Finite Group Databasebeta

← #13140392

Subgroups — #13140392

Order 1728.

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

normal subgroups only

This group has 400 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
1C2×#88544C211yesmaximal, characteristic, complemented
2C3×#63495C211yesmaximal, characteristic, complemented
3#11720873C211yesmaximal, characteristic, complemented
4#11720859C211yesmaximal, characteristic, complemented
5C9×#833C211yesmaximal, characteristic, complemented
6#11720288C211yesmaximal, characteristic, complemented
7#11720286C211yesmaximal, characteristic, complemented
8#10591508—31nomaximal
9#10588429—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C3×#3697C2211yescharacteristic
13#88544—21no—
14#88544—21no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×#3673—21no—
17C3×#3672—21no—
18C9×#208C2211yescharacteristic
19C9×#208C2211yescharacteristic
20C3×#3667C2211yescharacteristic
21#88417—21no—
22#88416—21no—
23#88413—21no—
24D9×C3⋊C8—21no—
25C2×#3753—31no—
26C3×#932—31no—
27#63825—31no—
28#63815—31no—
29C3×#833D311yescharacteristic
30#63578—31no—
31#63559—31no—
32#63495D311yescharacteristic, complemented
33C2×#3673—31no—
34C2×#3673—61no—
35#63470—31no—
36#63469—31no—
37C9×#100—31no—
38#63383—31no—
39#63381—31no—
40C23×C3×C9—21no—
41C22×C3⋊D9—21no—
42C3×D4×C9—21no—
43C3×D4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C3×#611—21no—
48C3×D36D411yescharacteristic
49C3×C4×D9—21no—
50C3×Dic18D411yescharacteristic
51C3⋊(C9⋊C8)D411yescharacteristic
52C3⋊(C9⋊C8)D411yescharacteristic
53C9×C3⋊C8—21no—
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#5199—91no—
57#4749—91no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)—31no—
60C3×#235—31no—
61#3753—61no—
62#3753—61no—
63C2×C3⋊(C3⋊C8)—31no—
64C3×#214—61no—
65C3×#213—61no—
66C3×#208—31no—
67C3×#208C2×D311yescharacteristic
68C3×#208C2×D311yescharacteristic
69#3717—61no—
70#3714—61no—
71#3711—61no—
72D3×C3⋊C8—61no—
73C2×D4×C9—21no—
74C2×D4×C9C2×D311yescharacteristic
75#3697C2×D311yescharacteristic
76C2×D36—31no—
77C2×D36—61no—
78C9×M16—31no—
79C9×M16—31no—
80#3673—21no—
81#3673—61no—
82#3673—61no—
83#3673—62no—
84#3673—62no—
85#3672—21no—
86#3667C2×D311yescharacteristic
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89D72—61no—
90#3664—61no—
91#3663—61no—
92C8×D9—61no—
93C22×C3×C9—41no—
94C22×C3×C9—81no—
95C22×C3×C9#4111yescharacteristic
96C2×C3⋊D9—41no—
97C2×C3⋊D9—81no—
98C2×C3×D9—21no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3×Dic9—21no—
102C3×AGL(1,ℤ/8)—91no—
103#949—91no—
104#948—91no—
105#932—31no—
106C2×#213—91no—
107C2×#213—91no—
108C2×#213—181no—
109#908—91no—
110#907—91no—
111C3×#100—31no—
112#833D911yescharacteristic, complemented
113#823—91no—
114#823—91no—
115#822—91no—
116#822—91no—
117C23×C32—21no—
118C22×C3⋊D3—61no—
119C32×D4—21no—
120C32×D4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C4⋊(C3⋊D3)—61no—
123C4⋊(C3⋊D3)—61no—
124C3×#67—61no—
125C3×D12—31no—
126C3×C4×D3—61no—
127C3×Dic6—31no—
128C23×C9—21no—
129C23×C9—41no—
130C22×D9—61no—
131C22×D9—121no—
132C3⋊(C3⋊C8)—31no—
133C3⋊(C3⋊C8)—31no—
134C3×C3⋊C8—21no—
135C3×C3⋊C8—21no—
136C3×C3⋊C8—61no—
137D4×C9—22no—
138D4×C9—21no—
139D4×C9—21no—
140D4×C9—41no—
141C2×C4×C9—21no—
142C2×C4×C9C22×D311yescharacteristic
143#611—21no—
144D36—62no—
145D36—61no—
146D36—61no—
147D36—121no—
148D36#6711yescharacteristic
149C4×D9—21no—
150Dic18#6711yescharacteristic
151C8×C9—61no—
152C8×C9—61no—
153C9⋊C8—21no—
154C9⋊C8—31no—
155C9⋊C8—31no—
156C9⋊C8—61no—
157C9⋊C8—61no—
158#472—271noSylow
159C2×C3×C9—21no—
160C2×C3×C9—81no—
161C2×C3×C9#12111yescharacteristic
162C3⋊D9—81no—
163C3×D9—41no—
164C3×#41—91no—
165C2×C3×D4—21no—
166C2×C3×D4C2×D911yescharacteristic
167C2×C3×D4D3211yescharacteristic
168#235—31no—
169C2×D12—91no—
170C2×D12—91no—
171C2×D12—181no—
172C3×SD16—181no—
173C3×D8—181no—
174C3×M16—31no—
175C3×M16—31no—
176C3×M16—91no—
177#215—181no—
178#214—61no—
179#213—61no—
180#213—181no—
181#213—181no—
182#213—181no—
183#213—181no—
184#213—181no—
185#213—182no—
186#213—182no—
187#208—31no—
