Finite Group Databasebeta

← #13140750

Subgroups — #13140750

Order 1728.

434 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 85 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#11721856C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3#11721045C211yesmaximal, characteristic
4C2×#88414C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720279C211yesmaximal, characteristic, complemented
8#10591590—31nomaximal
9#10588854—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×#6195—21no—
14#88536—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16#88534—21no—
17C9×#215—22no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic9—21no—
22#88414—22no—
23#64155—31no—
24C2×C3×#215D311yescharacteristic
25#63916—31no—
26C2×#3712—31no—
27C3×#835—31no—
28#63575—31no—
29#63563—31no—
30C2×C9×SD16—31no—
31#63508—31no—
32#63508—61no—
33#63500—31no—
34C2×#3663—31no—
35#63395D311yescharacteristic, complemented
36#63392—31no—
37#63378—31no—
38C22×C3⋊D9D411yescharacteristic
39C22×D3×C9—21no—
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9D411yescharacteristic
46C4×C3⋊D9—21no—
47C4×C3⋊D9—21no—
48C2×C3×Dic9—21no—
49C9×D12—21no—
50C9×D12—21no—
51#6195—42no—
52C9×C3⋊C8D412yescomplemented
53C3×C9⋊C8—21no—
54#5174—91no—
55#4812—91no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C4×C3⋊D3—31no—
58C2×C3×D12C2×D311yescharacteristic
59C2×#626—61no—
60#3752—61no—
61C4⋊(C3⋊Dic3)—31no—
62#3750—61no—
63C3×#215—22no—
64C3×C4⋊Dic3—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67(C2×D3)⋊Dic3—61no—
68#3712—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—31no—
72C2×#611—61no—
73C2×C4×D9—31no—
74C2×C4×D9—61no—
75C9×SD16—62no—
76C2×C8×C9—31no—
77#3676—61no—
78(C2×D9)⋊C4—61no—
79(C2×D9)⋊C4—62no—
80C4⋊Dic9—31no—
81C4⋊Dic9—61no—
82C4⋊Dic9C2×D311yescharacteristic
83Dic9⋊C4—61no—
84Dic9⋊C4—62no—
85C2×C9⋊C8C2×D311yescharacteristic
86#3663—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3⋊D9—21no—
89C2×C3⋊D9—21no—
90C2×C3⋊D9—21no—
91C2×D3×C9—41no—
92C2×D3×C9—41no—
93C2×D3×C9—41no—
94C3×C4×C9—41no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9#4111yescharacteristic
97C3⋊Dic9—21no—
98C3⋊Dic9—41no—
99C3×Dic9—41no—
100C2×C3×SD16—31no—
101#946—91no—
102#946—91no—
103#946—181no—
104C2×#215D911yescharacteristic, complemented
105#938—91no—
106C2×ΓL(1,𝔽25)—91no—
107C2×ΓL(1,𝔽25)—91no—
108#895—91no—
109C3×#103—91no—
110#835—31no—
111#832—91no—
112#821—91no—
113#818—91no—
114#807—91no—
115C22×C3⋊D3—31no—
116C22×C3×D3—21no—
117C32×Q8—21no—
118C32×Q8—21no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—61no—
123C4×C3⋊D3—61no—
124C4×C3⋊D3—61no—
125C2×C3×Dic3—61no—
126C3×D12—21no—
127C3×D12—21no—
128#626—122no—
129C23×C9—61no—
130C22×D9—31no—
131C22×D9—61no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8#6712yes—
134Q8×C9—21no—
135Q8×C9—21no—
136Q8×C9—41no—
137Q8×C9—41no—
138D4×C9—61no—
139D4×C9—61no—
140C2×C4×C9—22no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9C22×D311yescharacteristic
144#611—122no—
145C2×Dic9—21no—
146C2×Dic9—31no—
147C2×Dic9—61no—
148C2×Dic9—62no—
149C2×Dic9—61no—
150C4×D9—61no—
151C4×D9—61no—
152C4×D9—121no—
153C4×D9—121no—
154C8×C9—32no—
155C9⋊C8—21no—
156#468—271noSylow
157C2×C3×C9#12111yescharacteristic
158C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
159C2×C3×C9#13711yescharacteristic
160C3⋊D9—41no—
161D3×C9—81no—
162C2×C3×Q8—21no—
163C2×C3×Q8C2×D911yescharacteristic
164C2×C3×Q8D3211yescharacteristic
165C2×C3×D4—31no—
166C2×#67—181no—
167C2×D12—181no—
168C2×D12C2×D911yescharacteristic
169C2×C4×D3—91no—
170C2×C4×D3—91no—
171C2×C4×D3—181no—
172C3×SD16—62no—
173C2×C3×C8—31no—
174C2×C3×C8—91no—
175C3×C4⋊C4—91no—
176#217—181no—
177#215—22no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—182no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186Dic3⋊C4—181no—
187Dic3⋊C4—181no—
188Dic3⋊C4—182no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8C2×D911yescharacteristic
191ΓL(1,𝔽25)—182no—
192ΓL(1,𝔽25)—182no—
193C22×C32D3×D411yescharacteristic
194C2×C3⋊D3—61no—
195C2×C3⋊D3—61no—
196C2×C3⋊D3—61no—
197C2×C3×D3—41no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C32×C4—41no—
201C32×C4#23711yescharacteristic
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—61no—
204C3⋊Dic3—121no—
