Finite Group Databasebeta

← #13137802

Subgroups — #13137802

Order 1728.

364 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721748C211yesmaximal, characteristic
2#11720929C211yesmaximal, characteristic
3C9×#810C211yesmaximal, characteristic, complemented
4C3×#63367C211yesmaximal, characteristic, complemented
5#11720298C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic
7#11720279C211yesmaximal, characteristic, complemented
8#10590527—31nomaximal
9#10588494—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13#88536—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C4×C3⋊Dic9—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic9⋊Dic3—21no—
21#64101—31no—
22#63871—31no—
23C3×#810D311yescharacteristic
24C3×#808—31no—
25#63575—31no—
26#63566—31no—
27#63557—31no—
28#63508—31no—
29#63454—31no—
30#63454—61no—
31C9×#103—31no—
32#63395—31no—
33(C2×D9)⋊C8—31no—
34#63378—31no—
35#63367D311yescharacteristic, complemented
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C4×C3⋊D9—21no—
42C4×C3⋊D9—21no—
43C2×C9×Dic3—21no—
44C2×C3×Dic9—21no—
45C9×Dic6—21no—
46C9×Dic6—21no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4868—91no—
50#4814—91no—
51C2×C4×C3⋊D3—31no—
52C2×C3×Dic6C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54#3752—61no—
55C4⋊(C3⋊Dic3)—31no—
56C4×C3⋊Dic3—61no—
57C3×C4⋊Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60Dic3⋊Dic3—61no—
61#3722—61no—
62C2×Q8×C9—31no—
63C2×C4×D9—31no—
64C2×C4×D9—61no—
65C2×C8×C9—31no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C2×D311yescharacteristic
68(C2×D9)⋊C4—61no—
69(C2×D9)⋊C4—61no—
70(C2×D9)⋊C4—62no—
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—61no—
75C4×Dic9—61no—
76C4×Dic9—62no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—21no—
80C2×C3⋊D9—21no—
81C2×C3⋊D9—21no—
82C3×C4×C9—41no—
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—41no—
87C9×Dic3—41no—
88C3×Dic9—41no—
89#946—91no—
90#897—91no—
91#892—91no—
92#892—91no—
93#892—181no—
94C3×#106—91no—
95C3×#103—31no—
96#835—91no—
97(C2×D3)⋊C8—91no—
98(C2×D3)⋊C8—91no—
99#818—91no—
100#816—91no—
101#810D911yescharacteristic, complemented
102#808—31no—
103#807—91no—
104C22×C3⋊D3—31no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109C4×C3⋊D3—61no—
110C4×C3⋊D3—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—61no—
113C3×Dic6—21no—
114C3×Dic6—21no—
115C22×D9—31no—
116C22×D9—61no—
117C3×C3⋊C8—21no—
118C3×C3⋊C8—61no—
119Q8×C9—61no—
120Q8×C9—61no—
121C2×C4×C9—22no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9—31no—
128C2×Dic9—61no—
129C2×Dic9—62no—
130C2×Dic9—61no—
131C4×D9—62no—
132C4×D9—62no—
133C4×D9—61no—
134C4×D9—61no—
135C8×C9—61no—
136C9⋊C8—21no—
137#485—271noSylow
138C2×C3×C9#12311yescharacteristic
139C2×C3×C9#13511yescharacteristic
140C2×C3×C9#13711yescharacteristic
141C3⋊D9—41no—
142C2×C3×Q8—31no—
143C2×C4×D3—91no—
144C2×C4×D3—91no—
145C2×C4×D3—181no—
146C2×Dic6C2×D911yescharacteristic
147C2×C3×C8—31no—
148C2×C3×C8—91no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4C2×D911yescharacteristic
152C3×C4⋊C4D3211yescharacteristic
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—181no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—182no—
158C4⋊Dic3—31no—
159C4⋊Dic3—91no—
160C4⋊Dic3—91no—
161C4⋊Dic3—181no—
162C4⋊Dic3—181no—
163Dic3⋊C4—181no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166C4×Dic3—182no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8C2×D911yescharacteristic
169C22×C32D3×D411yescharacteristic
170C2×C3⋊D3—61no—
171C2×C3⋊D3—61no—
172C2×C3⋊D3—61no—
173C32×C4—41no—
174C32×C4#23511yescharacteristic
175C32×C4#23711yescharacteristic
176C3⋊Dic3—61no—
177C3⋊Dic3—121no—
178C3×Dic3—41no—
179C3×Dic3—121no—
