Finite Group Databasebeta

← #13155053

Subgroups — #13155053

Order 1728.

420 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 31 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
1C3×#63979C211yesmaximal, characteristic
2#11723111C211yesmaximal, characteristic, complemented
3#11722083C211yesmaximal, characteristic, complemented
4#10590080—31nomaximal
5C2×C3×#638C2211yescharacteristic
6C2×C32×#67—21no—
7C2×C3×#626—21no—
8C2×C3×#625C411yescharacteristic, complemented
9#88994—21no—
10C3×#3757—21no—
11C3×Dic32C411yescharacteristic, complemented
12C3×#940—61no—
13#63979D311yescharacteristic
14#63786—31no—
15#63581—31no—
16#63580—61no—
17C23×C33D411yescharacteristic
18C22×C3×C3⋊D3D411yescharacteristic
19C22×C32×D3—21no—
20C2×C3⋊#157—41no—
21C3×#638—41no—
22C2×C3×C3⋊Dic3—41no—
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C32×#67—42no—
25C2×C32×Dic3—21no—
26C3×#626—42no—
27C3×#625—41no—
28#4545—91nomaximal
29C2×C32×D4—61no—
30C2×#638C2×D311yescharacteristic
31C2×C3×#67—21no—
32C2×C3×#67—21no—
33C2×C3×#67—41no—
34C2×C3×#67—61no—
35C2×C3×#67—61no—
36C2×C3×#67—61no—
37C2×C3×#67—61no—
38C2×C3×D12—61no—
39C2×#626—21no—
40C2×#625Dic311yescharacteristic, complemented
41C22⋊#157—61no—
42#3757—21no—
43C3×#217—61no—
44C3×#217—61no—
45C3×#217—121no—
46C3×C4×Dic3—61no—
47Dic32Dic311yescharacteristic, complemented
48C22×C33—21no—
49C22×C33—41no—
50C22×C33C22⋊C411yescharacteristic
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—41no—
53C2×C32×D3—41no—
54C2×C32×D3—41no—
55C2×C32×D3—41no—
56C3⋊#157—81no—
57C3×C3⋊Dic3—21no—
58C3×C3⋊Dic3—81no—
59C32×Dic3—22no—
60C3×#127—91no—
61#940—61no—
62#834—91no—
63#834—91no—
64#806—181no—
65C23×C32—21no—
66C23×C32—21no—
67C23×C32—41no—
68C23×C32—41no—
69C23×C32—61no—
70C23×C32#6711yescharacteristic
71C22×C3⋊D3#6711yescharacteristic
72C22×C3×D3—21no—
73C22×C3×D3—21no—
74C22×C3×D3—41no—
75C22×C3×D3—61no—
76C22×C3×D3—61no—
77C2×#157—121no—
78C32×D4—122no—
79C2×C32×C4—61no—
80#638—41no—
81C2×C3⋊Dic3—41no—
82C2×C3⋊Dic3C2×Dic311yescharacteristic
83C3×#67—42no—
84C3×#67—42no—
85C3×#67—44no—
86C3×#67—122no—
87C3×#67—122no—
88C3×#67—122no—
89C3×#67—121no—
90C3×#67—121no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—121no—
97C2×C3×Dic3—121no—
98C2×C3×Dic3—122no—
99C3×D12—122no—
100#626—42no—
101#625—41no—
102#353—271noSylow
103C2×C33—21no—
104C2×C33—41no—
105C2×C33#9911yescharacteristic
106C3×C3⋊D3—41no—
107C32×D3—81no—
108C2×C3×D4—61no—
109C2×C3×D4—61no—
110C2×C3×D4—91no—
111C2×C3×D4—91no—
112C2×C3×D4—121no—
113C2×C3×D4—181no—
114C2×#67—21no—
115C2×#67—61no—
116C2×#67—61no—
117C2×#67—61no—
118C2×#67—61no—
119C2×D12—61no—
120C3×C22⋊C4—181no—
121C3×C42—91no—
122#217—61no—
123#217—61no—
124#217—121no—
125#217—181no—
126C4×Dic3—61no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—41no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32—42no—
141C22×C32—81no—
142C22×C32—81no—
143C22×C32—81no—
144C22×C32—82no—
145C22×C32—121no—
146C22×C32—121no—
147C22×C32—121no—
148C22×C32#21711yescharacteristic
149C2×C3⋊D3—21no—
150C2×C3⋊D3—41no—
151C2×C3×D3—41no—
152C2×C3×D3—41no—
153C2×C3×D3—41no—
154C2×C3×D3—41no—
155C2×C3×D3—41no—
156C2×C3×D3—42no—
157C2×C3×D3—41no—
158C2×C3×D3—81no—
