Finite Group Databasebeta

← #13140761

Subgroups — #13140761

Order 1728.

594 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q8×C3⋊D9C211yesmaximal, characteristic
2#11720863C212yesmaximal
3C9×#836C211yesmaximal, characteristic, complemented
4C3×#63394C211yesmaximal, characteristic, complemented
5#11720290C212yesmaximal, complemented
6#10591639—31nomaximal
7D9×#101—31nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9Q8×C3⋊D9—22no—
10Q8×C3⋊D9—22no—
11C2×C4×C3⋊D9C412yes—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×#6226C412yes—
14C2×#6226C2211yescharacteristic
15C9×#208C2212yes—
16C3×#3667C2212yes—
17#88414—22no—
18(C3⋊D9)⋊C8—22no—
19C2×Q8×C3⋊D3—31no—
20#63818—32no—
21C2×Q8×D9—31no—
22C2×Q8×D9—61no—
23C3×#836—31no—
24C3×#836D311yescharacteristic
25#63561—32no—
26M16×D9—32no—
27C9×#101—31no—
28#63394D311yescharacteristic, complemented
29#63382—32no—
30C22×C3⋊D9C2×C411yescharacteristic
31C3×Q8×C9—22no—
32C2×C3×C4×C9C2×C412yes—
33C2×C3×C4×C9C2311yescharacteristic
34C2×C3⋊Dic9C2×C412yes—
35C2×C3⋊Dic9C2×C411yescharacteristic
36C4×C3⋊D9—22no—
37C4×C3⋊D9—22no—
38C4×C3⋊D9D412yes—
39C4×C3⋊D9D412yes—
40#6226—22no—
41#6226—24no—
42C9×C3⋊C8—22no—
43C3×C9⋊C8—22no—
44D3×#101—91no—
45D3×#101—91no—
46C2×C32×Q8C2×D311yescharacteristic
47Q8×C3⋊D3—62no—
48Q8×C3⋊D3—62no—
49C2×C4×C3⋊D3—31no—
50C2×C4×C3⋊D3—32no—
51C2×#634—32no—
52C2×#634—31no—
53C3×#208—32no—
54C3×#208C2×D312yes—
55#3712—62no—
56#3710—62no—
57C2×Q8×C9—21no—
58C2×Q8×C9C2×D311yescharacteristic
59Q8×D9—62no—
60Q8×D9—62no—
61Q8×D9—64no—
62Q8×D9—64no—
63C2×C4×D9—31no—
64C2×C4×D9—32no—
65C2×C4×D9—62no—
66C2×C4×D9—61no—
67C2×Dic18—32no—
68C2×Dic18—31no—
69C2×Dic18—62no—
70C2×Dic18—61no—
71C9×M16—32no—
72#3667C2×D312yes—
73#3663—62no—
74C8×D9—62no—
75C22×C3×C9C22×C411yescharacteristic
76C2×C3⋊D9—22no—
77C2×C3⋊D9C22⋊C411yescharacteristic
78C2×C3⋊D9C22⋊C411yescharacteristic
79C3×C4×C9—22no—
80C3×C4×C9C2×D412yes—
81C3⋊Dic9—22no—
82C3⋊Dic9C22⋊C412yes—
83C2×D3×Q8—91no—
84C2×D3×Q8—91no—
85C2×D3×Q8—181no—
86D3×M16—92no—
87D3×M16—92no—
88C3×#101—31no—
89C3×#101—91no—
90#836—31no—
91#836D911yescharacteristic, complemented
92#824—92no—
93#824—92no—
94C22×C3⋊D3—31no—
95C32×Q8—22no—
96C2×C32×C4C4×D312yes—
97C2×C32×C4C22×D311yescharacteristic
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—32no—
100C4×C3⋊D3—32no—
101C4×C3⋊D3—32no—
102C4×C3⋊D3—62no—
103C4×C3⋊D3—62no—
104#634—64no—
105#634—62no—
106C22×D9—31no—
107C22×D9—61no—
108C3×C3⋊C8—22no—
109C3×C3⋊C8—62no—
110Q8×C9—24no—
111Q8×C9—22no—
112C2×C4×C9—22no—
113C2×C4×C9—21no—
114C2×C4×C9C4×D312yes—
115C2×C4×C9C22×D311yescharacteristic
116C2×Dic9—31no—
117C2×Dic9—32no—
118C2×Dic9—61no—
119C2×Dic9—62no—
120C4×D9—32no—
121C4×D9—32no—
122C4×D9—62no—
123C4×D9—62no—
124C4×D9—62no—
125C4×D9—62no—
126C4×D9—64no—
127C4×D9—64no—
128Dic18—64no—
129Dic18—68no—
130Dic18—64no—
131Dic18—62no—
132C8×C9—62no—
133C9⋊C8—22no—
134C2×#101—271noSylow
135C2×C3×C9—21no—
136C2×C3×C9C2×C22⋊C411yescharacteristic
137C3⋊D9—21no—
138C3⋊D9#10112yescomplemented
139C2×C3×Q8—21no—
140C2×C3×Q8C2×D911yescharacteristic
141C2×C3×Q8D3211yescharacteristic
142D3×Q8—182no—
143D3×Q8—182no—
144D3×Q8—182no—
145D3×Q8—182no—
146D3×Q8—184no—
147D3×Q8—184no—
148C2×C4×D3—91no—
149C2×C4×D3—91no—
150C2×C4×D3—92no—
151C2×C4×D3—92no—
152C2×C4×D3—181no—
153C2×C4×D3—182no—
154C2×Dic6—92no—
155C2×Dic6—91no—
156C2×Dic6—92no—
157C2×Dic6—91no—
158C2×Dic6—182no—
159C2×Dic6—181no—
160C3×M16—32no—
161C3×M16—92no—
162#208—32no—
163#208C2×D912yes—
164ΓL(1,𝔽25)—182no—
165ΓL(1,𝔽25)—182no—
166D3×C8—182no—
