Finite Group Databasebeta

← #13140387

Subgroups — #13140387

Order 1728.

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

normal subgroups only

This group has 794 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×D4×C3⋊D9C211yesmaximal, characteristic
2#11720863C212yesmaximal, complemented
3C9×#833C211yesmaximal, characteristic, complemented
4C3×#63391C211yesmaximal, characteristic, complemented
5#11720288C212yesmaximal, complemented
6#10591503—31nomaximal
7D9×#100—31nomaximal
8C23×C3⋊D9C412yes—
9C2×C3×D4×C9C2211yescharacteristic
10C2×#6233C412yes—
11D4×C3⋊D9—22no—
12D4×C3⋊D9—22no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×C4×C3⋊D9C2211yescharacteristic
15C9×#208C2212yescomplemented
16C3×#3667C2212yescomplemented
17#88414—22no—
18(C3⋊D9)⋊C8—22no—
19C2×D4×C3⋊D3—31no—
20#63818—32no—
21C2×D4×D9—31no—
22C2×D4×D9—61no—
23C3×#833—31no—
24C3×#833D311yescharacteristic
25#63559—32no—
26M16×D9—32no—
27C9×#100—31no—
28#63391D311yescharacteristic, complemented
29#63383—32no—
30C23×C3×C9C2×C412yes—
31C22×C3⋊D9—22no—
32C22×C3⋊D9—22no—
33C22×C3⋊D9—44no—
34C22×C3⋊D9C2×C411yescharacteristic
35C22×C3⋊D9C2×C412yes—
36C3×D4×C9—22no—
37C2×C3×C4×C9C2311yescharacteristic
38#6233—24no—
39C2×C3⋊Dic9C2×C411yescharacteristic
40C4⋊(C3⋊D9)—22no—
41C4×C3⋊D9D412yes—
42C4×C3⋊D9D412yes—
43C9×C3⋊C8—22no—
44C3×C9⋊C8—22no—
45D3×#100—91no—
46D3×#100—91no—
47C23×C3⋊D3—32no—
48C2×C32×D4C2×D311yescharacteristic
49C2×#638—32no—
50D4×C3⋊D3—62no—
51D4×C3⋊D3—62no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4×C3⋊D3—31no—
54C23×D9—32no—
55C23×D9—62no—
56C3×#208—32no—
57C3×#208C2×D312yes—
58#3712—62no—
59#3710—62no—
60C2×D4×C9—21no—
61C2×D4×C9C2×D311yescharacteristic
62C2×#611—32no—
63C2×#611—62no—
64D4×D9—62no—
65D4×D9—62no—
66D4×D9—64no—
67D4×D9—64no—
68C2×D36—31no—
69C2×D36—61no—
70C2×C4×D9—31no—
71C2×C4×D9—61no—
72C9×M16—32no—
73#3667C2×D312yescomplemented
74#3663—62no—
75C8×D9—62no—
76C22×C3×C9—22no—
77C22×C3×C9—42no—
78C22×C3×C9C22×C411yescharacteristic
79C2×C3⋊D9—22no—
80C2×C3⋊D9—22no—
81C2×C3⋊D9—44no—
82C2×C3⋊D9—44no—
83C2×C3⋊D9—42no—
84C2×C3⋊D9C22⋊C411yescharacteristic
85C2×C3⋊D9C22⋊C411yescharacteristic
86C3×C4×C9C2×D412yes—
87C3⋊Dic9C22⋊C412yes—
88C2×D3×D4—91no—
89C2×D3×D4—91no—
90C2×D3×D4—181no—
91D3×M16—92no—
92D3×M16—92no—
93C3×#100—31no—
94C3×#100—91no—
95#833—31no—
96#833D911yescharacteristic, complemented
97#823—92no—
98#823—92no—
99C23×C32C4×D312yes—
100C22×C3⋊D3—31no—
101C22×C3⋊D3—32no—
102C22×C3⋊D3—62no—
103C22×C3⋊D3—62no—
104C22×C3⋊D3—124no—
105C32×D4—22no—
106C2×C32×C4C22×D311yescharacteristic
107#638—64no—
108C2×C3⋊Dic3—31no—
109C4⋊(C3⋊D3)—62no—
110C4×C3⋊D3—32no—
111C4×C3⋊D3—32no—
112C23×C9—22no—
113C23×C9C4×D312yes—
114C22×D9—31no—
115C22×D9—32no—
116C22×D9—61no—
