Finite Group Databasebeta

← #13139168

Subgroups — #13139168

Order 1728.

401 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 73 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#64116⋊C3C211yesmaximal, characteristic, complemented
2#11721060C211yesmaximal, characteristic
3#11721054C211yesmaximal, characteristic, complemented
4#11720559C211yesmaximal, characteristic, complemented
5#63652⋊C3C211yesmaximal, characteristic, complemented
6#11720322C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10590995—31nomaximal
9#10590977—31nomaximal
10#3828⋊C3C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12M16×H(3,𝔽3)C2211yescharacteristic
13#88538C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#3743⋊C3—21no—
18C8×(C3⋊D3)⋊C3—21no—
19#88433C411yescharacteristic, complemented
20#88432C411yescharacteristic, complemented
21#88431C411yescharacteristic, complemented
22#88431C411yescharacteristic, complemented
23#88430—21no—
24((C3⋊D3)⋊C3)⋊C8—21no—
25#64116D311yescharacteristic, complemented
26C3×#907—31no—
27#63829—31no—
28#63829—31no—
29#63817—31no—
30#63815—31no—
31#63652D311yescharacteristic, complemented
32C3×#825—31no—
33C3×#822—31no—
34C3×#822—31no—
35C3×#822—61no—
36#63570—31no—
37#63568—31no—
38#63553—31no—
39Dic3×C3⋊C8—31no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#638⋊C3—21no—
42#638⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48#634⋊C3C2×C411yescharacteristic
49#634⋊C3C2×C411yescharacteristic
50C8×H(3,𝔽3)—21no—
51#6180—21no—
52#6180C2×C411yescharacteristic
53#6180C2×C411yescharacteristic
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56#5144—91no—
57#4895—91no—
58#4824—91no—
59#3828C2×D311yescharacteristic
60#3828C2×D311yescharacteristic
61C3×#235—31no—
62C3×#235—31no—
63C32×M16—21no—
64C32×M16C2×D311yescharacteristic
65C32×M16C2×D311yescharacteristic
66C4×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68#3743—21no—
69C8×C3⋊D3—21no—
70C3×C4×Dic3—31no—
71C3×#208—31no—
72C3×#208—31no—
73C3×#208—61no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78C3×ΓL(1,𝔽25)—61no—
79C3×D3×C8—61no—
80#3719—31no—
81#3719—31no—
82#3717—31no—
83#3717—31no—
84#3716—31no—
85#3716—31no—
86#3714—31no—
87#3714—31no—
88#3712—61no—
89#3711—61no—
90#3710—61no—
91D3×C3⋊C8—61no—
92C22×H(3,𝔽3)#4111yescharacteristic
93C2×(C3⋊D3)⋊C3—21no—
94C2×(C3⋊D3)⋊C3—21no—
95C4×H(3,𝔽3)C22×C411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—21no—
100C3×#131—91no—
101#950—91no—
102#950—91no—
103#948—91no—
104#907—31no—
105#907—91no—
106#907—91no—
107#904—91no—
108#901—91no—
109C3×#108—91no—
110C3×#104—91no—
111#837—91no—
112#825—31no—
113#825—91no—
114#825—91no—
115#822—31no—
116#822—31no—
117#822—61no—
118C8×Dic3—91no—
119C8×Dic3—91no—
120#805—91no—
121C4×C3⋊C8—91no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125#638—21no—
126#638—21no—
127C2×C3⋊Dic3—21no—
128C4⋊(C3⋊D3)C4×D311yescharacteristic
129C4⋊(C3⋊D3)C4×D311yescharacteristic
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
132#634C4×D311yescharacteristic
133#634C4×D311yescharacteristic
134C3×#67—61no—
135C3×#67—61no—
136C2×C3×Dic3—61no—
137C3×D12—31no—
138C3×D12—31no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×Dic6—31no—
142C3×Dic6—31no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—41no—
146C3⋊(C3⋊C8)#6711yescharacteristic
147C3⋊(C3⋊C8)#6711yescharacteristic
148C3×C3⋊C8—31no—
149C3×C3⋊C8—31no—
150C3×C3⋊C8—31no—
151C3×C3⋊C8—31no—
152C3×C3⋊C8—31no—
153C3×C3⋊C8—31no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—121no—
159#443—271noSylow
160C2×H(3,𝔽3)—21no—
161C2×H(3,𝔽3)C4×D411yescharacteristic
162(C3⋊D3)⋊C3—41no—
163(C3⋊D3)⋊C3—41no—
164C3×#41—91no—
165C3×#41—91no—
166#235—31no—
167#235—31no—
168#235—91no—
169#235—91no—
170C3×M16—31no—
171C3×M16—31no—
172C3×M16—61no—
173C3×M16—91no—
174C3×M16—181no—
175C3×M16D3211yescharacteristic
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C2×C3×C8—181no—
179C3×C42—91no—
180#216—91no—
181#216—91no—
182#215—91no—
183#215—91no—
184#214—91no—
185#214—91no—
186#213—91no—
187#213—91no—
188C4×Dic3—31no—
189C4×Dic3—91no—
190#208—31no—
191#208—31no—
192#208—61no—
