Finite Group Databasebeta

← #13138696

Subgroups — #13138696

Order 1728.

518 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 101 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×(C3⋊D9)⋊Q8C211yesmaximal, characteristic
2C2×#88523C211yesmaximal, characteristic
3#11720922C211yesmaximal, characteristic
4C9×#816C211yesmaximal, characteristic, complemented
5C3×#63376C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590691—31nomaximal
9#10588482—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14C2×#6189—21no—
15(C3⋊D9)⋊Q8—22no—
16(C3⋊D9)⋊Q8C414yescomplemented
17C2×C3×C8×C9C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19#88523—22no—
20C9×C4⋊Dic3C2211yescharacteristic
21C3×C4⋊Dic9C2211yescharacteristic
22#88445—21no—
23Dic3×Dic9—21no—
24C2×#3797—31no—
25C2×#3743—31no—
26#63849—31no—
27C2×Q8×D9—31no—
28C3×#816—31no—
29C3×#816D311yescharacteristic
30#63565—31no—
31#63565—31no—
32C2×#3663—31no—
33C2×#3663—61no—
34#63454—31no—
35C9×#103—31no—
36#63395—31no—
37#63376D311yescharacteristic, complemented
38#63368—31no—
39C22×C3⋊D9D411yescharacteristic
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C4×C3⋊D9C2×C412yes—
43C4×C3⋊D9D412yes—
44C2×C9×Dic3—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C2×C3×Dic9—21no—
48C9×Dic6—21no—
49C9×Dic6C2×C412yes—
50C3×Dic18—21no—
51C3×Dic18C2×C412yes—
52(C3⋊D9)⋊C4—24no—
53#6189—24no—
54C3×C8×C9—21no—
55C3⋊(C9⋊C8)—21no—
56#4802—91no—
57#4802—91no—
58C2×C4×C3⋊D3—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#627—61no—
62C2×#624—61no—
63#3797—34no—
64#3797—62no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67#3743—62no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3C2×D311yescharacteristic
70#3722—61no—
71Dic32—61no—
72C2×Q8×C9—31no—
73Q8×D9—34no—
74Q8×D9—62no—
75C2×C4×D9—31no—
76C2×C4×D9—61no—
77C2×C4×D9—61no—
78C2×Dic18—61no—
79C2×Dic18C2×D311yescharacteristic
80C2×C8×C9—21no—
81C2×C8×C9C2×D311yescharacteristic
82C9×C4⋊C4—31no—
83(C2×D9)⋊C4—61no—
84C4⋊Dic9C2×D311yescharacteristic
85C4×Dic9—61no—
86C2×C9⋊C8—31no—
87C2×C9⋊C8—61no—
88#3663—62no—
89#3663—64no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—21no—
92C2×C3⋊D9—21no—
93C2×C3⋊D9C22⋊C412yes—
94C3×C4×C9C22×C411yescharacteristic
95C3×C4×C9C2×D411yescharacteristic
96C3⋊Dic9C22⋊C412yes—
97C9×Dic3—22no—
98C9×Dic3—41no—
99C3×Dic9—22no—
100C3×Dic9—41no—
101C2×D3×Q8—91no—
102C2×D3×Q8—91no—
103C2×ΓL(1,𝔽25)—91no—
104C2×ΓL(1,𝔽25)—91no—
105C2×ΓL(1,𝔽25)—181no—
106#892—91no—
107#892—91no—
108C3×#103—31no—
109C3×#103—91no—
110#835—91no—
111#835—91no—
112#816—31no—
113#816D911yescharacteristic, complemented
114#810—91no—
115#810—91no—
116C22×C3⋊D3—31no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—31no—
119C4×C3⋊D3—32no—
120C4×C3⋊D3—32no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×Dic6—21no—
126C3×Dic6—32no—
127C3×Dic6—61no—
128C3×Dic6C4×D312yes—
129#627—64no—
130#624—64no—
131C22×D9—31no—
132C22×D9—61no—
133C32×C8—21no—
134C3⋊(C3⋊C8)—61no—
135Q8×C9—32no—
136Q8×C9—61no—
137C2×C4×C9—21no—
138C2×C4×C9—61no—
139C2×C4×C9—61no—
140C2×C4×C9C22×D311yescharacteristic
141C2×Dic9—21no—
142C2×Dic9—21no—
143C2×Dic9—31no—
144C2×Dic9—61no—
145C4×D9—32no—
146C4×D9—32no—
147C4×D9—62no—
148C4×D9—62no—
149C4×D9—64no—
150Dic18—21no—
151Dic18—64no—
152Dic18C4×D312yes—
153C8×C9—22no—
154C8×C9—21no—
155C9⋊C8—62no—
156C9⋊C8—61no—
157#419—271noSylow
158C2×C3×C9C2×C22⋊C411yescharacteristic
159C2×C3×C9#13711yescharacteristic
160C2×C3×C9#13711yescharacteristic
161C3⋊D9—22no—
162C2×C3×Q8—31no—
163C2×C3×Q8—91no—
164D3×Q8—94no—
165D3×Q8—94no—
166D3×Q8—182no—
167D3×Q8—182no—
168C2×C4×D3—91no—
169C2×C4×D3—91no—
170C2×C4×D3—181no—
171C2×C4×D3—181no—
172C2×C4×D3—181no—
173C2×Dic6—31no—
174C2×Dic6—181no—
175C2×Dic6—181no—
176C2×Dic6C2×D911yescharacteristic
177C2×C3×C8—21no—
178C2×C3×C8C2×D911yescharacteristic
179C2×C3×C8D3211yescharacteristic
180C3×C4⋊C4—31no—
181C3×C4⋊C4—91no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—181no—
184C4⋊Dic3—31no—
185C4⋊Dic3C2×D911yescharacteristic
186C4×Dic3—181no—
