Finite Group Databasebeta

← #13140747

Subgroups — #13140747

Order 1728.

404 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C2×C3×#3674C211yesmaximal, characteristic, complemented
3#11720936C211yesmaximal, characteristic
4C2×#88414C211yesmaximal, characteristic
5C9×#835C211yesmaximal, characteristic, complemented
6#11720276C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10591600—31nomaximal
9#10588496—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13#88536—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C3×#3674—22no—
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88445—21no—
21Dic3⋊Dic9—21no—
22#88414—22no—
23#64155—31no—
24C2×C3×#216—31no—
25#63871—31no—
26C2×#3712—31no—
27C3×#835D311yescharacteristic
28#63573—31no—
29#63566—31no—
30#63508—31no—
31#63508—61no—
32C2×#3674D311yescharacteristic, complemented
33C2×#3663—31no—
34#63459—31no—
35C9×#103—31no—
36#63376—31no—
37#63367—31no—
38C22×C3⋊D9D411yescharacteristic
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C4×C3⋊D9—21no—
46C4×C3⋊D9—21no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51C9×C3⋊C8—21no—
52C3×C9⋊C8D412yes—
53#5189—91no—
54#4816—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C4×C3⋊D3—31no—
57C2×C3×Dic6—31no—
58#3752—61no—
59C4⋊(C3⋊Dic3)—31no—
60#3750—61no—
61C3×#216—62no—
62C3×C4⋊Dic3C2×D311yescharacteristic
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65Dic3⋊Dic3—61no—
66#3722—61no—
67#3712—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C2×D311yescharacteristic
70C2×C4×D9—31no—
71C2×C4×D9—61no—
72C2×Dic18C2×D311yescharacteristic
73C2×C8×C9—31no—
74C9×C4⋊C4—31no—
75#3674—22no—
76(C2×D9)⋊C4—61no—
77(C2×D9)⋊C4—61no—
78(C2×D9)⋊C4—62no—
79C4⋊Dic9—31no—
80C4⋊Dic9—61no—
81C4⋊Dic9—61no—
82Dic9⋊C4—61no—
83Dic9⋊C4—62no—
84C2×C9⋊C8C2×D311yescharacteristic
85#3663—62no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—21no—
88C2×C3⋊D9—21no—
89C2×C3⋊D9—21no—
90C3×C4×C9—41no—
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C3⋊Dic9—21no—
94C3⋊Dic9—41no—
95C9×Dic3—41no—
96C3×Dic9—41no—
97C2×C3×Q16—91no—
98#946—91no—
99#946—91no—
100#946—91no—
101#946—181no—
102C2×#216—31no—
103C2×ΓL(1,𝔽25)—91no—
104C2×ΓL(1,𝔽25)—91no—
105#897—91no—
106C3×#103—31no—
107#835—91no—
108#835D911yescharacteristic, complemented
109#817—91no—
110#816—91no—
111#808—91no—
112C22×C3⋊D3—31no—
113C32×Q8—21no—
114C32×Q8—21no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—61no—
119C4×C3⋊D3—61no—
120C4×C3⋊D3—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—61no—
123C3×Dic6—61no—
124C3×Dic6—61no—
125C22×D9—31no—
126C22×D9—61no—
127C3×C3⋊C8—21no—
128C3×C3⋊C8—32no—
129Q8×C9—21no—
130Q8×C9—21no—
131Q8×C9—41no—
132Q8×C9—41no—
133C2×C4×C9—22no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—61no—
137C2×C4×C9C22×D311yescharacteristic
138C2×Dic9—21no—
139C2×Dic9—31no—
140C2×Dic9—61no—
141C2×Dic9—62no—
142C2×Dic9—61no—
143C4×D9—61no—
144C4×D9—61no—
145C4×D9—121no—
146C4×D9—121no—
147Dic18—21no—
148Dic18—21no—
149C8×C9—61no—
150C9⋊C8#6712yes—
151#470—271noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9#13711yescharacteristic
154C2×C3×C9#13711yescharacteristic
155C3⋊D9—41no—
156C2×C3×Q8—21no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×D911yescharacteristic
159C2×C3×Q8D3211yescharacteristic
160C2×C4×D3—91no—
161C2×C4×D3—91no—
162C2×C4×D3—181no—
163C2×Dic6—31no—
164C3×Q16—182no—
165C2×C3×C8—31no—
166C2×C3×C8—91no—
167C3×C4⋊C4—31no—
168#216—62no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173(C2×D3)⋊C4—182no—
174C4⋊Dic3—91no—
175C4⋊Dic3—91no—
176C4⋊Dic3—181no—
177C4⋊Dic3—181no—
178C4⋊Dic3C2×D911yescharacteristic
179Dic3⋊C4—181no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182Dic3⋊C4—182no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8C2×D911yescharacteristic
185ΓL(1,𝔽25)—182no—
186ΓL(1,𝔽25)—182no—
187C22×C32D3×D411yescharacteristic
188C2×C3⋊D3—61no—