167D3×C8—182no—
168C22×C32C2×C4×D311yescharacteristic
169C2×C3⋊D3—31no—
170C2×C3⋊D3—31no—
171C2×C3⋊D3—62no—
172C32×C4—22no—
173C32×C4D3×D412yes—
174C3⋊Dic3—32no—
175C3⋊Dic3—62no—
176C22×C9—21no—
177C22×C9C2×C4×D311yescharacteristic
178C2×D9—31no—
179C2×D9—31no—
180C2×D9—61no—
181C2×D9—61no—
182C2×D9—62no—
183C2×D9—64no—
184C4×C9—24no—
185C4×C9—22no—
186C4×C9—22no—
187C4×C9D3×D412yes—
188Dic9—32no—
189Dic9—62no—
190Dic9—64no—
191Dic9—62no—
192C22×Q8—271no—
193C2×M16—272no—
194#101—31no—
195#101—91no—
196#101—272no—
197C3×C9C2×#10111yescharacteristic, complemented, Sylow
198C22×D3—91no—
199C22×D3—91no—
200C22×D3—181no—
201C3×Q8—24no—
202C3×Q8—22no—
203C3×Q8—22no—
204C2×C3×C4—22no—
205C2×C3×C4—21no—
206C2×C3×C4C4×D912yes—
207C2×C3×C4C22×D911yescharacteristic
208C2×C3×C4#62412yes—
209C2×C3×C4C2×D3211yescharacteristic
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—92no—
213C2×Dic3—92no—
214C2×Dic3—181no—
215C2×Dic3—182no—
216C4×D3—92no—
217C4×D3—92no—
218C4×D3—92no—
219C4×D3—92no—
220C4×D3—182no—
221C4×D3—182no—
222C4×D3—182no—
223C4×D3—182no—
224C4×D3—182no—
225C4×D3—182no—
226C4×D3—184no—
227C4×D3—184no—
228Dic6—184no—
229Dic6—184no—
230Dic6—188no—
231Dic6—184no—
232Dic6—182no—
233Dic6—182no—
234C3×C8—62no—
235C3×C8—182no—
236C3⋊C8—22no—
237C3⋊C8—62no—
238C2×C32—21no—
239C2×C32D3×C22⋊C411yescharacteristic
240C3⋊D3—32no—
241C3⋊D3—61no—
242C2×C9—22no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9D3×C22⋊C411yescharacteristic
246D9—32no—
247D9—62no—
248D9—62no—
249D9—61no—
250C2×Q8—272no—
251C2×Q8—271no—
252C2×Q8—542no—
253C2×Q8—542no—
254C2×Q8D3×D911yescharacteristic
255C22×C4—271no—
256C22×C4—272no—
257M16—32no—
258M16—92no—
259M16—542no—
260C2×C8—542no—
261C22×C3—21no—
262C22×C3C2×C4×D911yescharacteristic
263C22×C3C2×#62411yescharacteristic
264C2×D3—91no—
265C2×D3—91no—
266C2×D3—91no—
267C2×D3—91no—
268C2×D3—181no—
269C2×D3—181no—
270C2×D3—182no—
271C2×D3—182no—
272C2×D3—184no—
273C3×C4—24no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4D4×D912yes—
278C3×C4#381112yes—
279Dic3—92no—
280Dic3—92no—
281Dic3—182no—
282Dic3—184no—
283Dic3—182no—
284Dic3—182no—
285C32D3×#10111yescharacteristic
286C9—21no—
287C9D3×#10111yescharacteristic, complemented
288C23—271no—
289Q8—22no—
290Q8—544no—
291Q8—542no—
292C2×C4—271no—
293C2×C4—272no—
294C2×C4—272no—
295C2×C4—272no—
296C2×C4—542no—
297C2×C4—542no—
298C2×C4(C3⋊D9)⋊C412yes—
299C2×C4C2×D3×D911yescharacteristic
300C8—62no—
301C8—182no—
302C2×C3—22no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3C22⋊C4×D911yescharacteristic
307C2×C3#6397411yescharacteristic
308D3—92no—
309D3—92no—
310D3—182no—
311D3—182no—
312D3—181no—
313D3—181no—
314C22—271no—
315C22—271no—
316C22—542no—
317C22C2×(C3⋊D9)⋊C411yescharacteristic
318C4—22no—
319C4—272no—
320C4—542no—
321C4#8866812yes—
322C3—21no—
323C3D9×#10111yescharacteristic, complemented
324C3#1059163911yescharacteristic
325C2—21no—
326C2—272no—
327C2—541no—
328C2#1172127411yescharacteristic
subgroupG/H
centerC2#11721274
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32D3×C22⋊C4
prime radicalC22×C3⋊D9C2×C4
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9C2×#101
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C9×#836C2semidirect
4. C3×#63394C2semidirect
5. #11720290C2semidirect
28. #63394D3semidirect
91. #836D9semidirect
138. C3⋊D9#101semidirect
197. C3×C9C2×#101semidirect
287. C9D3×#101semidirect
323. C3D9×#101semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D92
#117208632
C9×#8362
C3×#633942
#117202902
#105916393
D9×#1013