Finite Group Databasebeta

← #13166488

Subgroups — #13166488

Order 1728.

544 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 66 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×#88926C211yesmaximal, characteristic, complemented
2C2×#88924C211yesmaximal, characteristic, complemented
3#11723494C211yesmaximal, characteristic
4#11723102C211yesmaximal, characteristic, complemented
5C3×#63779C211yesmaximal, characteristic, complemented
6#11722726C211yesmaximal, characteristic, complemented
7#11722726C211yesmaximal, characteristic, complemented
8#10593751—31nomaximal
9C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
10C2×#6289C2211yescharacteristic
11C2×#6289C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16#88926—22no—
17#88924—22no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C411yescharacteristic
20#88804C411yescharacteristic
21#88804C411yescharacteristic
22C2×#3788—31no—
23C2×#3787—31no—
24#64102—31no—
25#63894—62no—
26#63779D311yescharacteristic, complemented
27#63778—31no—
28#63778—31no—
29#63777—31no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—22no—
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C2×C32×Dic3C2×C411yescharacteristic
37C2×C32×Dic3C2×C411yescharacteristic
38#6289—21no—
39#6289—21no—
40#6289—21no—
41#6289—21no—
42C3⋊#622—21no—
43C3⋊#622D412yescomplemented
44C3×#622—21no—
45C3×#622—21no—
46#4836—91nomaximal
47C2×C4⋊(C3⋊D3)—31no—
48C2×C4⋊(C3⋊D3)—62no—
49C2×C4⋊(C3⋊D3)—122no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—31no—
52C2×#624—62no—
53C2×#624—62no—
54C2×#622—31no—
55C2×#622—31no—
56C2×#622C2×D311yescharacteristic
57C2×#622C2×D311yescharacteristic
58#3788—62no—
59#3787—62no—
60C32×C4⋊C4—31no—
61#3752—31no—
62#3752—31no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C3×C4⋊Dic3—22no—
65C3×C4×Dic3—62no—
66#3722—62no—
67#3722—62no—
68C22×C33C22×C411yescharacteristic
69C2×C3⋊(C3⋊D3)—21no—
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—21no—
72C2×C3⋊(C3⋊D3)—21no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3⋊(C3⋊D3)M1612yescomplemented
75C33×C4—21no—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3C2×D411yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—21no—
80C32×Dic3—21no—
81C2×ΓL(1,𝔽25)—91no—
82C2×D3×C8—91no—
83#893—182no—
84C4×D12—91no—
85C3×C4⋊C8—91no—
86(C2×D3)⋊C8—91no—
87(C2×D3)⋊C8—91no—
88#803—91no—
89C22×C3⋊D3—31no—
90C22×C3⋊D3—31no—
91C22×C3⋊D3—62no—
92C22×C3⋊D3—62no—
93C22×C3⋊D3—122no—
94C22×C3⋊D3—122no—
95C2×C32×C4—22no—
96C2×C32×C4—31no—
97C2×C32×C4—31no—
98C2×C32×C4—42no—
99C2×C32×C4C4×D311yescharacteristic
100C2×C3⋊Dic3C4×D311yescharacteristic
101C2×C3⋊Dic3C22×D311yescharacteristic
102C4⋊(C3⋊D3)—62no—
103C4⋊(C3⋊D3)—68no—
104C4⋊(C3⋊D3)—128no—
105C4×C3⋊D3—61no—
106C4×C3⋊D3—61no—
107C4×C3⋊D3—61no—
108C4×C3⋊D3—61no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113#624—122no—
114#624—122no—
115#624—122no—
116#624—122no—
117#622—21no—
118#622—21no—
119#622—32no—
120#622—61no—
121#435—271noSylow
122C2×C33C4×D411yescharacteristic
123C2×C33C2×M1611yescharacteristic
124C2×C33#13111yescharacteristic
125C3⋊(C3⋊D3)—22no—
126C3⋊(C3⋊D3)—41no—
127C2×D12—91no—
128C2×D12—182no—
129C2×D12—362no—
130C2×C4×D3—91no—
131C2×C4×D3—91no—
132C2×C4×D3—182no—
133C2×C4×D3—182no—
134C2×C3×C8—91no—
135C2×C3×C8—91no—
136C3×C4⋊C4—62no—
137C3×C42—91no—
138(C2×D3)⋊C4—91no—
139(C2×D3)⋊C4—91no—
140(C2×D3)⋊C4—182no—
141(C2×D3)⋊C4—182no—
142C4⋊Dic3#15711yescharacteristic
143C4×Dic3—62no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146ΓL(1,𝔽25)—182no—
147D3×C8—182no—
148C22×C32—22no—
149C22×C32—42no—
150C22×C32C2×C4×D311yescharacteristic
151C2×C3⋊D3—32no—
152C2×C3⋊D3—64no—
153C2×C3⋊D3—61no—
154C2×C3⋊D3—64no—
155C2×C3⋊D3—61no—
156C2×C3⋊D3—61no—
157C2×C3⋊D3—61no—
158C2×C3⋊D3—61no—
159C2×C3⋊D3—124no—
