Finite Group Databasebeta

← #13166502

Subgroups — #13166502

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×#89022C211yesmaximal, characteristic
2#11724269C212yesmaximal, complemented
3#11723105C211yesmaximal, characteristic, complemented
4C3×#63780C211yesmaximal, characteristic, complemented
5#11722729C212yesmaximal, complemented
6#10593808—31nomaximal
7C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
8C2×C3×#634C2211yescharacteristic
9C2×C32×C4×D3C411yescharacteristic
10C2×#6291C411yescharacteristic
11C2×#6291C2211yescharacteristic
12C2×#6289C411yescharacteristic
13C2×#6289C2211yescharacteristic
14#89022—22no—
15#89022—22no—
16#88993C2212yes—
17C3×#3792C2212yes—
18#88926—22no—
19#88924—22no—
20C2×#3828—31no—
21C2×#3799—62no—
22#64283—32no—
23#63788—32no—
24#63780—31no—
25#63780D311yescharacteristic, complemented
26C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
27C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
28C22×C32×D3C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C4⋊(C3⋊(C3⋊D3))—22no—
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×#634—22no—
34C2×C32×Dic3C2×C411yescharacteristic
35C32×C4×D3—21no—
36C32×C4×D3—21no—
37#6291—22no—
38#6291—22no—
39#6289—21no—
40#6289—21no—
41#6289D412yes—
42#6289D412yes—
43C3⋊#622—22no—
44C3×#622—22no—
45#4751—91nomaximal
46C22×C32×C4—31no—
47C2×#638—31no—
48C2×#638—31no—
49#3828—62no—
50#3828—62no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4⋊(C3⋊D3)—62no—
53C2×C4⋊(C3⋊D3)—122no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—31no—
56C2×#634C2×D311yescharacteristic
57C2×C3×C4×D3—22no—
58C2×C3×Dic6—62no—
59C2×#626—62no—
60C2×#626—62no—
61C2×#624—62no—
62C2×#624—62no—
63#3799—68no—
64#3799—68no—
65#3792—32no—
66#3792C2×D312yes—
67#3788—62no—
68#3787—62no—
69C22×C33C22×C411yescharacteristic
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—41no—
72C2×C3⋊(C3⋊D3)—41no—
73C2×C3⋊(C3⋊D3)C22⋊C412yes—
74C2×C32×D3—21no—
75C2×C32×D3—41no—
76C33×C4—21no—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3C2×D412yes—
79C32×Dic3C22⋊C411yescharacteristic
80C32×Dic3C22⋊C411yescharacteristic
81C2×#239—91no—
82C2×#235—182no—
83D3×M16—92no—
84C3×#101—91no—
85#836—91no—
86#823—92no—
87C23×C32—31no—
88C22×C3⋊D3—31no—
89C22×C3⋊D3—31no—
90C22×C3⋊D3—62no—
91C22×C3⋊D3—62no—
92C22×C3⋊D3—122no—
93C22×C3⋊D3—122no—
94C22×C3×D3—22no—
95C2×C32×C4—22no—
96C2×C32×C4—31no—
97C2×C32×C4—42no—
98C2×C32×C4—61no—
99C2×C32×C4—61no—
100C2×C32×C4C4×D311yescharacteristic
101#638—62no—
102#638—62no—
103C2×C3⋊Dic3C4×D311yescharacteristic
104C2×C3⋊Dic3C22×D311yescharacteristic
105C4⋊(C3⋊D3)—62no—
106C4⋊(C3⋊D3)—68no—
107C4⋊(C3⋊D3)—128no—
108C4×C3⋊D3—32no—
109C4×C3⋊D3—32no—
110C4×C3⋊D3—61no—
111C4×C3⋊D3—61no—
112#634—22no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C3×C4×D3—24no—
117C3×C4×D3—24no—
118C3×Dic6—68no—
119#626—68no—
120#626—68no—
121#624—64no—
122#624—64no—
123#624—64no—
124#624—64no—
125#622—22no—
126#622—62no—
127#413—271noSylow
128C2×C33—21no—
129C2×C33C2×C22⋊C411yescharacteristic
130C3⋊(C3⋊D3)—22no—
131C3⋊(C3⋊D3)—41no—
132C32×D3—41no—
133C2×C3×Q8—91no—
134C22×C3×C4—62no—
135C2×#67—182no—
136C2×#67—182no—
137#239—182no—
138#239—182no—
139#235—188no—
140#235—188no—
141C2×D12—91no—
142C2×D12—91no—
143C2×D12—91no—
144C2×D12—182no—
145C2×D12—362no—
146C2×C4×D3—91no—
147C2×C4×D3—91no—
148C2×C4×D3—182no—
149C2×C4×D3—182no—
150C2×C4×D3#15711yescharacteristic
151C2×Dic6—62no—
152C3×M16—92no—
153#208—92no—
154ΓL(1,𝔽25)—182no—
155D3×C8—182no—
156C22×C32—22no—
157C22×C32—42no—
158C22×C32—61no—
159C22×C32—121no—
160C22×C32C2×C4×D311yescharacteristic
