Finite Group Databasebeta

← #13166501

Subgroups — #13166501

Order 1728.

736 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 64 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×#89015C211yesmaximal, characteristic
2D3×#3792C212yesmaximal, complemented
3#11723105C211yesmaximal, characteristic, complemented
4C3×#63780C211yesmaximal, characteristic, complemented
5#11722729C212yesmaximal, complemented
6#10593824—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
8C2×C3×#634C2211yescharacteristic
9C2×C32×D12C411yescharacteristic
10C2×#6291C411yescharacteristic
11C2×#6291C2211yescharacteristic
12C2×D3×C3⋊Dic3C411yescharacteristic
13C2×D3×C3⋊Dic3C2211yescharacteristic
14#89015—22no—
15#89015—22no—
16#88993C2212yes—
17C3×#3792C2212yes—
18#88925—22no—
19D3×#622—22no—
20C2×#3830—31no—
21C2×#3796—62no—
22C2×#3792—32no—
23#63788—32no—
24#63780—31no—
25#63780D311yescharacteristic, complemented
26C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
27C22×C32×D3C2×C411yescharacteristic
28C22×C32×D3C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
31C4×C3⋊(C3⋊D3)—21no—
32C4×C3⋊(C3⋊D3)—21no—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×#634—22no—
36C32×D12—22no—
37#6291—22no—
38#6291—22no—
39D3×C3⋊Dic3—21no—
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3D412yes—
42D3×C3⋊Dic3D412yes—
43C3⋊#622—22no—
44C3×#622—22no—
45#4751—91nomaximal
46C2×C32×D4—31no—
47C2×#638—31no—
48C2×#638—31no—
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—31no—
51#3830—62no—
52#3830—62no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—62no—
55C2×C4×C3⋊D3—122no—
56C2×#634C2×D311yescharacteristic
57C2×C3×D12—22no—
58C2×C3×Dic6—62no—
59C2×#626—62no—
60C2×#626—62no—
61C2×D3×Dic3—62no—
62C2×D3×Dic3—62no—
63#3796—68no—
64#3796—68no—
65#3792—32no—
66#3792—62no—
67#3792C2×D312yes—
68C2×#622—62no—
69C22×C33C22×C411yescharacteristic
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—41no—
72C2×C32×D3—21no—
73C2×C32×D3—41no—
74C2×C32×D3—41no—
75C2×C32×D3C22⋊C412yes—
76C33×C4—21no—
77C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
78C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3C2×D412yes—
81C2×#239—91no—
82C2×#237—182no—
83D3×M16—92no—
84C3×#101—91no—
85#836—91no—
86#823—92no—
87C23×C32—31no—
88C23×C32—31no—
89C22×C3⋊D3—31no—
90C22×C3⋊D3—62no—
91C22×C3⋊D3—122no—
92C22×C3×D3—22no—
93C22×C3×D3—22no—
94C32×D4—62no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4C4×D311yescharacteristic
98#638—62no—
99#638—62no—
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—32no—
102C2×C3⋊Dic3—32no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—122no—
107C2×C3⋊Dic3C4×D311yescharacteristic
108C2×C3⋊Dic3C22×D311yescharacteristic
109C4×C3⋊D3—61no—
110C4×C3⋊D3—61no—
111C4×C3⋊D3—64no—
112C4×C3⋊D3—64no—
113C4×C3⋊D3—124no—
114C4×C3⋊D3—124no—
115#634—22no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C3×D12—28no—
119C3×Dic6—68no—
120#626—68no—
121#626—68no—
122D3×Dic3—64no—
123D3×Dic3—64no—
124D3×Dic3—64no—
125D3×Dic3—64no—
126#622—22no—
127#622—62no—
128#413—271noSylow
129C2×C33—21no—
130C2×C33C2×C22⋊C411yescharacteristic
131C3⋊(C3⋊D3)—41no—
132C32×D3—22no—
133C32×D3—41no—
134C2×C3×Q8—91no—
135C2×C3×D4—62no—
136C2×#67—182no—
137C2×#67—182no—
138C22×Dic3—182no—
139C22×Dic3—182no—
140#239—182no—
141#239—182no—
142#237—188no—
143#237—188no—
144C2×D12—91no—
145C2×D12—91no—
146C2×D12#15711yescharacteristic
147C2×C4×D3—91no—
148C2×C4×D3—91no—
149C2×C4×D3—91no—
150C2×C4×D3—182no—
151C2×C4×D3—362no—
152C2×Dic6—62no—
153C3×M16—92no—
154#208—92no—
155ΓL(1,𝔽25)—182no—
156D3×C8—182no—
157C22×C32—22no—
158C22×C32—32no—
159C22×C32—42no—
160C22×C32—61no—
161C22×C32—121no—
162C22×C32—121no—
163C22×C32C2×C4×D311yescharacteristic