188#208—181no—
189#208C2×D911yescharacteristic
190#208C2×D911yescharacteristic
191C2×C3⋊C8—91no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—181no—
194C2×C3⋊C8—181no—
195D24—181no—
196#204—181no—
197ΓL(1,𝔽25)—181no—
198D3×C8—181no—
199C22×C32—41no—
200C22×C32—81no—
201C22×C32#23511yescharacteristic
202C2×C3⋊D3—121no—
203C2×C3⋊D3—241no—
204C2×C3×D3—61no—
205C32×C4D3×D411yescharacteristic
206C32×C4D3×D411yescharacteristic
207C3×Dic3—61no—
208C22×C9—21no—
209C22×C9—42no—
210C22×C9—41no—
211C22×C9—81no—
212C22×C9—82no—
213C22×C9#23711yescharacteristic
214C2×D9—21no—
215C2×D9—122no—
216C2×D9—121no—
217C2×D9—242no—
218C2×D9—241no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9D3×D411yescharacteristic
222C4×C9C2×#6711yescharacteristic
223Dic9—21no—
224AGL(1,ℤ/8)—91no—
225AGL(1,ℤ/8)—271no—
226C2×D8—271no—
227#131—271no—
228#108—271no—
229#100—31no—
230#100—271no—
231C3×C9#47211yescharacteristic, complemented, Sylow
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—41no—
235C22×D3—181no—
236C22×D3—181no—
237C22×D3—361no—
238C3×Q8—91no—
239C3×D4—22no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—41no—
245C3×D4—91no—
246C3×D4—181no—
247C2×C3×C4—21no—
248C2×C3×C4—181no—
249C2×C3×C4C22×D911yescharacteristic
250C2×C3×C4C2×D3211yescharacteristic
251#67—61no—
252D12—31no—
253D12—182no—
254D12—181no—
255D12—181no—
256D12—181no—
257D12—181no—
258D12—361no—
259C4×D3—61no—
260Dic6—31no—
261C3×C8—61no—
262C3×C8—61no—
263C3×C8—181no—
264C3⋊C8—21no—
265C3⋊C8—21no—
266C3⋊C8—61no—
267C3⋊C8—91no—
268C3⋊C8—91no—
269C3⋊C8—91no—
270C3⋊C8—91no—
271C3⋊C8—181no—
272C3⋊C8—181no—
273C2×C32—21no—
274C2×C32—81no—
275C2×C32#88411yescharacteristic
276C3⋊D3—241no—
277C3×D3—121no—
278C2×C9—22no—
279C2×C9—21no—
280C2×C9—21no—
281C2×C9—82no—
282C2×C9—81no—
283C2×C9#93911yescharacteristic
284D9—41no—
285D9—242no—
286D9—241no—
287#41—91no—
288C2×D4—271no—
289C2×D4D3×D911yescharacteristic
290SD16—181no—
291SD16—541no—
292D8—181no—
293D8—541no—
294D8—541no—
295D8—541no—
296M16—31no—
297M16—31no—
298M16—91no—
299M16—541no—
300C2×C8—271no—
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—181no—
310C22×C3#369711yescharacteristic
311C22×C3#380411yescharacteristic
312C2×D3—61no—
313C2×D3—362no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—722no—
317C2×D3—721no—
318C2×D3—721no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—181no—
322C3×C4D4×D911yescharacteristic
323C3×C4D4×D911yescharacteristic
324C3×C4D3×#6711yescharacteristic
325C3×C4#381111yescharacteristic
326Dic3—61no—
327C32#474911yescharacteristic
328C9—21no—
329C9#519911yescharacteristic, complemented
330C23—21no—
331C23—541no—
332Q8—91no—
333D4—21no—
334D4—21no—
335D4—91no—
336D4—181no—
337D4—541no—
338D4—541no—
339C2×C4—181no—
340C2×C4C2×D3×D911yescharacteristic
341C8—61no—
342C8—61no—
343C8—181no—
344C8—271no—
345C8—271no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—82no—
351C2×C3—81no—
352C2×C3—81no—
353C2×C3—361no—
354C2×C3#6344611yescharacteristic
355C2×C3#6397111yescharacteristic
356D3—121no—
357D3—722no—
358D3—721no—
359D3—721no—
360C22—41no—
361C22—81no—
362C22—181no—
363C22—1081no—
364C22—2161no—
365C22#8865811yescharacteristic
366C4—181no—
367C4D9×#6711yescharacteristic
368C4#8866811yescharacteristic
369C3—21no—
370C3#1058842911yescharacteristic, complemented
371C3#1059150811yescharacteristic
372C2—21no—
373C2—81no—
374C2—361no—
375C2—2161no—
376C2#1172127111yescharacteristic
subgroupG/H
centerC2#11721271
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#884
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. C2×#88544C2semidirect
2. C3×#63495C2semidirect
3. #11720873C2semidirect
4. #11720859C2semidirect
5. C9×#833C2semidirect
6. #11720288C2semidirect
7. #11720286C2semidirect
32. #63495D3semidirect
112. #833D9semidirect
231. C3×C9#472semidirect
329. C9#5199semidirect
370. C3#10588429semidirect

Maximal subgroups

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

subgroupindex
C2×#885442
C3×#634952
#117208732
#117208592
C9×#8332
#117202882
#117202862
#105915083
#105884293