180C22×C9—21no—
181C22×C9D3×D411yescharacteristic
182C2×D9—62no—
183C2×D9—61no—
184C2×D9—61no—
185C2×D9—61no—
186C2×D9—121no—
187C2×D9—121no—
188C4×C9—21no—
189C4×C9—21no—
190C4×C9—42no—
191C4×C9—41no—
192C4×C9—121no—
193C4×C9#23511yescharacteristic
194C4×C9#23911yescharacteristic
195Dic9—41no—
196Dic9—62no—
197Dic9—61no—
198Dic9—122no—
199Dic9—121no—
200C22⋊Q8—271no—
201#117—271no—
202#107—271no—
203#106—91no—
204#103—31no—
205#103—271no—
206C22⋊C8—271no—
207C3×C9#48511yescharacteristic, complemented, Sylow
208C22×D3—91no—
209C22×D3—91no—
210C22×D3—181no—
211C3×Q8—61no—
212C3×Q8—61no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—61no—
218C2×C3×C4—181no—
219C2×C3×C4C22×D911yescharacteristic
220C2×C3×C4C2×D3211yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—61no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—181no—
226C2×Dic3—182no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C4×D3—182no—
230C4×D3—182no—
231C4×D3—181no—
232C4×D3—181no—
233C4×D3—181no—
234C4×D3—181no—
235Dic6—21no—
236Dic6—21no—
237C3×C8—61no—
238C3×C8—181no—
239C3⋊C8—21no—
240C3⋊C8—61no—
241C2×C32#88311yescharacteristic
242C2×C32#91511yescharacteristic
243C2×C32#91911yescharacteristic
244C3⋊D3—121no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9#89411yescharacteristic
249C2×C9#91511yescharacteristic
250C2×C9#91611yescharacteristic
251D9—122no—
252D9—121no—
253C2×Q8—31no—
254C22×C4—271no—
255C2×C8—31no—
256C2×C8—91no—
257C4⋊C4—91no—
258C4⋊C4—271no—
259C4⋊C4—541no—
260C4⋊C4D3×D911yescharacteristic
261C22⋊C4—541no—
262C22⋊C4—541no—
263C42—541no—
264C22×C3—21no—
265C22×C3D4×D911yescharacteristic
266C22×C3#381111yescharacteristic
267C2×D3—182no—
268C2×D3—181no—
269C2×D3—181no—
270C2×D3—181no—
271C2×D3—181no—
272C2×D3—181no—
273C2×D3—181no—
274C2×D3—361no—
275C2×D3—361no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—42no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—121no—
282C3×C4—361no—
283C3×C4#369711yescharacteristic
284C3×C4#369911yescharacteristic
285C3×C4#379611yescharacteristic
286C3×C4#379811yescharacteristic
287Dic3—41no—
288Dic3—121no—
289Dic3—182no—
290Dic3—181no—
291Dic3—181no—
292Dic3—362no—
293Dic3—361no—
294Dic3—361no—
295C32#481411yescharacteristic
296C9—21no—
297C9#486811yescharacteristic, complemented
298C23—271no—
299Q8—61no—
300Q8—61no—
301C2×C4—21no—
302C2×C4—61no—
303C2×C4—181no—
304C2×C4—271no—
305C2×C4—541no—
306C2×C4—541no—
307C2×C4—541no—
308C2×C4C2×D3×D911yescharacteristic
309C8—61no—
310C8—181no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3#6344511yescharacteristic
315C2×C3#6347711yescharacteristic
316C2×C3#6348111yescharacteristic
317C2×C3#6385311yescharacteristic
318C2×C3#6393111yescharacteristic
319C2×C3#6395011yescharacteristic
320D3—362no—
321D3—361no—
322D3—361no—
323C22—541no—
324C22—541no—
325C22—541no—
326C22#8866811yescharacteristic
327C4—41no—
328C4—121no—
329C4—361no—
330C4—541no—
331C4—1081no—
332C4#8863911yescharacteristic
333C4#8864011yescharacteristic
334C3—21no—
335C3#1058849411yescharacteristic, complemented
336C3#1059052711yescharacteristic
337C2—1081no—
338C2#1172095711yescharacteristic
339C2#1172118311yescharacteristic
340C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#485
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C9×#810C2semidirect
4. C3×#63367C2semidirect
5. #11720298C2semidirect
7. #11720279C2semidirect
35. #63367D3semidirect
101. #810D9semidirect
207. C3×C9#485semidirect
297. C9#4868semidirect
335. C3#10588494semidirect

Maximal subgroups

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

subgroupindex
#117217482
#117209292
C9×#8102
C3×#633672
#117202982
#117202822
#117202792
#105905273
#105884943