Finite Group Databasebeta

← #13140696

Subgroups — #13140696

Order 1728.

342 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#63509C211yesmaximal, characteristic
2#11720969C211yesmaximal, characteristic, complemented
3#11720582C211yesmaximal, characteristic, complemented
4C9×#835C211yesmaximal, characteristic, complemented
5#11720283C211yesmaximal, characteristic
6#11720274C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic
8#10591624—31nomaximal
9#10588474—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×#6189—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C3×(C2×D9)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20Dic9⋊Dic3—21no—
21C3×#947—31no—
22#63918—31no—
23#63662—31no—
24C3×#835D311yescharacteristic
25#63563—31no—
26#63562—31no—
27#63554—31no—
28#63509D311yescharacteristic
29#63450—31no—
30C9×#103—31no—
31#63395—31no—
32#63395—61no—
33#63380—31no—
34#63375—31no—
35#63369—31no—
36C22×C3×D9—21no—
37C3×Q8×C9—21no—
38C3×Q8×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45C3×D36—21no—
46C3×D36—21no—
47#6189—42no—
48C3⋊(C9⋊C8)—21no—
49C9×C3⋊C8—21no—
50#5055—91no—
51#4794—91no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C3×D12—31no—
54C2×#627—61no—
55C4⋊(C3⋊Dic3)—31no—
56C2×C3⋊(C3⋊C8)—31no—
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63C2×Q8×C9—21no—
64C2×Q8×C9C2×D311yescharacteristic
65C2×D36C2×D311yescharacteristic
66C2×Dic18—61no—
67C2×C8×C9—31no—
68C9×C4⋊C4—31no—
69(C2×D9)⋊C4—21no—
70(C2×D9)⋊C4—21no—
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73Dic9⋊C4—61no—
74C4×Dic9C2×D311yescharacteristic
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C22×C3×C9C2×D411yescharacteristic
78C2×C3×D9—41no—
79C2×C3×D9—41no—
80C2×C3×D9—41no—
81C3×C4×C9—41no—
82C3×C4×C9#4111yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—41no—
85C9×Dic3—41no—
86C3×Dic9—21no—
87C3×Dic9—21no—
88C3×#124—91no—
89#947—31no—
90#888—91no—
91#869—91no—
92C3×#103—31no—
93#835—91no—
94#835—91no—
95#835—181no—
96#835D911yescharacteristic, complemented
97#832—91no—
98#832—91no—
99#821—91no—
100#815—91no—
101#809—91no—
102#803—91no—
103C22×C3×D3—61no—
104C32×Q8—21no—
105C32×Q8—21no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—61no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—31no—
112C3×D12—61no—
113C3×D12—61no—
114#627—122no—
115C22×D9—21no—
116C3⋊(C3⋊C8)—61no—
117C3×C3⋊C8—21no—
118Q8×C9—22no—
119Q8×C9—21no—
120Q8×C9—21no—
121Q8×C9—41no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—41no—
125C2×C4×C9—61no—
126C2×C4×C9C22×D311yescharacteristic
127C2×Dic9—61no—
128C2×Dic9—121no—
129C2×Dic9#6711yescharacteristic
130C2×Dic9#6711yescharacteristic
131D36—21no—
132D36—21no—
133Dic18—122no—
134C8×C9—61no—
135C9⋊C8—61no—
136C9⋊C8—121no—
137#489—271noSylow
138C2×C3×C9#12411yescharacteristic
139C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
140C2×C3×C9#13711yescharacteristic
141C3×D9—81no—
142C2×C3×Q8—21no—
143C2×C3×Q8C2×D911yescharacteristic
144C2×C3×Q8D3211yescharacteristic
145C2×C3×D4—91no—
146C2×D12—31no—
147C2×Dic6—181no—
148C2×Dic6—181no—
149C2×C3×C8—31no—
150C3×C4⋊C4—31no—
151C3×C22⋊C4—181no—
152C3×C22⋊C4—181no—
153C3×C42—91no—
154(C2×D3)⋊C4—61no—
155(C2×D3)⋊C4—61no—
156C4⋊Dic3—91no—
157C4⋊Dic3—91no—
158C4⋊Dic3—181no—
159C4⋊Dic3—181no—
160C4⋊Dic3C2×D911yescharacteristic
161Dic3⋊C4—181no—
162C4×Dic3—31no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—181no—
166C2×C3⋊C8C2×D911yescharacteristic
167C22×C32D3×D411yescharacteristic
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C32×C4—41no—
172C32×C4#23511yescharacteristic
173C32×C4#23711yescharacteristic
