Finite Group Databasebeta

← #13166641

Subgroups — #13166641

Order 1728.

544 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 72 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×#88926C211yesmaximal, characteristic, complemented
3#11723622C211yesmaximal, characteristic
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5Dic3⋊#622C211yesmaximal, characteristic, complemented
6#11722726C211yesmaximal, characteristic, complemented
7#11722724C211yesmaximal, characteristic, complemented
8#10593740—31nomaximal
9C2×#6312C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×#6289C2211yescharacteristic
12C2×C3⋊#622—22no—
13C2×C3⋊#622C2211yescharacteristic
14C2×C3⋊#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17#88926—22no—
18C32×#217C411yescharacteristic
19#88806C411yescharacteristic
20#88804C411yescharacteristic
21Dic3×C3⋊Dic3C2211yescharacteristic
22C22×#622—31no—
23C2×#3788—31no—
24#64090—31no—
25#63952—62no—
26C22⋊#622D311yescharacteristic, complemented
27#63778—31no—
28#63777—31no—
29C4⋊#622—31no—
30C23×C33C2×C411yescharacteristic
31C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
32#6312—22no—
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C2×C32×Dic3C2×C411yescharacteristic
39C2×C32×Dic3C2×C411yescharacteristic
40#6289—21no—
41#6289—21no—
42C3⋊#622—21no—
43C3⋊#622D412yescomplemented
44C3×#622—21no—
45C3×#622—21no—
46#4730—91nomaximal
47C2×#638—31no—
48C2×#638—62no—
49C2×#638—122no—
50C22×C3⋊Dic3C2×D311yescharacteristic
51C2×C4×C3⋊D3—31no—
52C22×C3×Dic3—62no—
53C2×#624—62no—
54C2×#622—31no—
55C2×#622—31no—
56C2×#622—62no—
57C2×#622C2×D311yescharacteristic
58C2×#622C2×D311yescharacteristic
59#3788—62no—
60C32×C22⋊C4—31no—
61#3752—31no—
62#3750—31no—
63C4×C3⋊Dic3—31no—
64C3×#217—22no—
65#3724—62no—
66#3722—62no—
67Dic32—62no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33M1612yescomplemented
71C22×C33C22×C411yescharacteristic
72C2×C3⋊(C3⋊D3)—21no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3⋊(C3⋊D3)—21no—
75C3⋊(C3⋊Dic3)—21no—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3C2×D411yescharacteristic
78C3×C3⋊Dic3#4111yescharacteristic
79C32×Dic3—21no—
80C32×Dic3—21no—
81C4×#67—91no—
82C22×C3⋊C8—91no—
83C2×ΓL(1,𝔽25)—91no—
84#67⋊C4—182no—
85C3×C22⋊C8—91no—
86(C2×D3)⋊C8—91no—
87#815—91no—
88Dic3⋊C8—91no—
89C23×C32—22no—
90C23×C32—42no—
91C23×C32C4×D311yescharacteristic
92C22×C3⋊D3—31no—
93C22×C3⋊D3—62no—
94C22×C3⋊D3—122no—
95C2×C32×C4—31no—
96C2×C32×C4—31no—
97#638—62no—
98#638—68no—
99#638—128no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—62no—
104C2×C3⋊Dic3—122no—
105C2×C3⋊Dic3C4×D311yescharacteristic
106C2×C3⋊Dic3C22×D311yescharacteristic
107C4×C3⋊D3—61no—
108C4×C3⋊D3—61no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C2×C3×Dic3—122no—
115#624—122no—
116#624—122no—
117#622—21no—
118#622—21no—
119#622—32no—
120#622—61no—
121#435—271noSylow
122C2×C33—22no—
123C2×C33C4×D411yescharacteristic
124C2×C33C2×M1611yescharacteristic
125C2×C33#13111yescharacteristic
126C3⋊(C3⋊D3)—41no—
127C22×C3×C4—91no—
128C2×#67—91no—
129C2×#67—182no—
130C2×#67—362no—
131C22×Dic3—62no—
132C2×C4×D3—91no—
133C2×C4×D3—182no—
134C2×C3×C8—91no—
135C2×C3×C8—91no—
136C3×C22⋊C4—62no—
137#217#15711yescharacteristic
138(C2×D3)⋊C4—91no—
139(C2×D3)⋊C4—182no—
140Dic3⋊C4—91no—
141Dic3⋊C4—182no—
142C4×Dic3—91no—
143C4×Dic3—182no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—182no—
147ΓL(1,𝔽25)—182no—
148C22×C32—22no—
149C22×C32—24no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—42no—
153C22×C32—44no—
154C22×C32—42no—
155C22×C32—42no—
156C22×C32—82no—
157C22×C32—82no—
158C22×C32ΓL(1,𝔽25)12yescomplemented
159C22×C32C2×C4×D311yescharacteristic
