Finite Group Databasebeta

← #13139085

Subgroups — #13139085

Order 1728.

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

normal subgroups only

This group has 392 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#11721783C211yesmaximal, characteristic, complemented
2#11720890C211yesmaximal, characteristic, complemented
3C2×#88422C211yesmaximal, characteristic, complemented
4#11720861C211yesmaximal, characteristic, complemented
5C9×#823C211yesmaximal, characteristic, complemented
6C3×#63381C211yesmaximal, characteristic, complemented
7#11720288C211yesmaximal, characteristic, complemented
8#10590987—31nomaximal
9#10588429—31nomaximal
10#88736C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C3×C9×M16C2211yescharacteristic
14#88525—21no—
15#88524—21no—
16C9×#208C2211yescharacteristic
17C3×#3667C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88424—21no—
20#88422—21no—
21#88422—21no—
22#88422—21no—
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25#64117—31no—
26#63830—31no—
27C2×#3714—31no—
28#63817—31no—
29C3×#823D311yescharacteristic
30C3×#822—31no—
31#63559—31no—
32#63495—31no—
33C2×#3673—31no—
34#63470—31no—
35#63470—61no—
36#63469—31no—
37C9×#100—31no—
38#63383—31no—
39#63381D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×D3×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43#6233—21no—
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)D411yescharacteristic
47C4×C3⋊D9—21no—
48#6226D411yescharacteristic
49C9×D12—21no—
50C9×D12—21no—
51C3×C8×C9—21no—
52C9×C3⋊C8—21no—
53C3×C9⋊C8—21no—
54C3×C9⋊C8D411yescharacteristic
55C3×C9⋊C8D411yescharacteristic
56#4897—91no—
57#4749—91no—
58#3828—31no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C3×D12C2×D311yescharacteristic
61C32×M16C2×D311yescharacteristic
62#3745—61no—
63#3744—61no—
64C3×#208—31no—
65C3×#208C2×D311yescharacteristic
66C2×C3×C3⋊C8—31no—
67#3716—61no—
68#3714—61no—
69#3714—61no—
70#3714—61no—
71#3712—61no—
72#3710—61no—
73C2×D4×C9—31no—
74#3697—31no—
75#3697—61no—
76C2×D36—31no—
77C2×D36—61no—
78C9×M16—21no—
79C9×M16—31no—
80C9×M16C2×D311yescharacteristic
81#3673—61no—
82#3673—61no—
83#3673—61no—
84#3672—61no—
85#3667C2×D311yescharacteristic
86C2×C9⋊C8C2×D311yescharacteristic
87D72—61no—
88D72—62no—
89#3664—61no—
90#3664—62no—
91#3663—61no—
92C8×D9—61no—
93C22×C3×C9#4111yescharacteristic
94C2×C3⋊D9—21no—
95C2×C3⋊D9—41no—
96C2×C3⋊D9—81no—
97C2×D3×C9—41no—
98C2×D3×C9—81no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—21no—
102#932—91no—
103C2×#213—91no—
104#908—91no—
105#908—91no—
106#908—91no—
107#908—181no—
108#907—91no—
109C2×D24—91no—
110#901—91no—
111C3×#108—91no—
112C3×#100—31no—
113#823—91no—
114#823—91no—
115#823D911yescharacteristic, complemented
116#822—31no—
117C22×C3⋊D3—61no—
118C22×C3×D3—21no—
119C2×C32×C4C22×D311yescharacteristic
120#638—61no—
121C4⋊(C3⋊D3)—31no—
122C4⋊(C3⋊D3)—61no—
123C4⋊(C3⋊D3)—61no—
124C4×C3⋊D3—61no—
125#634—31no—
126C3×D12—21no—
127C3×D12—21no—
128C23×C9—61no—
129C22×D9—61no—
130C22×D9—121no—
131C32×C8—21no—
132C3×C3⋊C8—21no—
133C3×C3⋊C8—31no—
134C3×C3⋊C8—31no—
135C3×C3⋊C8—61no—
136D4×C9—61no—
137D4×C9—61no—
138C2×C4×C9—21no—
139C2×C4×C9C22×D311yescharacteristic
140#611—61no—
141#611—121no—
142D36—31no—
143D36—61no—
144D36—62no—
145D36—61no—
146D36—61no—
147D36—121no—
148C4×D9—61no—
149C4×D9—121no—
150Dic18—31no—
151Dic18—61no—
152C8×C9—22no—
153C8×C9—21no—
154C8×C9—61no—
155C9⋊C8—21no—
156C9⋊C8D1211yescharacteristic
157C9⋊C8D1211yescharacteristic
158#472—271noSylow
159C2×C3×C9—21no—
160C2×C3×C9#12111yescharacteristic
161C3⋊D9—41no—
162C3⋊D9—81no—
163D3×C9—81no—
164C2×C3×D4—31no—
165#235—91no—
166#235—91no—
167#235—181no—
168C2×D12—91no—
169C2×D12—91no—
170C2×D12—181no—
171C2×D12C2×D911yescharacteristic
172C3×M16—21no—
173C3×M16—31no—
174C3×M16—91no—
175C3×M16C2×D911yescharacteristic
176C3×M16D3211yescharacteristic
177C2×C3×C8—91no—
178#214—181no—
179#213—181no—
180#213—181no—
181#213—181no—
182#208—31no—
183#208C2×D911yescharacteristic
184C2×C3⋊C8—31no—
185D24—181no—
186D24—181no—
187D24—182no—
188D24—181no—
189D24—181no—
190D24—181no—