174C3⋊Dic3—121no—
175C3×Dic3—41no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C22×C9—21no—
179C22×C9C2×#6711yescharacteristic
180C2×D9—41no—
181C2×D9—41no—
182C2×D9—41no—
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—42no—
186C4×C9—41no—
187C4×C9—121no—
188C4×C9#23711yescharacteristic
189C4×C9#23711yescharacteristic
190Dic9—21no—
191Dic9—21no—
192Dic9—121no—
193Dic9—241no—
194C4⋊Q8—271no—
195#124—91no—
196#103—31no—
197#103—271no—
198#102—271no—
199#102—271no—
200#97—271no—
201C3×C9#48911yescharacteristic, complemented, Sylow
202C23×C3—181no—
203C22×D3—61no—
204C3×Q8—22no—
205C3×Q8—21no—
206C3×Q8—21no—
207C3×Q8—21no—
208C3×Q8—21no—
209C3×Q8—41no—
210C3×D4—181no—
211C3×D4—181no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—41no—
216C2×C3×C4—61no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4C22×D911yescharacteristic
220C2×C3×C4C2×D3211yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226C2×Dic3—361no—
227D12—61no—
228D12—61no—
229Dic6—362no—
230Dic6—362no—
231C3×C8—61no—
232C3⋊C8—21no—
233C3⋊C8—181no—
234C3⋊C8—181no—
235C3⋊C8—361no—
236C2×C32#88511yescharacteristic
237C2×C32#91411yescharacteristic
238C2×C32#91811yescharacteristic
239C3×D3—241no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9#93611yescharacteristic
244C2×C9#94911yescharacteristic
245C2×C9#95111yescharacteristic
246D9—81no—
247C2×Q8—541no—
248C2×Q8D3×D911yescharacteristic
249C2×D4—91no—
250C2×C8—31no—
251C2×C8—271no—
252C4⋊C4—31no—
253C4⋊C4—271no—
254C4⋊C4—541no—
255C22⋊C4—181no—
256C22⋊C4—181no—
257C42—91no—
258C22×C3—21no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3D4×D911yescharacteristic
263C22×C3D3×#6711yescharacteristic
264C2×D3—121no—
265C2×D3—121no—
266C2×D3—121no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—42no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—121no—
273C3×C4—181no—
274C3×C4—181no—
275C3×C4#369711yescharacteristic
276C3×C4#369911yescharacteristic
277C3×C4#380411yescharacteristic
278C3×C4#380511yescharacteristic
279Dic3—41no—
280Dic3—61no—
281Dic3—61no—
282Dic3—361no—
283Dic3—361no—
284Dic3—721no—
285C32#479411yescharacteristic
286C9—21no—
287C9#505511yescharacteristic, complemented
288C23—181no—
289Q8—21no—
290Q8—21no—
291Q8—1082no—
292D4—181no—
293D4—181no—
294C2×C4—21no—
295C2×C4—61no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—541no—
299C2×C4C2×D3×D911yescharacteristic
300C8—61no—
301C8—541no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—721no—
306C2×C3#6344711yescharacteristic
307C2×C3#6347611yescharacteristic
308C2×C3#6348011yescharacteristic
309C2×C3#6393911yescharacteristic
310C2×C3#6394311yescharacteristic
311C2×C3#6395911yescharacteristic
312D3—241no—
313C22—361no—
314C22—361no—
315C22—361no—
316C22D9×#6711yescharacteristic
317C4—41no—
318C4—121no—
319C4—181no—
320C4—181no—
321C4—1081no—
322C4#8865811yescharacteristic
323C4#8866311yescharacteristic
324C3—21no—
325C3#1058847411yescharacteristic, complemented
326C3#1059162411yescharacteristic
327C2—721no—
328C2#1172121111yescharacteristic
329C2#1172121911yescharacteristic
330C2#1172125711yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#489
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. #11720969C2semidirect
3. #11720582C2semidirect
4. C9×#835C2semidirect
6. #11720274C2semidirect
96. #835D9semidirect
201. C3×C9#489semidirect
287. C9#5055semidirect
325. C3#10588474semidirect

Maximal subgroups

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

subgroupindex
C3×#635092
#117209692
#117205822
C9×#8352
#117202832
#117202742
#117202662
#105916243
#105884743