Finite Group Databasebeta

← #13140762

Subgroups — #13140762

Order 1728.

614 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 93 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×#88746C211yesmaximal, characteristic
2#11720863C212yesmaximal, complemented
3C9×#836C211yesmaximal, characteristic, complemented
4C3×#63394C211yesmaximal, characteristic, complemented
5#11720288C212yesmaximal, complemented
6#10591640—31nomaximal
7#10588431—31nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9#88746—22no—
10#88746—22no—
11C2×C4⋊(C3⋊D9)C412yes—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C4×C3⋊D9C412yes—
14C2×C4×C3⋊D9C2211yescharacteristic
15C9×#208C2212yes—
16C3×#3667C2212yes—
17#88414—22no—
18(C3⋊D9)⋊C8—22no—
19C2×#3832—31no—
20#63818—32no—
21C2×#3701—31no—
22C2×#3701—61no—
23C3×#836—31no—
24C3×#836D311yescharacteristic
25#63559—32no—
26M16×D9—32no—
27C9×#101—31no—
28#63394D311yescharacteristic, complemented
29#63383—32no—
30C22×C3⋊D9C2×C412yes—
31C22×C3⋊D9C2×C411yescharacteristic
32C3×Q8×C9—22no—
33C2×C3×C4×C9C2×C412yes—
34C2×C3×C4×C9C2311yescharacteristic
35C2×C3⋊Dic9C2×C411yescharacteristic
36C4⋊(C3⋊D9)—24no—
37C4⋊(C3⋊D9)—22no—
38C4×C3⋊D9—22no—
39C4×C3⋊D9—22no—
40C4×C3⋊D9D412yes—
41C4×C3⋊D9D412yes—
42C9×C3⋊C8—22no—
43C3×C9⋊C8—22no—
44#4751—91no—
45#4751—91no—
46C2×C32×Q8C2×D311yescharacteristic
47#3832—62no—
48#3832—62no—
49C2×C4⋊(C3⋊D3)—32no—
50C2×C4⋊(C3⋊D3)—31no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—32no—
53C3×#208—32no—
54C3×#208C2×D312yes—
55#3712—62no—
56#3710—62no—
57C2×Q8×C9—21no—
58C2×Q8×C9C2×D311yescharacteristic
59#3701—62no—
60#3701—62no—
61#3701—64no—
62#3701—64no—
63C2×D36—32no—
64C2×D36—31no—
65C2×D36—62no—
66C2×D36—61no—
67C2×C4×D9—31no—
68C2×C4×D9—32no—
69C2×C4×D9—62no—
70C2×C4×D9—61no—
71C9×M16—32no—
72#3667C2×D312yes—
73#3663—62no—
74C8×D9—62no—
75C22×C3×C9C22×C411yescharacteristic
76C2×C3⋊D9—22no—
77C2×C3⋊D9—41no—
78C2×C3⋊D9—42no—
79C2×C3⋊D9C22⋊C412yes—
80C3×C4×C9—22no—
81C3×C4×C9C2×D412yes—
82C3⋊Dic9C22⋊C411yescharacteristic
83C3⋊Dic9C22⋊C411yescharacteristic
84C2×#239—91no—
85C2×#239—91no—
86C2×#239—181no—
87D3×M16—92no—
88D3×M16—92no—
89C3×#101—31no—
90C3×#101—91no—
91#836—31no—
92#836D911yescharacteristic, complemented
93#823—92no—
94#823—92no—
95C22×C3⋊D3—31no—
96C22×C3⋊D3—32no—
97C32×Q8—22no—
98C2×C32×C4C4×D312yes—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3—31no—
101C4⋊(C3⋊D3)—64no—
102C4⋊(C3⋊D3)—62no—
103C4×C3⋊D3—32no—
104C4×C3⋊D3—32no—
105C4×C3⋊D3—62no—
106C4×C3⋊D3—62no—
107C22×D9—31no—
108C22×D9—32no—
109C22×D9—61no—
110C22×D9—62no—
111C3×C3⋊C8—22no—
112C3×C3⋊C8—62no—
113Q8×C9—24no—
114Q8×C9—22no—
115C2×C4×C9—22no—
116C2×C4×C9—21no—
117C2×C4×C9C4×D312yes—
118C2×C4×C9C22×D311yescharacteristic
119C2×Dic9—31no—
120C2×Dic9—61no—
121D36—64no—
122D36—68no—
123D36—64no—
124D36—62no—
125C4×D9—32no—
126C4×D9—32no—
127C4×D9—62no—
128C4×D9—62no—
129C4×D9—62no—
130C4×D9—62no—
131C4×D9—64no—
132C4×D9—64no—
133C8×C9—62no—
134C9⋊C8—22no—
135#413—271noSylow
136C2×C3×C9—21no—
137C2×C3×C9C2×C22⋊C411yescharacteristic
138C3⋊D9—22no—
139C3⋊D9—42no—
140C2×C3×Q8—21no—
141C2×C3×Q8C2×D911yescharacteristic
142C2×C3×Q8D3211yescharacteristic
143#239—182no—
144#239—182no—
145#239—182no—
146#239—182no—
147#239—184no—
148#239—184no—
149C2×D12—92no—
150C2×D12—91no—
151C2×D12—92no—
152C2×D12—91no—
153C2×D12—182no—
154C2×D12—181no—
155C2×C4×D3—91no—
156C2×C4×D3—91no—
157C2×C4×D3—92no—
158C2×C4×D3—92no—
159C2×C4×D3—181no—
160C2×C4×D3—182no—
161C3×M16—32no—
162C3×M16—92no—
163#208—32no—
164#208C2×D912yes—
165ΓL(1,𝔽25)—182no—
166ΓL(1,𝔽25)—182no—
167D3×C8—182no—
168D3×C8—182no—
169C22×C32C2×C4×D311yescharacteristic