205C3×Dic3—121no—
206C22×C9—21no—
207C22×C9—121no—
208C22×C9—121no—
209C22×C9—121no—
210C22×C9D3×D411yescharacteristic
211C2×D9—62no—
212C2×D9—61no—
213C2×D9—61no—
214C2×D9—61no—
215C2×D9—121no—
216C2×D9—121no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—42no—
220C4×C9—41no—
221C4×C9#23511yescharacteristic
222C4×C9D3×D411yescharacteristic
223Dic9—41no—
224Dic9—61no—
225Dic9—121no—
226Dic9—122no—
227Dic9—121no—
228C2×SD16—31no—
229C2×M16—271no—
230C22⋊Q8—271no—
231#121—271no—
232#107—271no—
233#103—91no—
234#102—271no—
235C3×C9#46811yescharacteristic, complemented, Sylow
236C23×C3—61no—
237C22×D3—21no—
238C22×D3—91no—
239C22×D3—91no—
240C22×D3—181no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×D4—61no—
248C3×D4—61no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—181no—
254C2×C3×C4C22×D911yescharacteristic
255C2×C3×C4C2×D3211yescharacteristic
256#67—362no—
257C2×Dic3—61no—
258C2×Dic3—91no—
259C2×Dic3—91no—
260C2×Dic3—181no—
261C2×Dic3—182no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264D12—21no—
265D12—21no—
266D12—362no—
267C4×D3—181no—
268C4×D3—181no—
269C4×D3—181no—
270C4×D3—181no—
271C4×D3—361no—
272C4×D3—361no—
273C3×C8—32no—
274C3×C8—181no—
275C3⋊C8—61no—
276C3⋊C8#61112yescomplemented
277C2×C32#91411yescharacteristic
278C2×C32#91511yescharacteristic
279C2×C32#93811yescharacteristic
280C3⋊D3—121no—
281C3×D3—81no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—241no—
286C2×C9#88411yescharacteristic
287C2×C9#91411yescharacteristic
288C2×C9#91811yescharacteristic
289D9—122no—
290D9—121no—
291C2×Q8D3×D911yescharacteristic
292C2×D4—31no—
293C2×D4—541no—
294C22×C4—271no—
295SD16—62no—
296M16—542no—
297C2×C8—31no—
298C2×C8—91no—
299C4⋊C4—91no—
300C4⋊C4—271no—
301C4⋊C4—541no—
302C22⋊C4—541no—
303C22⋊C4—541no—
304C22×C3—21no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3—121no—
308C22×C3D4×D911yescharacteristic
309C22×C3#381111yescharacteristic
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—182no—
314C2×D3—181no—
315C2×D3—181no—
316C2×D3—181no—
317C2×D3—181no—
318C2×D3—181no—
319C2×D3—181no—
320C2×D3—361no—
321C2×D3—361no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—41no—
326C3×C4—41no—
327C3×C4—361no—
328C3×C4#369911yescharacteristic
329C3×C4C2×#61111yescharacteristic
330C3×C4#380411yescharacteristic
331C3×C4D3×#6711yescharacteristic
332Dic3—121no—
333Dic3—181no—
334Dic3—181no—
335Dic3—361no—
336Dic3—362no—
337Dic3—361no—
338Dic3—361no—
339C32#517411yescharacteristic
340C9—21no—
341C9#481211yescharacteristic, complemented
342C23—61no—
343C23—271no—
344Q8—21no—
345Q8—21no—
346D4—61no—
347D4—61no—
348D4—1082no—
349C2×C4—21no—
350C2×C4—181no—
351C2×C4—271no—
352C2×C4—541no—
353C2×C4—541no—
354C2×C4—541no—
355C2×C4C2×D3×D911yescharacteristic
356C8—32no—
357C8—181no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—241no—
362C2×C3#6347611yescharacteristic
363C2×C3#6347711yescharacteristic
364C2×C3#6350011yescharacteristic
365C2×C3#6394111yescharacteristic
366C2×C3#6395111yescharacteristic
367C2×C3#6395811yescharacteristic
368D3—81no—
369D3—362no—
370D3—361no—
371D3—361no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—541no—
376C22—541no—
377C22—541no—
378C22#8866811yescharacteristic
379C4—41no—
380C4—361no—
381C4—541no—
382C4—1081no—
383C4#8866211yescharacteristic
384C4D3×#61111yescharacteristic
385C3—21no—
386C3#1058885411yescharacteristic, complemented
387C3#1059159011yescharacteristic
388C2—241no—
389C2—1081no—
390C2#1172121311yescharacteristic
391C2#1172121711yescharacteristic
392C2#1172126211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radical#11721045C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#468
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11721856C2semidirect
2. C2×C9×#215C2semidirect
4. C2×#88414C2semidirect
5. C3×#63395C2semidirect
6. #11720295C2semidirect
7. #11720279C2semidirect
35. #63395D3semidirect
52. C9×C3⋊C8D4semidirect
104. C2×#215D9semidirect
235. C3×C9#468semidirect
276. C3⋊C8#611semidirect
341. C9#4812semidirect
386. C3#10588854semidirect

Maximal subgroups

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

subgroupindex
#117218562
C2×C9×#2152
#117210452
C2×#884142
C3×#633952
#117202952
#117202792
#105915903
#105888543