8—61no—
289C3×C8—91no—
290C3×C8—91no—
291C3×C8—181no—
292C3×C8—181no—
293C3×C8#62611yescharacteristic
294C3×C8#62611yescharacteristic
295C3⋊C8—31no—
296C3⋊C8—31no—
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×C32C4×D1211yescharacteristic
309C2×C32C4×#6711yescharacteristic
310C3⋊D3—41no—
311C3×D3—122no—
312C3×D3—121no—
313C3×D3—121no—
314C3×D3—121no—
315#41—91no—
316#41—91no—
317Q16—271no—
318SD16—271no—
319SD16—271no—
320D8—271no—
321M16—91no—
322M16—91no—
323M16—181no—
324M16—181no—
325C2×C8—91no—
326C2×C8—181no—
327C2×C8—181no—
328C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C42—91no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—61no—
333C22×C3—181no—
334C22×C3—181no—
335C22×C3#379811yescharacteristic
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—62no—
339C2×D3—61no—
340C2×D3—181no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—61no—
346C3×C4—61no—
347C3×C4—181no—
348C3×C4—181no—
349C3×C4—361no—
350C3×C4C4×D3211yescharacteristic
351C3×C4C2×#62611yescharacteristic
352Dic3—61no—
353Dic3—61no—
354Dic3—62no—
355Dic3—61no—
356Dic3—121no—
357Dic3—181no—
358Dic3—361no—
359C32—21no—
360C32#471711yescharacteristic, complemented
361C32#514511yescharacteristic, complemented
362Q8—91no—
363Q8—91no—
364D4—91no—
365D4—91no—
366D4—181no—
367D4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C8—91no—
373C8—91no—
374C8—181no—
375C8—181no—
376C8#620011yescharacteristic
377C8#620011yescharacteristic
378C2×C3—21no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—121no—
385C2×C3—361no—
386C2×C3—361no—
387C2×C3C4×#62611yescharacteristic
388D3—121no—
389D3—121no—
390D3—122no—
391D3—121no—
392D3—361no—
393C22—181no—
394C22—181no—
395C22#8865011yescharacteristic
396C4—181no—
397C4—181no—
398C4—361no—
399C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
400C4C2×#620011yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3#1059092411yescharacteristic
405C2—21no—
406C2—361no—
407C2—361no—
408C2C4×#620011yescharacteristic
subgroupG/H
centerC4C2×#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C4×#626
prime radical#11721073C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#444
LCSG ▷ C4×H(3,𝔽3) ▷ 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
1. #11721776C2semidirect
2. #64110⋊C3C2semidirect
4. #63642⋊C3C2semidirect
5. #11720328C2semidirect
6. #11720322C2semidirect
7. #11720319C2semidirect
21. #88439C4semidirect
22. #88438C4semidirect
23. #88437C4semidirect
24. #88436C4semidirect
25. #64110D3semidirect
32. #63642D3semidirect
247. H(3,𝔽3)#444semidirect
360. C32#4717semidirect
361. C32#5145semidirect

Maximal subgroups

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

subgroupindex
#117217762
#64110⋊C32
#117210732
#63642⋊C32
#117203282
#117203222
#117203192
#105909243
#105909223