117C22×D9—62no—
118C22×D9—62no—
119C22×D9—62no—
120C22×D9—64no—
121C22×D9—64no—
122C22×D9—128no—
123C22×D9—124no—
124C3×C3⋊C8—22no—
125C3×C3⋊C8—62no—
126D4×C9—24no—
127D4×C9—22no—
128C2×C4×C9—21no—
129C2×C4×C9C22×D311yescharacteristic
130#611—64no—
131#611—68no—
132C2×Dic9—31no—
133C2×Dic9—61no—
134D36—64no—
135D36—62no—
136C4×D9—32no—
137C4×D9—32no—
138C4×D9—62no—
139C4×D9—62no—
140C8×C9—62no—
141C9⋊C8—22no—
142C2×#100—271noSylow
143C2×C3×C9—21no—
144C2×C3×C9—42no—
145C2×C3×C9C2×C22⋊C411yescharacteristic
146C3⋊D9—21no—
147C3⋊D9—42no—
148C3⋊D9#10012yescomplemented
149C23×D3—92no—
150C23×D3—92no—
151C23×D3—182no—
152C2×C3×D4—21no—
153C2×C3×D4C2×D911yescharacteristic
154C2×C3×D4D3211yescharacteristic
155C2×#67—92no—
156C2×#67—92no—
157C2×#67—182no—
158D3×D4—182no—
159D3×D4—182no—
160D3×D4—182no—
161D3×D4—182no—
162D3×D4—184no—
163D3×D4—184no—
164C2×D12—91no—
165C2×D12—91no—
166C2×D12—181no—
167C2×C4×D3—91no—
168C2×C4×D3—91no—
169C2×C4×D3—181no—
170C3×M16—32no—
171C3×M16—92no—
172#208—32no—
173#208C2×D912yescomplemented
174ΓL(1,𝔽25)—182no—
175ΓL(1,𝔽25)—182no—
176D3×C8—182no—
177D3×C8—182no—
178C22×C32—22no—
179C22×C32—42no—
180C22×C32C2×C4×D311yescharacteristic
181C2×C3⋊D3—31no—
182C2×C3⋊D3—31no—
183C2×C3⋊D3—62no—
184C2×C3⋊D3—62no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—124no—
187C2×C3⋊D3—122no—
188C32×C4D3×D412yes—
189C3⋊Dic3—32no—
190C22×C9—24no—
191C22×C9—22no—
192C22×C9—21no—
193C22×C9—42no—
194C22×C9—44no—
195C22×C9C2×C4×D311yescharacteristic
196C2×D9—31no—
197C2×D9—31no—
198C2×D9—61no—
199C2×D9—61no—
200C2×D9—62no—
201C2×D9—64no—
202C2×D9—64no—
203C2×D9—62no—
204C2×D9—124no—
205C2×D9—128no—
206C2×D9—128no—
207C2×D9—124no—
208C2×D9—124no—
209C2×D9—122no—
210C4×C9—22no—
211C4×C9D3×D412yes—
212Dic9—32no—
213Dic9—62no—
214C22×D4—271no—
215C2×M16—272no—
216#100—31no—
217#100—91no—
218#100—272no—
219C3×C9C2×#10011yescharacteristic, complemented, Sylow
220C23×C3—22no—
221C23×C3C4×D912yes—
222C23×C3#62412yes—
223C22×D3—91no—
224C22×D3—91no—
225C22×D3—92no—
226C22×D3—92no—
227C22×D3—181no—
228C22×D3—182no—
229C22×D3—182no—
230C22×D3—182no—
231C22×D3—182no—
232C22×D3—182no—
233C22×D3—184no—
234C22×D3—184no—
235C22×D3—368no—
236C22×D3—364no—
237C22×D3—364no—
238C3×D4—24no—
239C3×D4—22no—
240C3×D4—22no—
241C2×C3×C4—21no—
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4C2×D3211yescharacteristic
244#67—184no—
245#67—184no—
246#67—188no—
247C2×Dic3—91no—
248C2×Dic3—91no—
249C2×Dic3—181no—
250D12—184no—
251D12—182no—
252D12—182no—
253C4×D3—92no—
254C4×D3—92no—
255C4×D3—92no—
256C4×D3—92no—
257C4×D3—182no—
258C4×D3—182no—
259C3×C8—62no—
260C3×C8—182no—
261C3⋊C8—22no—
262C3⋊C8—62no—
263C2×C32—21no—
264C2×C32—42no—
265C2×C32D3×C22⋊C411yescharacteristic
266C3⋊D3—32no—
