Finite Group Databasebeta

← #13173336

Subgroups — #13173336

Order 1728.

524 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×C3⋊#622C211yesmaximal, characteristic, complemented
2C2×#88988C211yesmaximal, characteristic, complemented
3C3×#64090C211yesmaximal, characteristic
4#11723112C211yesmaximal, characteristic, complemented
5#11723104C211yesmaximal, characteristic, complemented
6#11723103C211yesmaximal, characteristic, complemented
7C4×C3⋊#622C211yesmaximal, characteristic, complemented
8#10593739—31nomaximal
9C2×C3×#638C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3⋊#622—22no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3⋊#622C2211yescharacteristic
15C2×C3⋊#622C2211yescharacteristic
16C2×C3⋊#622C2211yescharacteristic
17#88988—22no—
18C33×C22⋊C4C411yescharacteristic
19C3×#3752C411yescharacteristic
20C3×#3750C411yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C22×#622—31no—
23C2×#3787—31no—
24#64090D311yescharacteristic
25C3×#67⋊C4—62no—
26C22⋊#622—31no—
27#63779—31no—
28#63778—31no—
29C4×#622—31no—
30C23×C33C2×C411yescharacteristic
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C3×#638C814yescomplemented
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C2×C3×C3⋊Dic3C2×C411yescharacteristic
39C2×C3×C3⋊Dic3C2311yescharacteristic
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3—21no—
42C3⋊#622—21no—
43C3⋊#622—21no—
44C3⋊#622—21no—
45C3⋊#622D412yescomplemented
46D4×C3⋊C8—91nomaximal
47C2×#638Dic311yescharacteristic
48C22×C3⋊Dic3C2×D311yescharacteristic
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×C3×#67—62no—
51C22×C3×Dic3—62no—
52C2×C3×C4×D3—62no—
53C2×#622—31no—
54C2×#622—31no—
55C2×#622—31no—
56C2×#622—31no—
57C2×#622—62no—
58#3787—62no—
59C32×C22⋊C4—22no—
60C32×C22⋊C4—42no—
61C32×C22⋊C4Dic311yescharacteristic
62#3752Dic311yescharacteristic
63#3750Dic311yescharacteristic
64C4×C3⋊Dic3C2×D311yescharacteristic
65C3×(C2×D3)⋊C4—62no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—62no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33C2×C812yescomplemented
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3C2×C812yescomplemented
75C33×C4—21no—
76C33×C4—21no—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3C2×C812yescomplemented
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C3×C4×D4—91no—
82C22×C3⋊C8—91no—
83C22×C3⋊C8—91no—
84#67⋊C4—62no—
85#830—91no—
86#830—91no—
87C4⋊(C3⋊C8)—91no—
88C4×C3⋊C8—91no—
89C23×C32—22no—
90C23×C32—42no—
91C23×C32C2×Dic311yescharacteristic
92C22×C3⋊D3C2×Dic311yescharacteristic
93C22×C3×D3—62no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4—42no—
98C2×C32×C4C2×Dic311yescharacteristic
99C2×C32×C4C2×Dic311yescharacteristic
100#638C3⋊C814yescomplemented
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3C2×Dic311yescharacteristic
104C2×C3⋊Dic3C2×Dic311yescharacteristic
105C2×C3⋊Dic3C22×D311yescharacteristic
106C4×C3⋊D3—21no—
107C4×C3⋊D3—21no—
108C3×#67—68no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—122no—
113C2×C3×Dic3—122no—
114C3×C4×D3—122no—
115C3×C4×D3—122no—
116#622—32no—
117#622—61no—
118#622—61no—
119#622—61no—
120C8×D4—271noSylow
121C2×C33—22no—
122C2×C33C4×D411yescharacteristic
123C2×C33C22×C811yescharacteristic
124C2×C33#13111yescharacteristic
125C3×C3⋊D3—22no—
126C2×C3×D4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C2×#67—62no—
130C22×Dic3—62no—
131C2×C4×D3—62no—
132C3×C4⋊C4—91no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—91no—
136C3×C22⋊C4#15711yescharacteristic
137C3×C42—91no—
138(C2×D3)⋊C4—62no—
139Dic3⋊C4—62no—
140C4×Dic3—62no—
141C2×C3⋊C8—91no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—182no—
146C2×C3⋊C8—182no—
147C22×C32—22no—
148C22×C32—24no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—42no—
152C22×C32—44no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32—82no—
156C22×C32—82no—
157C22×C32C2×C3⋊C812yescomplemented
