Finite Group Databasebeta

← #13140377

Subgroups — #13140377

Order 1728.

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

normal subgroups only

This group has 392 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#11721833C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic, complemented
3#11720890C211yesmaximal, characteristic, complemented
4#11720862C211yesmaximal, characteristic, complemented
5C3×#63391C211yesmaximal, characteristic, complemented
6#11720296C211yesmaximal, characteristic, complemented
7#11720280C211yesmaximal, characteristic, complemented
8#10591406—31nomaximal
9#10588840—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11#88736C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13#88544—21no—
14#88543—21no—
15#88532C2211yescharacteristic
16C9×#213—21no—
17C9×#213—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19C3×#3667C2211yescharacteristic
20C3×#3667C2211yescharacteristic
21#88424—21no—
22#88422—21no—
23#88414—21no—
24(C3⋊D9)⋊C8—21no—
25#64141—31no—
26C2×C3×#213D311yescharacteristic
27#63830—31no—
28#63817—31no—
29C3×#833—31no—
30#63577—31no—
31#63558—31no—
32C2×C9×D8—31no—
33#63495—31no—
34#63495—31no—
35#63495—61no—
36#63464—31no—
37#63391—31no—
38#63391D311yescharacteristic, complemented
39#63372—31no—
40C23×C3×C9—21no—
41C22×D3×C9—21no—
42C3×D4×C9—21no—
43C3×D4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45#6233—21no—
46C4⋊(C3⋊D9)D411yescharacteristic
47C4×C3⋊D9—21no—
48#6226D411yescharacteristic
49C9×D12—21no—
50C9×D12—21no—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8D411yescharacteristic
53C9×C3⋊C8D411yescharacteristic
54C3×C9⋊C8—21no—
55C3×C9⋊C8—21no—
56#5160—91no—
57#4749—91no—
58C2×C32×D4C2×D311yescharacteristic
59#3828—31no—
60C2×C3×D12C2×D311yescharacteristic
61#3754—61no—
62#3753—61no—
63#3748—31no—
64C3×#213—21no—
65C3×#213—21no—
66C3×#208—31no—
67C3×#208—31no—
68C2×C3×C3⋊C8C2×D311yescharacteristic
69#3716—61no—
70#3714—61no—
71#3712—61no—
72#3710—61no—
73C2×D4×C9—21no—
74C2×D4×C9—31no—
75C2×D4×C9C2×D311yescharacteristic
76#3697—31no—
77#3697—61no—
78C9×D8—61no—
79C9×D8—61no—
80C2×C8×C9—31no—
81#3673—61no—
82#3673—61no—
83#3673—62no—
84#3672—61no—
85#3672—61no—
86#3672—62no—
87#3667—31no—
88#3667—61no—
89#3667C2×D311yescharacteristic
90#3667C2×D311yescharacteristic
91#3663—61no—
92C8×D9—61no—
93C22×C3×C9—41no—
94C22×C3×C9—81no—
95C22×C3×C9#4111yescharacteristic
96C2×C3⋊D9—21no—
97C2×D3×C9—41no—
98C2×D3×C9—81no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—21no—
102C2×C3×D8—31no—
103#932—91no—
104#932—91no—
105#932—91no—
106#932—181no—
107C2×#213D911yescharacteristic, complemented
108#908—91no—
109#907—91no—
110#901—91no—
111C3×#100—91no—
112#833—31no—
113#833—91no—
114#823—91no—
115#822—91no—
116#819—91no—
117C23×C32—21no—
118C22×C3×D3—21no—
119C32×D4—21no—
120C32×D4—21no—
121C2×C32×C4C22×D311yescharacteristic
122#638—61no—
123C4⋊(C3⋊D3)—31no—
124C4×C3⋊D3—61no—
125#634—31no—
126C3×D12—21no—
127C3×D12—21no—
128C23×C9—21no—
129C23×C9—41no—
130C23×C9—61no—
131C3⋊(C3⋊C8)—61no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8#6711yescharacteristic
135C3×C3⋊C8#6711yescharacteristic
136D4×C9—22no—
137D4×C9—21no—
138D4×C9—21no—
139D4×C9—41no—
140D4×C9—61no—
141D4×C9—61no—
142C2×C4×C9—21no—
143C2×C4×C9C22×D311yescharacteristic
144#611—61no—
145#611—121no—
146D36—31no—
147D36—61no—
148C4×D9—61no—
149C4×D9—121no—
150Dic18—31no—
151Dic18—61no—
152C8×C9—31no—
153C8×C9—31no—
154C9⋊C8—21no—
155C9⋊C8—21no—
156C9⋊C8—62no—
157C9⋊C8—61no—
158#472—271noSylow
159C2×C3×C9—21no—
160C2×C3×C9—81no—
161C2×C3×C9#12111yescharacteristic
162C3⋊D9—41no—
163D3×C9—81no—
164C2×C3×D4—21no—
165C2×C3×D4—31no—
166C2×C3×D4C2×D911yescharacteristic
167C2×C3×D4D3211yescharacteristic
168#235—91no—
169#235—91no—
170#235—181no—
171C2×D12C2×D911yescharacteristic
172C3×D8—61no—
173C3×D8—61no—
174C3×M16—91no—
175C3×M16—91no—
176C2×C3×C8—31no—
177#214—181no—
178#214—181no—
179#214—181no—
180#214—182no—
181#213—21no—
182#213—21no—
183#213—181no—
184#213—181no—
185#213—181no—
186#213—182no—
187#208—31no—
188#208—31no—
189#208—91no—
190#208—91no—