164C2×C3⋊D3—61no—
165C2×C3⋊D3—64no—
166C2×C3⋊D3—124no—
167C2×C3⋊D3—124no—
168C2×C3⋊D3—128no—
169C2×C3⋊D3—121no—
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C2×C3×D3—44no—
173C2×C3×D3—44no—
174C32×C4—21no—
175C32×C4—24no—
176C32×C4—44no—
177C3⋊Dic3—21no—
178C3⋊Dic3—31no—
179C3⋊Dic3—31no—
180C3⋊Dic3—62no—
181C3⋊Dic3—62no—
182C3⋊Dic3—122no—
183C3⋊Dic3—122no—
184C3⋊Dic3D3×D412yes—
185C3×Dic3—64no—
186C3×Dic3—64no—
187C2×#41—271no—
188C2×M16—272no—
189#101—91no—
190#101—271no—
191#100—272no—
192C33#41311yescharacteristic, complemented, Sylow
193C23×C3—62no—
194C23×C3—62no—
195C22×D3—91no—
196C22×D3—182no—
197C22×D3—362no—
198C22×D3C2×#15711yescharacteristic
199C22×D3C2×#15711yescharacteristic
200C3×Q8—182no—
201C3×D4—68no—
202C2×C3×C4—22no—
203C2×C3×C4—42no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4C2×#15711yescharacteristic
207#67—188no—
208#67—188no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—91no—
212C2×Dic3—184no—
213C2×Dic3—184no—
214C2×Dic3—182no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217C2×Dic3—362no—
218D12—22no—
219D12—182no—
220D12—182no—
221C4×D3—92no—
222C4×D3—92no—
223C4×D3—181no—
224C4×D3—181no—
225C4×D3—181no—
226C4×D3—181no—
227C4×D3—184no—
228C4×D3—184no—
229C4×D3—364no—
230C4×D3—364no—
231Dic6—68no—
232C3×C8—182no—
233C3⋊C8—182no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—22no—
237C2×C32—44no—
238C2×C32—42no—
239C2×C32—62no—
240C2×C32—121no—
241C2×C32D3×C22⋊C411yescharacteristic
242C3⋊D3—124no—
243C3⋊D3—128no—
244C3⋊D3—121no—
245C3×D3—44no—
246C3×D3—44no—
247#41—542no—
248#41—542no—
249C2×Q8—91no—
250C2×D4—31no—
251C2×D4—271no—
252C2×D4—271no—
253C22×C4—271no—
254C22×C4—271no—
255C22×C4—271no—
256M16—92no—
257M16—272no—
258M16—542no—
259C2×C8—542no—
260C22×C3—22no—
261C22×C3—42no—
262C22×C3—64no—
263C22×C3—64no—
264C22×C3—124no—
265C22×C3—124no—
266C22×C3C22×#15711yescharacteristic
267C2×D3—21no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—184no—
271C2×D3—181no—
272C2×D3—364no—
273C2×D3—364no—
274C2×D3—368no—
275C2×D3—361no—
276C2×D3C22⋊#15712yes—
277C3×C4—24no—
278C3×C4—21no—
279C3×C4—44no—
280C3×C4—92no—
281C3×C4—181no—
282Dic3—64no—
283Dic3—64no—
284Dic3—91no—
285Dic3—91no—
286Dic3—182no—
287Dic3—182no—
288Dic3—362no—
289Dic3—362no—
290C32—22no—
291C32—42no—
292C32#475111yescharacteristic, complemented
293C23—31no—
294C23—31no—
295C23—271no—
296Q8—182no—
297D4—62no—
298D4—542no—
299D4—542no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—272no—
303C2×C4—272no—
304C2×C4—271no—
305C2×C4—541no—
306C2×C4—541no—
307C2×C4—541no—
308C2×C4—541no—
309C2×C4D3×#15711yescharacteristic
310C8—182no—
311C8—542no—
312C2×C3—24no—
313C2×C3—21no—
314C2×C3—22no—
315C2×C3—44no—
316C2×C3—42no—
317C2×C3—124no—
318C2×C3—124no—
319C2×C3C2×C22⋊#15711yescharacteristic
320D3—22no—
321D3—41no—
322D3—364no—
323D3—368no—
324D3—361no—
325C22—32no—
326C22—61no—
327C22—121no—
328C22—121no—
329C22—541no—
330C22—1081no—
331C22C2×D3×#15711yescharacteristic
332C4—21no—
333C4—92no—
334C4—181no—
335C4—271no—
336C4—271no—
337C3—22no—
338C3—42no—
339C3#1059382411yescharacteristic, complemented
340C2—21no—
341C2—62no—
342C2—121no—
343C2—1081no—
344C2D3×C22⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C22⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89015C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. D3×#3792C2semidirect
3. #11723105C2semidirect
4. C3×#63780C2semidirect
5. #11722729C2semidirect
25. #63780D3semidirect
192. C33#413semidirect
292. C32#4751semidirect
339. C3#10593824semidirect

Maximal subgroups

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

subgroupindex
C2×#890152
D3×#37922
#117231052
C3×#637802
#117227292
#105938243
#47519