160C2×C3⋊D3—124no—
161C2×C3⋊D3—122no—
162C2×C3⋊D3—122no—
163C2×C3⋊D3—122no—
164C2×C3⋊D3—122no—
165C2×C3⋊D3—242no—
166C2×C3⋊D3—242no—
167C2×C3⋊D3—242no—
168C2×C3⋊D3—242no—
169C32×C4—21no—
170C32×C4—24no—
171C32×C4—44no—
172C32×C4—61no—
173C32×C4—61no—
174C3⋊Dic3—21no—
175C3⋊Dic3D3×D411yescharacteristic
176C3⋊Dic3#23911yescharacteristic
177C3×Dic3—42no—
178C3×Dic3—42no—
179C3×Dic3—62no—
180C3×Dic3—62no—
181C3×Dic3—122no—
182C2×M16—271no—
183C22×C8—271no—
184C4×D4—271no—
185C4⋊C8—91no—
186C22⋊C8—271no—
187C22⋊C8—271no—
188#97—271no—
189C33#43511yescharacteristic, complemented, Sylow
190C22×D3—91no—
191C22×D3—91no—
192C22×D3—182no—
193C22×D3—182no—
194C22×D3—362no—
195C22×D3—362no—
196C2×C3×C4—22no—
197C2×C3×C4—42no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—91no—
201C2×C3×C4—91no—
202C2×C3×C4C2×#15711yescharacteristic
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3C2×#15711yescharacteristic
206C2×Dic3C2×#15711yescharacteristic
207D12—182no—
208D12—188no—
209D12—368no—
210C4×D3—181no—
211C4×D3—181no—
212C4×D3—181no—
213C4×D3—181no—
214C4×D3—362no—
215C4×D3—362no—
216C4×D3—362no—
217C4×D3—362no—
218C3×C8—181no—
219C3×C8—181no—
220C3⋊C8—92no—
221C3⋊C8—181no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—22no—
225C2×C32—42no—
226C2×C32—42no—
227C2×C32—42no—
228C2×C32#89311yescharacteristic
229C2×C32#90111yescharacteristic
230C2×C32D3×M1611yescharacteristic
231C3⋊D3—62no—
232C3⋊D3—124no—
233C3⋊D3—121no—
234C3⋊D3—124no—
235C3⋊D3—244no—
236C3⋊D3—244no—
237C2×D4—271no—
238C22×C4—271no—
239C22×C4—271no—
240M16—542no—
241C2×C8—91no—
242C2×C8—91no—
243C2×C8—271no—
244C2×C8—271no—
245C2×C8—542no—
246C4⋊C4—31no—
247C22⋊C4—271no—
248C22⋊C4—271no—
249C42—91no—
250C22×C3—22no—
251C22×C3—42no—
252C22×C3C22×#15711yescharacteristic
253C2×D3—92no—
254C2×D3—184no—
255C2×D3—181no—
256C2×D3—181no—
257C2×D3—184no—
258C2×D3—181no—
259C2×D3—181no—
260C2×D3—181no—
261C2×D3—364no—
262C2×D3—362no—
263C2×D3—362no—
264C2×D3—362no—
265C2×D3—362no—
266C2×D3—364no—
267C2×D3—722no—
268C2×D3—722no—
269C2×D3—722no—
270C2×D3—722no—
271C3×C4—24no—
272C3×C4—21no—
273C3×C4—44no—
274C3×C4—91no—
275C3×C4—91no—
276C3×C4—122no—
277C3×C4—122no—
278C3×C4—181no—
279Dic3—21no—
280Dic3—21no—
281Dic3—62no—
282Dic3—62no—
283Dic3—122no—
284C32—22no—
285C32—42no—
286C32#483611yescharacteristic, complemented
287C23—271no—
288C23—271no—
289D4—542no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—541no—
295C2×C4—541no—
296C2×C4—541no—
297C2×C4—541no—
298C2×C4D3×#15711yescharacteristic
299C8—181no—
300C8—181no—
301C8—272no—
302C8—541no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3—42no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3#6428311yescharacteristic
310C2×C3D4×#15711yescharacteristic
311C2×C3#6428911yescharacteristic
312D3—182no—
313D3—364no—
314D3—364no—
315D3—361no—
316D3—724no—
317D3—724no—
318C22—272no—
319C22—541no—
320C22—541no—
321C22—541no—
322C22—541no—
323C22—541no—
324C22C2×D3×#15711yescharacteristic
325C4—21no—
326C4—61no—
327C4—61no—
328C4—91no—
329C4—91no—
330C4—181no—
331C3—22no—
332C3—42no—
333C3#1059375111yescharacteristic, complemented
334C2—542no—
335C2—1081no—
336C2#1172425211yescharacteristic
337C2#1172425911yescharacteristic
338C2#1172426911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88926C2semidirect
2. C2×#88924C2semidirect
4. #11723102C2semidirect
5. C3×#63779C2semidirect
6. #11722726C2semidirect
7. #11722726C2semidirect
26. #63779D3semidirect
43. C3⋊#622D4semidirect
74. C2×C3⋊(C3⋊D3)M16semidirect
189. C33#435semidirect
286. C32#4836semidirect
333. C3#10593751semidirect

Maximal subgroups

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

subgroupindex
C2×#889262
C2×#889242
#117234942
#117231022
C3×#637792
#117227262
#105937513
#48369