Finite Group Databasebeta

← #13139191

Subgroups — #13139191

Order 1728.

433 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721786C211yesmaximal, characteristic, complemented
2#11721073C211yesmaximal, characteristic
3#11721055C211yesmaximal, characteristic, complemented
4#63652⋊C3C211yesmaximal, characteristic, complemented
5#63649⋊C3C211yesmaximal, characteristic, complemented
6#11720328C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10590995—31nomaximal
9#10590975—31nomaximal
10#88740C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12M16×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#88527—21no—
15C8×#272—21no—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3748⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88439C411yescharacteristic, complemented
20#88438C411yescharacteristic, complemented
21#88437C411yescharacteristic, complemented
22#88436C411yescharacteristic, complemented
23#88430—21no—
24((C3⋊D3)⋊C3)⋊C8—21no—
25C3×#907—31no—
26C3×#907—31no—
27C3×#907—61no—
28#63829—31no—
29#63822—31no—
30#63817—31no—
31#63814—31no—
32#63652D311yescharacteristic, complemented
33#63649D311yescharacteristic, complemented
34C3×#825—31no—
35C3×#822—31no—
36#63569—31no—
37#63568—31no—
38#63553—31no—
39Dic3×C3⋊C8—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)—21no—
51#6180D411yescharacteristic
52#6180D411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
56#5144—91no—
57#4895—91no—
58#4824—91no—
59#3828C2×D311yescharacteristic
60C3×#235—31no—
61C3×#235—31no—
62C3×#235—31no—
63C3×#235—61no—
64C32×M16—21no—
65C32×M16C2×D311yescharacteristic
66C32×M16C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68#3748C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4×Dic3—31no—
71C3×#208—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76C3×ΓL(1,𝔽25)—61no—
77C3×ΓL(1,𝔽25)—61no—
78C3×ΓL(1,𝔽25)—62no—
79C3×D3×C8—61no—
80C3×D3×C8—61no—
81C3×D3×C8—62no—
82#3719—31no—
83#3718—31no—
84#3717—31no—
85#3716—31no—
86#3715—31no—
87#3715—31no—
88#3714—31no—
89#3713—31no—
90#3712—61no—
91#3711—61no—
92#3710—61no—
93D3×C3⋊C8—61no—
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—22no—
101(C3⋊Dic3)⋊C3—21no—
102C3×#131—91no—
103#950—91no—
104#950—91no—
105#948—91no—
106#907—31no—
107#907—31no—
108#907—61no—
109#907—91no—
110#904—91no—
111#901—91no—
112C3×#108—91no—
113C3×#104—91no—
114#837—91no—
115#825—31no—
116#825—91no—
117#825—91no—
118#822—31no—
119#822—91no—
120C8×Dic3—91no—
121C8×Dic3—91no—
122#805—91no—
123C4×C3⋊C8—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—22no—
150C32×C8—21no—
151C32×C8—21no—
152C3⋊(C3⋊C8)—21no—
153C3⋊(C3⋊C8)C4×D311yescharacteristic
154C3⋊(C3⋊C8)C4×D311yescharacteristic
155C3×C3⋊C8—31no—
156C3×C3⋊C8—31no—
157C3×C3⋊C8—31no—
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—31no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#443—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—31no—
177C3×M16—31no—
178C3×M16—61no—
179C3×M16—91no—
180C3×M16—181no—
181C3×M16D3211yescharacteristic
182C2×C3×C8—91no—
183C2×C3×C8—91no—
184C2×C3×C8—181no—
185C3×C42—91no—
186#216—91no—
187#216—91no—
188#215—91no—
189#215—91no—
190#214—91no—
191#214—91no—
192#213—91no—
193#213—91no—
194C4×Dic3—31no—
195C4×Dic3—91no—
196#208—31no—
197#208—91no—
198#208—181no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—181no—
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)—181no—
213ΓL(1,𝔽25)—181no—
214D3×C8—61no—
215D3×C8—61no—
216D3×C8—62no—
217D3×C8—181no—
218D3×C8—181no—
219C22×C32—21no—
220C22×C32#23711yescharacteristic
221C22×C32#23911yescharacteristic
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×C4D3×D411yescharacteristic
232C32×C4D3×D411yescharacteristic
233C3⋊Dic3—22no—
234C3⋊Dic3—21no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240#135—271no—
241#131—91no—
242#131—271no—
243#108—91no—
244#104—91no—
245#104—271no—
246C4×C8—271no—
247H(3,𝔽3)#44311yescharacteristic, 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—21no—
284C3×C8—61no—
285C3×C8—61no—
286C3×C8—62no—
287C3×C8—91no—
288C3×C8—91no—
289C3×C8—91no—
290C3×C8—91no—
291C3×C8—181no—
292C3⋊C8—31no—
293C3⋊C8—31no—
294C3⋊C8—31no—
295C3⋊C8—31no—
296C3⋊C8—31no—
297C3⋊C8—31no—
298C3⋊C8—61no—
299C3⋊C8—61no—
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#67⋊C411yescharacteristic
309C2×C32#89311yescharacteristic
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—181no—
323M16—541no—
324M16((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C2×C8—91no—
326C2×C8—91no—
327C2×C8—181no—
328C2×C8—541no—
329C42—91no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—61no—
333C22×C3—181no—
334C22×C3—181no—
335C22×C3#379611yescharacteristic
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—182no—
349C3×C4—181no—
350C3×C4C4×D3211yescharacteristic
351C3×C4#381111yescharacteristic
352Dic3—62no—
353Dic3—61no—
354Dic3—61no—
355Dic3—62no—
356Dic3—61no—
357Dic3—181no—
358Dic3—182no—
359C32—21no—
360C32#482411yescharacteristic, complemented
361C32#489511yescharacteristic, 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—21no—
373C8—91no—
374C8—91no—
375C8—91no—
376C8—91no—
377C8—181no—
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×C3#6388111yescharacteristic
388D3—121no—
389D3—121no—
390D3—122no—
391D3—121no—
392D3—361no—
393C22—181no—
394C22—181no—
395C22#8864911yescharacteristic
396C4—182no—
397C4—181no—
398C4—181no—
399C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
400C4#8867711yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3#1059097711yescharacteristic
405C2—21no—
406C2—361no—
407C2—361no—
408C2#1172112411yescharacteristic
subgroupG/H
centerC4#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3#63881
prime radical#11721073C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #11721786C2semidirect
3. #11721055C2semidirect
4. #63652⋊C3C2semidirect
5. #63649⋊C3C2semidirect
6. #11720328C2semidirect
7. #11720310C2semidirect
19. #88439C4semidirect
20. #88438C4semidirect
21. #88437C4semidirect
22. #88436C4semidirect
32. #63652D3semidirect
33. #63649D3semidirect
247. H(3,𝔽3)#443semidirect
360. C32#4824semidirect
361. C32#4895semidirect

Maximal subgroups

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

subgroupindex
#117217862
#117210732
#117210552
#63652⋊C32
#63649⋊C32
#117203282
#117203102
#105909953
#105909753