187C4×Dic3—181no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—181no—
191ΓL(1,𝔽25)—182no—
192ΓL(1,𝔽25)—182no—
193ΓL(1,𝔽25)—184no—
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—32no—
196C2×C3⋊D3—61no—
197C2×C3⋊D3—61no—
198C32×C4C2×C4×D311yescharacteristic
199C32×C4D3×D411yescharacteristic
200C3⋊Dic3—32no—
201C3×Dic3—22no—
202C3×Dic3—41no—
203C3×Dic3—62no—
204C3×Dic3—121no—
205C22×C9—21no—
206C22×C9D3×D411yescharacteristic
207C2×D9—32no—
208C2×D9—62no—
209C2×D9—61no—
210C2×D9—61no—
211C2×D9—121no—
212C2×D9—121no—
213C4×C9—21no—
214C4×C9—21no—
215C4×C9—62no—
216C4×C9—121no—
217C4×C9C2×C4×D311yescharacteristic
218C4×C9D3×D411yescharacteristic
219Dic9—22no—
220Dic9—32no—
221Dic9—41no—
222Dic9—62no—
223C22×Q8—271no—
224C2×M16—271no—
225#117—271no—
226#103—31no—
227#103—91no—
228#103—271no—
229#103—271no—
230C3×C9#41911yescharacteristic, complemented, Sylow
231C22×D3—91no—
232C22×D3—91no—
233C22×D3—181no—
234C3×Q8—32no—
235C3×Q8—61no—
236C3×Q8—92no—
237C3×Q8—181no—
238C2×C3×C4—21no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—181no—
252C4×D3—92no—
253C4×D3—92no—
254C4×D3—92no—
255C4×D3—92no—
256C4×D3—182no—
257C4×D3—182no—
258C4×D3—184no—
259C4×D3—184no—
260Dic6—21no—
261Dic6—32no—
262Dic6—61no—
263Dic6—184no—
264Dic6—184no—
265Dic6C4×D912yes—
266C3×C8—22no—
267C3×C8—21no—
268C3×C8—21no—
269C3⋊C8—182no—
270C3⋊C8—181no—
271C3⋊C8—181no—
272C2×C32D3×C22⋊C411yescharacteristic
273C2×C32#91511yescharacteristic
274C2×C32#91811yescharacteristic
275C3⋊D3—62no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9—21no—
279C2×C9D3×C22⋊C411yescharacteristic
280C2×C9#91511yescharacteristic
281C2×C9#91811yescharacteristic
282D9—62no—
283D9—122no—
284C2×Q8—31no—
285C2×Q8—91no—
286C2×Q8—274no—
287C2×Q8—541no—
288C2×Q8—542no—
289C22×C4—271no—
290C22×C4—541no—
291M16—542no—
292C2×C8—271no—
293C2×C8D3×D911yescharacteristic
294C4⋊C4—31no—
295C4⋊C4—91no—
296C22⋊C4—541no—
297C42—541no—
298C22×C3—21no—
299C22×C3D4×D911yescharacteristic
300C22×C3#380211yescharacteristic
301C2×D3—92no—
302C2×D3—92no—
303C2×D3—182no—
304C2×D3—181no—
305C2×D3—181no—
306C2×D3—181no—
307C2×D3—181no—
308C2×D3—361no—
309C2×D3—361no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—62no—
313C3×C4—121no—
314C3×C4—182no—
315C3×C4—361no—
316C3×C4C2×C4×D911yescharacteristic
317C3×C4D4×D911yescharacteristic
318C3×C4C4×D3211yescharacteristic
319C3×C4#381111yescharacteristic
320Dic3—22no—
321Dic3—41no—
322Dic3—62no—
323Dic3—92no—
324Dic3—92no—
325Dic3—121no—
326Dic3—182no—
327C32#480211yescharacteristic
328C9—21no—
329C9#480211yescharacteristic, complemented
330C23—271no—
331Q8—32no—
332Q8—61no—
333Q8—92no—
334Q8—181no—
335Q8—544no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—271no—
341C2×C4—272no—
342C2×C4—272no—
343C2×C4—544no—
344C2×C4C2×D3×D911yescharacteristic
345C8—21no—
346C8—541no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3C22⋊C4×D911yescharacteristic
351C2×C3#6347711yescharacteristic
352C2×C3#6348011yescharacteristic
353C2×C3#6381111yescharacteristic
354C2×C3#6381111yescharacteristic
355C2×C3#6392111yescharacteristic
356D3—182no—
357D3—182no—
358D3—362no—
359C22—272no—
360C22—541no—
361C22—541no—
362C22#8864611yescharacteristic
363C4—62no—
364C4—121no—
365C4—182no—
366C4—272no—
367C4—361no—
368C4C4×D3×D911yescharacteristic
369C4#8866811yescharacteristic
370C3—21no—
371C3#1058848211yescharacteristic, complemented
372C3#1059069111yescharacteristic
373C2—542no—
374C2#1172082511yescharacteristic
375C2#1172083211yescharacteristic
376C2#1172104811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#419
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C9×#816C2semidirect
5. C3×#63376C2semidirect
6. #11720297C2semidirect
7. #11720268C2semidirect
16. (C3⋊D9)⋊Q8C4semidirect
37. #63376D3semidirect
113. #816D9semidirect
230. C3×C9#419semidirect
329. C9#4802semidirect
371. C3#10588482semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
C2×#885232
#117209222
C9×#8162
C3×#633762
#117202972
#117202682
#105906913
#105884823