161C2×C3⋊D3—32no—
162C2×C3⋊D3—64no—
163C2×C3⋊D3—61no—
164C2×C3⋊D3—64no—
165C2×C3⋊D3—124no—
166C2×C3⋊D3—124no—
167C2×C3⋊D3—121no—
168C2×C3⋊D3—121no—
169C2×C3⋊D3—124no—
170C2×C3⋊D3—124no—
171C2×C3⋊D3—244no—
172C2×C3⋊D3—244no—
173C2×C3×D3—24no—
174C2×C3×D3—44no—
175C32×C4—21no—
176C32×C4—24no—
177C32×C4—31no—
178C32×C4—31no—
179C32×C4—44no—
180C3⋊Dic3—21no—
181C3⋊Dic3D3×D412yes—
182C3×Dic3—22no—
183C3×Dic3—22no—
184C3×Dic3—64no—
185C3×Dic3—64no—
186C2×#41—271no—
187C2×M16—272no—
188#101—91no—
189#101—271no—
190#100—272no—
191C33#41311yescharacteristic, complemented, Sylow
192C23×C3—62no—
193C22×D3—91no—
194C22×D3—91no—
195C22×D3—182no—
196C22×D3—182no—
197C22×D3—362no—
198C22×D3—362no—
199C22×D3C2×#15711yescharacteristic
200C3×Q8—182no—
201C2×C3×C4—22no—
202C2×C3×C4—42no—
203C2×C3×C4—62no—
204C2×C3×C4—64no—
205C2×C3×C4—64no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4C2×#15711yescharacteristic
209#67—188no—
210#67—188no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3C2×#15711yescharacteristic
214D12—182no—
215D12—188no—
216D12—182no—
217D12—182no—
218D12—368no—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—92no—
222C4×D3—92no—
223C4×D3—184no—
224C4×D3—184no—
225C4×D3—184no—
226C4×D3—184no—
227C4×D3—181no—
228C4×D3—181no—
229Dic6—68no—
230C3×C8—182no—
231C3⋊C8—182no—
232C2×C32—21no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—44no—
236C2×C32—42no—
237C2×C32—121no—
238C2×C32D3×C22⋊C411yescharacteristic
239C3⋊D3—62no—
240C3⋊D3—124no—
241C3⋊D3—121no—
242C3⋊D3—124no—
243C3⋊D3—244no—
244C3⋊D3—244no—
245C3×D3—44no—
246#41—542no—
247#41—542no—
248C2×Q8—91no—
249C2×D4—271no—
250C2×D4—271no—
251C2×D4—271no—
252C22×C4—31no—
253C22×C4—271no—
254C22×C4—271no—
255M16—92no—
256M16—272no—
257M16—542no—
258C2×C8—542no—
259C22×C3—22no—
260C22×C3—42no—
261C22×C3—64no—
262C22×C3—124no—
263C22×C3C22×#15711yescharacteristic
264C2×D3—21no—
265C2×D3—41no—
266C2×D3—92no—
267C2×D3—184no—
268C2×D3—184no—
269C2×D3—181no—
270C2×D3—364no—
271C2×D3—364no—
272C2×D3—361no—
273C2×D3—364no—
274C2×D3—364no—
275C2×D3—361no—
276C2×D3—724no—
277C2×D3—724no—
278C3×C4—24no—
279C3×C4—21no—
280C3×C4—44no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4—92no—
284C3×C4—181no—
285Dic3—64no—
286Dic3—64no—
287Dic3C22⋊#15711yescharacteristic
288Dic3C22⋊#15711yescharacteristic
289C32—22no—
290C32—42no—
291C32#475111yescharacteristic, complemented
292C23—31no—
293C23—271no—
294C23—271no—
295Q8—182no—
296D4—542no—
297D4—542no—
298D4—542no—
299C2×C4—31no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—272no—
305C2×C4—272no—
306C2×C4—541no—
307C2×C4—541no—
308C2×C4D3×#15711yescharacteristic
309C8—182no—
310C8—542no—
311C2×C3—24no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—44no—
315C2×C3—42no—
316C2×C3—124no—
317C2×C3C2×C22⋊#15711yescharacteristic
318D3—41no—
319D3—182no—
320D3—364no—
321D3—364no—
322D3—361no—
323D3—724no—
324D3—724no—
325C22—61no—
326C22—121no—
327C22—272no—
328C22—541no—
329C22—1081no—
330C22—1081no—
331C22C2×D3×#15711yescharacteristic
332C4—21no—
333C4—31no—
334C4—31no—
335C4—92no—
336C4—181no—
337C3—22no—
338C3—42no—
339C3#1059380811yescharacteristic, complemented
340C2—21no—
341C2—121no—
342C2—542no—
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×#89022C2
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. #11724269C2semidirect
3. #11723105C2semidirect
4. C3×#63780C2semidirect
5. #11722729C2semidirect
25. #63780D3semidirect
191. C33#413semidirect
291. C32#4751semidirect
339. C3#10593808semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
#117242692
#117231052
C3×#637802
#117227292
#105938083
#47519