191#208—181no—
192C2×C3⋊C8C2×D911yescharacteristic
193D24—181no—
194#204—181no—
195ΓL(1,𝔽25)—181no—
196ΓL(1,𝔽25)—181no—
197D3×C8—181no—
198D3×C8—181no—
199C22×C32—41no—
200C22×C32—81no—
201C22×C32#23711yescharacteristic
202C2×C3⋊D3—61no—
203C2×C3×D3—41no—
204C2×C3×D3—81no—
205C32×C4D3×D411yescharacteristic
206C32×C4C2×#6711yescharacteristic
207C3⋊Dic3—61no—
208C22×C9—21no—
209C22×C9—42no—
210C22×C9—41no—
211C22×C9—81no—
212C22×C9—82no—
213C22×C9—121no—
214C22×C9—241no—
215C22×C9#23511yescharacteristic
216C2×D9—61no—
217C2×D9—121no—
218C4×C9—21no—
219C4×C9—21no—
220C4×C9D3×D411yescharacteristic
221C4×C9D3×D411yescharacteristic
222Dic9—61no—
223Dic9—121no—
224AGL(1,ℤ/8)—271no—
225AGL(1,ℤ/8)—271no—
226C2×D8—31no—
227#131—271no—
228#108—271no—
229#100—91no—
230#100—271no—
231C3×C9#47211yescharacteristic, complemented, Sylow
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—41no—
235C23×C3—61no—
236C22×D3—21no—
237C3×D4—22no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—41no—
243C3×D4—61no—
244C3×D4—61no—
245C2×C3×C4—21no—
246C2×C3×C4C22×D911yescharacteristic
247C2×C3×C4C2×D3211yescharacteristic
248#67—181no—
249#67—181no—
250#67—361no—
251D12—21no—
252D12—21no—
253D12—91no—
254D12—91no—
255D12—181no—
256C4×D3—181no—
257C4×D3—181no—
258C4×D3—361no—
259Dic6—91no—
260Dic6—91no—
261Dic6—181no—
262C3×C8—31no—
263C3×C8—31no—
264C3×C8—181no—
265C3×C8—181no—
266C3⋊C8—61no—
267C3⋊C8—61no—
268C3⋊C8—182no—
269C3⋊C8—181no—
270C3⋊C8—181no—
271C3⋊C8#61111yescharacteristic
272C3⋊C8#61111yescharacteristic
273C2×C32—21no—
274C2×C32—81no—
275C2×C32#93811yescharacteristic
276C3⋊D3—121no—
277C3×D3—81no—
278C2×C9—21no—
279C2×C9—21no—
280C2×C9—41no—
281C2×C9—82no—
282C2×C9—81no—
283C2×C9—241no—
284C2×C9#88411yescharacteristic
285D9—121no—
286D9—241no—
287#41—271no—
288C2×D4—31no—
289C2×D4D3×D911yescharacteristic
290SD16—541no—
291SD16—541no—
292D8—61no—
293D8—61no—
294D8—541no—
295D8—541no—
296M16—91no—
297M16—91no—
298M16—271no—
299M16—541no—
300C2×C8—31no—
301C2×C8—541no—
302C22×C3—21no—
303C22×C3—42no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—82no—
309C22×C3—121no—
310C22×C3—241no—
311C22×C3#369911yescharacteristic
312C22×C3#380411yescharacteristic
313C2×D3—41no—
314C2×D3—81no—
315C2×D3—181no—
316C2×D3—181no—
317C2×D3—361no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4D4×D911yescharacteristic
321C3×C4C2×#61111yescharacteristic
322C3×C4D3×#6711yescharacteristic
323C3×C4#381111yescharacteristic
324Dic3—181no—
325Dic3—181no—
326Dic3—361no—
327C32#516011yescharacteristic
328C9—21no—
329C9#474911yescharacteristic, complemented
330C23—21no—
331C23—61no—
332Q8—271no—
333D4—21no—
334D4—21no—
335D4—61no—
336D4—61no—
337D4—271no—
338D4—541no—
339C2×C4—541no—
340C2×C4C2×D3×D911yescharacteristic
341C8—31no—
342C8—31no—
343C8—181no—
344C8—181no—
345C8—541no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—41no—
350C2×C3—82no—
351C2×C3—81no—
352C2×C3—81no—
353C2×C3—241no—
354C2×C3#6350011yescharacteristic
355C2×C3#6395811yescharacteristic
356D3—81no—
357D3—361no—
358D3—361no—
359D3—721no—
360C22—41no—
361C22—81no—
362C22—121no—
363C22—241no—
364C22—541no—
365C22#8866211yescharacteristic
366C4—541no—
367C4D3×#61111yescharacteristic
368C4#8866811yescharacteristic
369C3—21no—
370C3#1058884011yescharacteristic, complemented
371C3#1059140611yescharacteristic
372C2—21no—
373C2—81no—
374C2—241no—
375C2—1081no—
376C2#1172126211yescharacteristic
subgroupG/H
centerC2#11721262
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#938
hypercenterC2×D4D3×D9
hypocenterC3×C9#472
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721833C2semidirect
2. C2×C9×#213C2semidirect
3. #11720890C2semidirect
4. #11720862C2semidirect
5. C3×#63391C2semidirect
6. #11720296C2semidirect
7. #11720280C2semidirect
38. #63391D3semidirect
107. C2×#213D9semidirect
231. C3×C9#472semidirect
329. C9#4749semidirect
370. C3#10588840semidirect

Maximal subgroups

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

subgroupindex
#117218332
C2×C9×#2132
#117208902
#117208622
C3×#633912
#117202962
#117202802
#105914063
#105888403