160C2×C3⋊D3—61no—
161C2×C3⋊D3—64no—
162C2×C3⋊D3—61no—
163C2×C3⋊D3—61no—
164C2×C3⋊D3—124no—
165C2×C3⋊D3—122no—
166C2×C3⋊D3—122no—
167C2×C3⋊D3—242no—
168C2×C3⋊D3—242no—
169C32×C4—61no—
170C32×C4—61no—
171C3⋊Dic3—21no—
172C3⋊Dic3—61no—
173C3⋊Dic3—64no—
174C3⋊Dic3—124no—
175C3⋊Dic3D3×D411yescharacteristic
176C3⋊Dic3#23711yescharacteristic
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—271no—
186C22⋊C8—91no—
187C22⋊C8—271no—
188#97—271no—
189C33#43511yescharacteristic, complemented, Sylow
190C23×C3—22no—
191C23×C3—42no—
192C23×C3C2×#15711yescharacteristic
193C22×D3—91no—
194C22×D3—182no—
195C22×D3—362no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202#67—188no—
203#67—182no—
204#67—368no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—91no—
208C2×Dic3—122no—
209C2×Dic3—122no—
210C2×Dic3—182no—
211C2×Dic3—362no—
212C2×Dic3C2×#15711yescharacteristic
213C2×Dic3C2×#15711yescharacteristic
214C4×D3—181no—
215C4×D3—181no—
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—22no—
226C2×C32—42no—
227C2×C32—42no—
228C2×C32—42no—
229C2×C32—44no—
230C2×C32—84no—
231C2×C32#67⋊C411yescharacteristic
232C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
233C2×C32#90711yescharacteristic
234C3⋊D3—121no—
235C3⋊D3—124no—
236C3⋊D3—244no—
237C2×D4—271no—
238C22×C4—91no—
239C22×C4—271no—
240M16—542no—
241C2×C8—91no—
242C2×C8—91no—
243C2×C8—271no—
244C2×C8—271no—
245C2×C8—542no—
246C4⋊C4—271no—
247C22⋊C4—31no—
248C22⋊C4—271no—
249C42—271no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—22no—
253C22×C3—24no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—44no—
258C22×C3—82no—
259C22×C3—82no—
260C22×C3#378812yescomplemented
261C22×C3C22×#15711yescharacteristic
262C2×D3—184no—
263C2×D3—181no—
264C2×D3—181no—
265C2×D3—181no—
266C2×D3—362no—
267C2×D3—362no—
268C2×D3—364no—
269C2×D3—722no—
270C2×D3—722no—
271C3×C4—91no—
272C3×C4—91no—
273C3×C4—122no—
274C3×C4—122no—
275C3×C4—181no—
276Dic3—21no—
277Dic3—21no—
278Dic3—62no—
279Dic3—62no—
280Dic3—122no—
281Dic3—184no—
282Dic3—181no—
283Dic3—364no—
284C32—22no—
285C32—42no—
286C32#473011yescharacteristic, complemented
287C23—271no—
288C23D3×#15711yescharacteristic
289D4—542no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—271no—
297C2×C4—541no—
298C2×C4—541no—
299C8—181no—
300C8—181no—
301C8—272no—
302C8—541no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3—22no—
307C2×C3—44no—
308C2×C3—42no—
309C2×C3—42no—
310C2×C3—42no—
311C2×C3—84no—
312C2×C3C2×#378811yescharacteristic
313C2×C3#6428711yescharacteristic
314C2×C3D4×#15711yescharacteristic
315D3—364no—
316D3—361no—
317D3—724no—
318C22—21no—
319C22—21no—
320C22—541no—
321C22—541no—
322C22—541no—
323C22#8892612yescomplemented
324C22C2×D3×#15711yescharacteristic
325C4—61no—
326C4—61no—
327C4—91no—
328C4—91no—
329C4—181no—
330C4—541no—
331C3—22no—
332C3—42no—
333C3#1059374011yescharacteristic, complemented
334C2—22no—
335C2—1081no—
336C2#1172426711yescharacteristic
337C2C2×#8892611yescharacteristic
338C2#67⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC23×C33C2×C4
prime radicalC2×#6312C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊#622C2semidirect
2. C2×#88926C2semidirect
4. C3×C22⋊#622C2semidirect
5. Dic3⋊#622C2semidirect
6. #11722726C2semidirect
7. #11722724C2semidirect
26. C22⋊#622D3semidirect
43. C3⋊#622D4semidirect
70. C22×C33M16semidirect
158. C22×C32ΓL(1,𝔽25)semidirect
189. C33#435semidirect
260. C22×C3#3788semidirect
286. C32#4730semidirect
323. C22#88926semidirect
333. C3#10593740semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×#889262
#117236222
C3×C22⋊#6222
Dic3⋊#6222
#117227262
#117227242
#105937403
#47309