193#208—181no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—61no—
198C2×C3⋊C8—91no—
199C2×C3⋊C8—181no—
200Dic12—91no—
201D24—91no—
202#204—91no—
203#204—91no—
204ΓL(1,𝔽25)—61no—
205ΓL(1,𝔽25)—181no—
206ΓL(1,𝔽25)—181no—
207ΓL(1,𝔽25)—181no—
208D3×C8—61no—
209D3×C8—181no—
210D3×C8—181no—
211D3×C8—181no—
212C22×C32—21no—
213C22×C32#23511yescharacteristic
214C22×C32#23711yescharacteristic
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4C2×C4×D311yescharacteristic
222C32×C4C2×C4×D311yescharacteristic
223C32×C4D3×D411yescharacteristic
224C32×C4C2×#6711yescharacteristic
225C3⋊Dic3—22no—
226C3⋊Dic3—21no—
227C3⋊Dic3—21no—
228C3×Dic3—62no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231#135—271no—
232#131—91no—
233#131—271no—
234#108—91no—
235#104—91no—
236#104—271no—
237C4×C8—271no—
238H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
239C3×Q8—91no—
240C3×Q8—91no—
241C3×D4—91no—
242C3×D4—91no—
243C3×D4—181no—
244C3×D4—181no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3211yescharacteristic
252#67—61no—
253#67—61no—
254#67—181no—
255#67—181no—
256C2×Dic3—61no—
257C2×Dic3—181no—
258D12—31no—
259D12—31no—
260D12—91no—
261D12—91no—
262C4×D3—61no—
263C4×D3—61no—
264C4×D3—181no—
265C4×D3—181no—
266Dic6—31no—
267Dic6—31no—
268Dic6—91no—
269Dic6—91no—
270C3×C8—21no—
271C3×C8—61no—
272C3×C8—61no—
273C3×C8—91no—
274C3×C8—91no—
275C3×C8—91no—
276C3×C8—91no—
277C3×C8—121no—
278C3×C8—181no—
279C3⋊C8—31no—
280C3⋊C8—31no—
281C3⋊C8—31no—
282C3⋊C8—31no—
283C3⋊C8—31no—
284C3⋊C8—31no—
285C3⋊C8—61no—
286C3⋊C8—61no—
287C3⋊C8—61no—
288C3⋊C8—61no—
289C3⋊C8—91no—
290C3⋊C8—91no—
291C3⋊C8—121no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—41no—
296C2×C32#67⋊C411yescharacteristic
297C2×C32C4×#6711yescharacteristic
298C3⋊D3—41no—
299C3⋊D3—41no—
300C3×D3—121no—
301C3×D3—121no—
302#41—91no—
303#41—91no—
304Q16—271no—
305SD16—271no—
306SD16—271no—
307D8—271no—
308M16—91no—
309M16—181no—
310M16—541no—
311M16((C3⋊D3)⋊C3)⋊C211yescharacteristic
312C2×C8—91no—
313C2×C8—91no—
314C2×C8—181no—
315C2×C8—541no—
316C42—91no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—61no—
320C22×C3—181no—
321C22×C3—181no—
322C22×C3#379511yescharacteristic
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—181no—
326C2×D3—181no—
327C3×C4—31no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—61no—
332C3×C4—61no—
333C3×C4—181no—
334C3×C4—181no—
335C3×C4—182no—
336C3×C4C4×D3211yescharacteristic
337C3×C4D3×#6711yescharacteristic
338Dic3—62no—
339Dic3—61no—
340Dic3—61no—
341Dic3—181no—
342Dic3—181no—
343Dic3—182no—
344C32—21no—
345C32#482411yescharacteristic, complemented
346C32#514411yescharacteristic, complemented
347Q8—91no—
348Q8—91no—
349D4—91no—
350D4—91no—
351D4—181no—
352D4—181no—
353C2×C4—181no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C8—21no—
358C8—91no—
359C8—91no—
360C8—91no—
361C8—91no—
362C8—181no—
363C2×C3—21no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—61no—
369C2×C3—121no—
370C2×C3—361no—
371C2×C3—361no—
372C2×C3#6387911yescharacteristic
373D3—121no—
374D3—121no—
375D3—361no—
376D3—361no—
377C22—181no—
378C22—181no—
379C22#8864811yescharacteristic
380C4—182no—
381C4—181no—
382C4—181no—
383C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C4#8867611yescharacteristic
385C3—31no—
386C3—31no—
387C3—61no—
388C3#1059097511yescharacteristic
389C2—21no—
390C2—361no—
391C2—361no—
392C2#1172112211yescharacteristic
subgroupG/H
centerC4#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3#63879
prime radical#11721060C2
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. #64116⋊C3C2semidirect
3. #11721054C2semidirect
4. #11720559C2semidirect
5. #63652⋊C3C2semidirect
6. #11720322C2semidirect
7. #11720310C2semidirect
19. #88433C4semidirect
20. #88432C4semidirect
21. #88431C4semidirect
22. #88431C4semidirect
25. #64116D3semidirect
31. #63652D3semidirect
238. H(3,𝔽3)#443semidirect
345. C32#4824semidirect
346. C32#5144semidirect

Maximal subgroups

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

subgroupindex
#64116⋊C32
#117210602
#117210542
#117205592
#63652⋊C32
#117203222
#117203102
#105909953
#105909773