158C22×C32C22×Dic311yescharacteristic
159C2×C3⋊D3—21no—
160C2×C3⋊D3—21no—
161C2×C3⋊D3C2×C3⋊C812yescomplemented
162C2×C3×D3—64no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4—82no—
170C32×C4—82no—
171C3⋊Dic3—21no—
172C3⋊Dic3C2×C3⋊C812yescomplemented
173C3⋊Dic3D3×D411yescharacteristic
174C3⋊Dic3#23711yescharacteristic
175C3×Dic3—64no—
176C3×Dic3—62no—
177C3×Dic3—62no—
178C3×Dic3—122no—
179C22×C8—271no—
180C22×C8—271no—
181C4×D4—91no—
182C4⋊C8—271no—
183C22⋊C8—271no—
184C22⋊C8—271no—
185C4×C8—271no—
186C33C8×D411yescharacteristic, complemented, Sylow
187C23×C3—22no—
188C23×C3—42no—
189C23×C3—91no—
190C23×C3C2×#15711yescharacteristic
191C22×D3—62no—
192C3×D4—94no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×C3×C4C2×#15711yescharacteristic
205C2×C3×C4C2×#15711yescharacteristic
206#67—68no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—122no—
211C2×Dic3—122no—
212C4×D3—122no—
213C4×D3—122no—
214C3⋊C8—92no—
215C3⋊C8—181no—
216C3⋊C8—181no—
217C3⋊C8—181no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—22no—
222C2×C32—42no—
223C2×C32—42no—
224C2×C32—42no—
225C2×C32—44no—
226C2×C32—84no—
227C2×C32C22×C3⋊C811yescharacteristic
228C2×C32D4×Dic311yescharacteristic
229C2×C32#94811yescharacteristic
230C3⋊D3—22no—
231C3×D3—124no—
232C2×D4—91no—
233C22×C4—91no—
234C22×C4—91no—
235C2×C8—271no—
236C2×C8—271no—
237C2×C8—271no—
238C2×C8—271no—
239C2×C8—542no—
240C2×C8—542no—
241C4⋊C4—91no—
242C22⋊C4—91no—
243C22⋊C4C3⋊#15711yescharacteristic
244C42—91no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—82no—
254C22×C3—82no—
255C22×C3—92no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3#378712yescomplemented
259C22×C3C22×#15711yescharacteristic
260C2×D3—64no—
261C2×D3—122no—
262C2×D3—122no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—42no—
266C3×C4—42no—
267C3×C4—82no—
268C3×C4—82no—
269C3×C4—92no—
270C3×C4—91no—
271C3×C4—91no—
272C3×C4—181no—
273Dic3—62no—
274Dic3—62no—
275Dic3—64no—
276Dic3—122no—
277C32—22no—
278C32—42no—
279C32D4×C3⋊C811yescharacteristic, complemented
280C23—91no—
281C23C2×C3⋊#15711yescharacteristic
282D4—94no—
283C2×C4—91no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4C2×C3⋊#15711yescharacteristic
291C2×C4C2×C3⋊#15711yescharacteristic
292C8—272no—
293C8—541no—
294C8—541no—
295C8—541no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—44no—
301C2×C3—42no—
302C2×C3—42no—
303C2×C3—42no—
304C2×C3—84no—
305C2×C3—182no—
306C2×C3C2×#378711yescharacteristic
307C2×C3#6428711yescharacteristic
308C2×C3D4×#15711yescharacteristic
309D3—124no—
310C22—21no—
311C22—21no—
312C22—92no—
313C22—181no—
314C22—181no—
315C22#8898812yescomplemented
316C22C22×C3⋊#15711yescharacteristic
317C4—21no—
318C4—21no—
319C4—92no—
320C4—91no—
321C4—91no—
322C4—181no—
323C3—22no—
324C3—42no—
325C3#1059373911yescharacteristic, complemented
326C2—22no—
327C2—182no—
328C2C2×#8898811yescharacteristic
329C2#1172451511yescharacteristic
330C2D4×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutatorC2×C33C22×C8
FittingC33×C22⋊C4C4
socleC22×C33C22×C4
prime radicalC2×C3×#638C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22⋊C4C3⋊#157
hypocenterC33C8×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊#622C2semidirect
2. C2×#88988C2semidirect
4. #11723112C2semidirect
5. #11723104C2semidirect
6. #11723103C2semidirect
7. C4×C3⋊#622C2semidirect
34. C3×#638C8semidirect
45. C3⋊#622D4semidirect
70. C22×C33C2×C8semidirect
74. C2×C3×C3⋊D3C2×C8semidirect
78. C3×C3⋊Dic3C2×C8semidirect
100. #638C3⋊C8semidirect
157. C22×C32C2×C3⋊C8semidirect
161. C2×C3⋊D3C2×C3⋊C8semidirect
172. C3⋊Dic3C2×C3⋊C8semidirect
186. C33C8×D4semidirect
258. C22×C3#3787semidirect
279. C32D4×C3⋊C8semidirect
315. C22#88988semidirect
325. C3#10593739semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×#889882
C3×#640902
#117231122
#117231042
#117231032
C4×C3⋊#6222
#105937393
D4×C3⋊C89