267C3⋊D3—61no—
268C3⋊D3—122no—
269C2×C9—22no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—44no—
273C2×C9—42no—
274C2×C9D3×C22⋊C411yescharacteristic
275D9—32no—
276D9—62no—
277D9—62no—
278D9—61no—
279D9—124no—
280D9—122no—
281C24—272no—
282C2×D4—272no—
283C2×D4—271no—
284C2×D4—542no—
285C2×D4—542no—
286C2×D4D3×D911yescharacteristic
287C22×C4—271no—
288M16—32no—
289M16—92no—
290M16—542no—
291C2×C8—542no—
292C22×C3—24no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—21no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—44no—
299C22×C3C2×C4×D911yescharacteristic
300C22×C3C2×#62411yescharacteristic
301C2×D3—91no—
302C2×D3—91no—
303C2×D3—91no—
304C2×D3—91no—
305C2×D3—181no—
306C2×D3—181no—
307C2×D3—182no—
308C2×D3—182no—
309C2×D3—184no—
310C2×D3—184no—
311C2×D3—182no—
312C2×D3—182no—
313C2×D3—364no—
314C2×D3—364no—
315C2×D3—364no—
316C2×D3—368no—
317C2×D3—368no—
318C2×D3—364no—
319C2×D3—364no—
320C2×D3—362no—
321C2×D3—362no—
322C3×C4—22no—
323C3×C4D4×D912yes—
324C3×C4#381112yes—
325Dic3—92no—
326Dic3—92no—
327Dic3—182no—
328C32D3×#10011yescharacteristic
329C9—21no—
330C9D3×#10011yescharacteristic, complemented
331C23—271no—
332C23—272no—
333C23—542no—
334C23—542no—
335C23—1084no—
336C23(C3⋊D9)⋊C412yes—
337D4—22no—
338D4—544no—
339D4—542no—
340C2×C4—271no—
341C2×C4—272no—
342C2×C4—272no—
343C2×C4C2×D3×D911yescharacteristic
344C8—62no—
345C8—182no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—44no—
351C2×C3—42no—
352C2×C3—42no—
353C2×C3C22⋊C4×D911yescharacteristic
354C2×C3#6397411yescharacteristic
355D3—92no—
356D3—92no—
357D3—182no—
358D3—182no—
359D3—181no—
360D3—181no—
361D3—364no—
362D3—362no—
363D3—362no—
364C22—22no—
365C22—42no—
366C22—271no—
367C22—271no—
368C22—542no—
369C22—542no—
370C22—1082no—
371C22—1084no—
372C22—1084no—
373C22C2×(C3⋊D9)⋊C411yescharacteristic
374C4—272no—
375C4#8866812yes—
376C3—21no—
377C3D9×#10011yescharacteristic, complemented
378C3#1059150311yescharacteristic
379C2—21no—
380C2—42no—
381C2—272no—
382C2—541no—
383C2—1082no—
384C2#1172127411yescharacteristic
subgroupG/H
centerC2#11721274
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32D3×C22⋊C4
prime radicalC2×D4×C3⋊D9C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#100
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720863C2semidirect
3. C9×#833C2semidirect
4. C3×#63391C2semidirect
5. #11720288C2semidirect
15. C9×#208C22semidirect
16. C3×#3667C22semidirect
28. #63391D3semidirect
73. #3667C2×D3semidirect
96. #833D9semidirect
148. C3⋊D9#100semidirect
173. #208C2×D9semidirect
219. C3×C9C2×#100semidirect
330. C9D3×#100semidirect
377. C3D9×#100semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D92
#117208632
C9×#8332
C3×#633912
#117202882
#105915033
D9×#1003