189C2×C3⋊D3—61no—
190C2×C3⋊D3—61no—
191C32×C4—41no—
192C32×C4#23511yescharacteristic
193C32×C4D3×D411yescharacteristic
194C3⋊Dic3—61no—
195C3⋊Dic3—121no—
196C3×Dic3—41no—
197C3×Dic3—121no—
198C22×C9—21no—
199C22×C9D3×D411yescharacteristic
200C2×D9—62no—
201C2×D9—61no—
202C2×D9—61no—
203C2×D9—61no—
204C2×D9—121no—
205C2×D9—121no—
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—42no—
209C4×C9—41no—
210C4×C9—121no—
211C4×C9#23711yescharacteristic
212C4×C9C2×#6711yescharacteristic
213Dic9—41no—
214Dic9—61no—
215Dic9—121no—
216Dic9—122no—
217Dic9—121no—
218C2×Q16—91no—
219C2×M16—271no—
220C22⋊Q8—271no—
221C22⋊Q8—271no—
222#106—271no—
223#103—31no—
224#103—271no—
225C3×C9#47011yescharacteristic, complemented, Sylow
226C22×D3—91no—
227C22×D3—91no—
228C22×D3—181no—
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×Q8—181no—
236C3×Q8—181no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—61no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—61no—
247C2×Dic3—91no—
248C2×Dic3—91no—
249C2×Dic3—181no—
250C2×Dic3—182no—
251C2×Dic3—181no—
252C2×Dic3—181no—
253C4×D3—181no—
254C4×D3—181no—
255C4×D3—181no—
256C4×D3—181no—
257C4×D3—361no—
258C4×D3—361no—
259Dic6—61no—
260Dic6—61no—
261C3×C8—61no—
262C3×C8—92no—
263C3⋊C8—21no—
264C3⋊C8—32no—
265C2×C32#88411yescharacteristic
266C2×C32#91511yescharacteristic
267C2×C32#91811yescharacteristic
268C3⋊D3—121no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9#91811yescharacteristic
273C2×C9#91811yescharacteristic
274C2×C9#93811yescharacteristic
275D9—122no—
276D9—121no—
277C2×Q8—91no—
278C2×Q8D3×D911yescharacteristic
279C22×C4—271no—
280Q16—182no—
281M16—542no—
282C2×C8—31no—
283C2×C8—91no—
284C4⋊C4—31no—
285C4⋊C4—271no—
286C4⋊C4—541no—
287C4⋊C4—541no—
288C22⋊C4—541no—
289C22⋊C4—541no—
290C22×C3—21no—
291C22×C3D4×D911yescharacteristic
292C22×C3#381111yescharacteristic
293C2×D3—182no—
294C2×D3—181no—
295C2×D3—181no—
296C2×D3—181no—
297C2×D3—181no—
298C2×D3—181no—
299C2×D3—181no—
300C2×D3—361no—
301C2×D3—361no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—121no—
308C3×C4—361no—
309C3×C4#369711yescharacteristic
310C3×C4D4×D911yescharacteristic
311C3×C4#380411yescharacteristic
312C3×C4D3×#6711yescharacteristic
313Dic3—41no—
314Dic3—121no—
315Dic3—181no—
316Dic3—181no—
317Dic3—361no—
318Dic3—362no—
319Dic3—361no—
320Dic3—361no—
321C32#481611yescharacteristic
322C9—21no—
323C9#518911yescharacteristic, complemented
324C23—271no—
325Q8—21no—
326Q8—21no—
327Q8—181no—
328Q8—181no—
329C2×C4—21no—
330C2×C4—61no—
331C2×C4—181no—
332C2×C4—271no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4—541no—
336C2×C4C2×D3×D911yescharacteristic
337C8—61no—
338C8—92no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3#6344611yescharacteristic
343C2×C3#6347711yescharacteristic
344C2×C3#6348011yescharacteristic
345C2×C3#6394511yescharacteristic
346C2×C3#6395111yescharacteristic
347C2×C3#6395811yescharacteristic
348D3—362no—
349D3—361no—
350D3—361no—
351C22—541no—
352C22—541no—
353C22—541no—
354C22#8866811yescharacteristic
355C4—41no—
356C4—121no—
357C4—361no—
358C4—541no—
359C4—1081no—
360C4#8865811yescharacteristic
361C4D9×#6711yescharacteristic
362C3—21no—
363C3#1058849611yescharacteristic, complemented
364C3#1059160011yescharacteristic
365C2—1081no—
366C2#1172121711yescharacteristic
367C2#1172122111yescharacteristic
368C2#1172125611yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#470
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
2. C2×C3×#3674C2semidirect
5. C9×#835C2semidirect
6. #11720276C2semidirect
7. #11720264C2semidirect
32. C2×#3674D3semidirect
108. #835D9semidirect
225. C3×C9#470semidirect
323. C9#5189semidirect
363. C3#10588496semidirect

Maximal subgroups

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

subgroupindex
#117218562
C2×C3×#36742
#117209362
C2×#884142
C9×#8352
#117202762
#117202642
#105916003
#105884963