191#204—181no—
192#204—181no—
193#204—182no—
194#204—181no—
195ΓL(1,𝔽25)—181no—
196ΓL(1,𝔽25)—181no—
197D3×C8—181no—
198D3×C8—181no—
199C22×C32#23511yescharacteristic
200C2×C3⋊D3—61no—
201C2×C3⋊D3—121no—
202C2×C3⋊D3—241no—
203C2×C3×D3—41no—
204C2×C3×D3—81no—
205C32×C4D3×D411yescharacteristic
206C32×C4D3×D411yescharacteristic
207C3⋊Dic3—61no—
208C22×C9—21no—
209C22×C9—121no—
210C22×C9—241no—
211C22×C9#23911yescharacteristic
212C2×D9—61no—
213C2×D9—121no—
214C2×D9—122no—
215C2×D9—121no—
216C2×D9—242no—
217C2×D9—241no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9C2×D1211yescharacteristic
221C4×C9D3×D411yescharacteristic
222Dic9—61no—
223Dic9—121no—
224AGL(1,ℤ/8)—271no—
225AGL(1,ℤ/8)—271no—
226C2×D8—271no—
227#131—271no—
228#108—91no—
229#100—31no—
230#100—271no—
231C3×C9#47211yescharacteristic, complemented, Sylow
232C23×C3—61no—
233C22×D3—21no—
234C22×D3—181no—
235C22×D3—181no—
236C22×D3—361no—
237C3×D4—61no—
238C3×D4—61no—
239C2×C3×C4—21no—
240C2×C3×C4C22×D911yescharacteristic
241C2×C3×C4C2×D3211yescharacteristic
242#67—181no—
243#67—181no—
244#67—361no—
245D12—21no—
246D12—21no—
247D12—91no—
248D12—91no—
249D12—181no—
250D12—182no—
251D12—181no—
252D12—181no—
253D12—181no—
254D12—181no—
255D12—361no—
256C4×D3—181no—
257C4×D3—181no—
258C4×D3—361no—
259Dic6—91no—
260Dic6—91no—
261Dic6—181no—
262C3×C8—22no—
263C3×C8—21no—
264C3×C8—21no—
265C3×C8—61no—
266C3×C8—91no—
267C3×C8—91no—
268C3×C8—181no—
269C3⋊C8—21no—
270C3⋊C8—31no—
271C3⋊C8—31no—
272C3⋊C8—61no—
273C2×C32—21no—
274C2×C32#88411yescharacteristic
275C3⋊D3—121no—
276C3⋊D3—241no—
277C3×D3—81no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9—41no—
281C2×C9—241no—
282C2×C9#89511yescharacteristic
283D9—121no—
284D9—241no—
285D9—242no—
286D9—241no—
287#41—271no—
288C2×D4—31no—
289C2×D4—271no—
290SD16—541no—
291SD16—541no—
292D8—541no—
293D8—541no—
294D8—541no—
295D8—541no—
296M16—31no—
297M16—91no—
298M16—541no—
299M16D3×D911yescharacteristic
300C2×C8—91no—
301C2×C8—541no—
302C22×C3—21no—
303C22×C3—121no—
304C22×C3—241no—
305C22×C3#369711yescharacteristic
306C22×C3#379911yescharacteristic
307C2×D3—41no—
308C2×D3—81no—
309C2×D3—181no—
310C2×D3—181no—
311C2×D3—361no—
312C2×D3—362no—
313C2×D3—361no—
314C2×D3—361no—
315C2×D3—722no—
316C2×D3—721no—
317C2×D3—721no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4D4×D911yescharacteristic
321C3×C4D4×D911yescharacteristic
322C3×C4D3×D1211yescharacteristic
323C3×C4#381111yescharacteristic
324Dic3—181no—
325Dic3—181no—
326Dic3—361no—
327C32#474911yescharacteristic
328C9—21no—
329C9#489711yescharacteristic, complemented
330C23—61no—
331C23—541no—
332Q8—271no—
333D4—61no—
334D4—61no—
335D4—271no—
336D4—541no—
337D4—541no—
338D4—541no—
339C2×C4—541no—
340C2×C4C2×D3×D911yescharacteristic
341C8—21no—
342C8—61no—
343C8—91no—
344C8—91no—
345C8—181no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—41no—
350C2×C3—241no—
351C2×C3#6344611yescharacteristic
352C2×C3#6391011yescharacteristic
353D3—81no—
354D3—361no—
355D3—361no—
356D3—721no—
357D3—722no—
358D3—721no—
359D3—721no—
360C22—121no—
361C22—241no—
362C22—541no—
363C22—1081no—
364C22—2161no—
365C22#8864211yescharacteristic
366C4—541no—
367C4D9×D1211yescharacteristic
368C4#8866811yescharacteristic
369C3—21no—
370C3#1058842911yescharacteristic, complemented
371C3#1059098711yescharacteristic
372C2—21no—
373C2—241no—
374C2—1081no—
375C2—2161no—
376C2#1172103911yescharacteristic
subgroupG/H
centerC2#11721039
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×M16C22
socleC2×C32#884
hypercenterM16D3×D9
hypocenterC3×C9#472
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #11721783C2semidirect
2. #11720890C2semidirect
3. C2×#88422C2semidirect
4. #11720861C2semidirect
5. C9×#823C2semidirect
6. C3×#63381C2semidirect
7. #11720288C2semidirect
39. #63381D3semidirect
115. #823D9semidirect
231. C3×C9#472semidirect
329. C9#4897semidirect
370. C3#10588429semidirect

Maximal subgroups

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

subgroupindex
#117217832
#117208902
C2×#884222
#117208612
C9×#8232
C3×#633812
#117202882
#105909873
#105884293