170C2×C3⋊D3—32no—
171C2×C3⋊D3—62no—
172C2×C3⋊D3—121no—
173C2×C3⋊D3—122no—
174C32×C4—22no—
175C32×C4D3×D412yes—
176C3⋊Dic3—31no—
177C3⋊Dic3—31no—
178C22×C9—21no—
179C22×C9C2×C4×D311yescharacteristic
180C2×D9—32no—
181C2×D9—62no—
182C2×D9—64no—
183C2×D9—62no—
184C2×D9—121no—
185C2×D9—122no—
186C2×D9—124no—
187C2×D9—122no—
188C4×C9—24no—
189C4×C9—22no—
190C4×C9—22no—
191C4×C9D3×D412yes—
192Dic9—31no—
193Dic9—31no—
194Dic9—61no—
195Dic9—61no—
196C2×#41—271no—
197C2×M16—272no—
198#101—31no—
199#101—91no—
200#100—272no—
201C3×C9#41311yescharacteristic, complemented, Sylow
202C22×D3—91no—
203C22×D3—91no—
204C22×D3—92no—
205C22×D3—92no—
206C22×D3—181no—
207C22×D3—182no—
208C3×Q8—24no—
209C3×Q8—22no—
210C3×Q8—22no—
211C2×C3×C4—22no—
212C2×C3×C4—21no—
213C2×C3×C4C4×D912yes—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4#62412yes—
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—91no—
218C2×Dic3—91no—
219C2×Dic3—181no—
220D12—184no—
221D12—184no—
222D12—188no—
223D12—184no—
224D12—182no—
225D12—182no—
226C4×D3—92no—
227C4×D3—92no—
228C4×D3—92no—
229C4×D3—92no—
230C4×D3—182no—
231C4×D3—182no—
232C4×D3—182no—
233C4×D3—182no—
234C4×D3—182no—
235C4×D3—182no—
236C4×D3—184no—
237C4×D3—184no—
238C3×C8—62no—
239C3×C8—182no—
240C3⋊C8—22no—
241C3⋊C8—62no—
242C2×C32—21no—
243C2×C32D3×C22⋊C411yescharacteristic
244C3⋊D3—62no—
245C3⋊D3—122no—
246C2×C9—22no—
247C2×C9—21no—
248C2×C9—21no—
249C2×C9D3×C22⋊C411yescharacteristic
250D9—62no—
251D9—122no—
252D9—124no—
253D9—122no—
254#41—542no—
255#41—542no—
256C2×Q8D3×D911yescharacteristic
257C2×D4—272no—
258C2×D4—271no—
259C22×C4—271no—
260C22×C4—272no—
261M16—32no—
262M16—92no—
263M16—542no—
264C2×C8—542no—
265C22×C3—21no—
266C22×C3C2×C4×D911yescharacteristic
267C22×C3C2×#62411yescharacteristic
268C2×D3—92no—
269C2×D3—92no—
270C2×D3—182no—
271C2×D3—184no—
272C2×D3—182no—
273C2×D3—182no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—362no—
277C2×D3—364no—
278C2×D3—362no—
279C2×D3—362no—
280C3×C4—24no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—22no—
284C3×C4D4×D912yes—
285C3×C4#381112yes—
286Dic3—91no—
287Dic3—91no—
288Dic3—91no—
289Dic3—91no—
290Dic3—181no—
291Dic3—181no—
292C32#475111yescharacteristic
293C9—21no—
294C9#475111yescharacteristic, complemented
295C23—271no—
296C23—272no—
297Q8—22no—
298D4—544no—
299D4—542no—
300C2×C4—271no—
301C2×C4—272no—
302C2×C4—272no—
303C2×C4—542no—
304C2×C4—542no—
305C2×C4(C3⋊D9)⋊C412yes—
306C2×C4C2×D3×D911yescharacteristic
307C8—62no—
308C8—182no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3C22⋊C4×D911yescharacteristic
314C2×C3#6397411yescharacteristic
315D3—182no—
316D3—182no—
317D3—362no—
318D3—364no—
319D3—362no—
320D3—362no—
321C22—272no—
322C22—542no—
323C22—1081no—
324C22—1082no—
325C22C2×(C3⋊D9)⋊C411yescharacteristic
326C4—22no—
327C4—271no—
328C4—271no—
329C4#8866812yes—
330C3—21no—
331C3#1058843111yescharacteristic, complemented
332C3#1059164011yescharacteristic
333C2—21no—
334C2—542no—
335C2—1082no—
336C2#1172127411yescharacteristic
subgroupG/H
centerC2#11721274
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32D3×C22⋊C4
prime radicalC2×#88746C2
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#413
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
2. #11720863C2semidirect
3. C9×#836C2semidirect
4. C3×#63394C2semidirect
5. #11720288C2semidirect
28. #63394D3semidirect
92. #836D9semidirect
201. C3×C9#413semidirect
294. C9#4751semidirect
331. C3#10588431semidirect

Maximal subgroups

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

subgroupindex
C2×#887462
#117208632
C9×#8362
C3×#633942
#117202882
#105916403
#105884313