159C2×C3×D3—81no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C2×C3×D3—241no—
165#157—241no—
166C32×C4—62no—
167C3⋊Dic3—21no—
168C3⋊Dic3—81no—
169C3×Dic3—22no—
170C3×Dic3—22no—
171C3×Dic3—42no—
172C3×Dic3—62no—
173C3×Dic3—121no—
174C3×Dic3—241no—
175C3×Dic3—242no—
176C3×Dic3—241no—
177#127—91no—
178#99—271no—
179#99—271no—
180C33#35311yescharacteristic, complemented, Sylow
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—41no—
184C23×C3—41no—
185C23×C3—61no—
186C23×C3—61no—
187C23×C3—91no—
188C23×C3—121no—
189C23×C3GO+(4,𝔽2)11yescharacteristic
190C22×D3—21no—
191C22×D3—61no—
192C22×D3—61no—
193C3×D4—122no—
194C3×D4—122no—
195C3×D4—124no—
196C3×D4—362no—
197C3×D4—361no—
198C3×D4—361no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—91no—
202C2×C3×C4—121no—
203C2×C3×C4—361no—
204#67—42no—
205#67—122no—
206#67—122no—
207#67—122no—
208#67—121no—
209#67—121no—
210C2×Dic3—21no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—121no—
214C2×Dic3—121no—
215C2×Dic3—122no—
216C2×Dic3—361no—
217D12—122no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—42no—
231C2×C32—41no—
232C2×C32—81no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—82no—
236C2×C32—241no—
237C2×C32#83411yescharacteristic
238C3⋊D3—41no—
239C3×D3—81no—
240C3×D3—82no—
241C3×D3—81no—
242C3×D3—241no—
243C3×D3—241no—
244C2×D4—61no—
245C2×D4—91no—
246C2×D4—91no—
247C2×D4—181no—
248C22⋊C4—181no—
249C22⋊C4—541no—
250C42—91no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—42no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—81no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3—82no—
269C22×C3—121no—
270C22×C3—121no—
271C22×C3—121no—
272C22×C3—121no—
273C22×C3—121no—
274C22×C3—122no—
275C22×C3—121no—
276C22×C3—181no—
277C22×C3—241no—
278C22×C3—241no—
279C22×C3—361no—
280C22×C3D32⋊C411yescharacteristic
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—121no—
285C2×D3—121no—
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—241no—
289C3×C4—62no—
290C3×C4—62no—
291C3×C4—122no—
292C3×C4—181no—
293C3×C4—721no—
294Dic3—22no—
295Dic3—62no—
296Dic3—121no—
297Dic3—241no—
298Dic3—242no—
299Dic3—241no—
300Dic3—721no—
301C32—21no—
302C32—21no—
303C32—41no—
304C32—41no—
305C32#454511yescharacteristic, complemented
306C23—61no—
307C23—91no—
308C23D32⋊D311yescharacteristic
309D4—122no—
310D4—362no—
311D4—361no—
312D4—361no—
313C2×C4—61no—
314C2×C4—91no—
315C2×C4—361no—
316C2×C4—1081no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—42no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3—81no—
333C2×C3—82no—
334C2×C3—81no—
335C2×C3—241no—
336C2×C3—242no—
337C2×C3—241no—
338C2×C3—361no—
339C2×C3#6373811yescharacteristic
340D3—81no—
341D3—241no—
342D3—241no—
343C22—21no—
344C22—41no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22—181no—
349C22—361no—
350C22D32⋊Dic311yescharacteristic
351C4—62no—
352C4—181no—
353C4—721no—
354C4—2161no—
355C3—21no—
356C3—21no—
357C3—41no—
358C3—41no—
359C3#1059008011yescharacteristic, complemented
360C2—21no—
361C2—41no—
362C2—241no—
363C2—361no—
364C2#1172277111yescharacteristic
subgroupG/H
centerC2#11722771
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC23D32⋊D3
FittingC23×C33D4
socleC2×C33#99
prime radicalC3×#63979C2
cosocleC2×C3×#638C22
hypocenterC33#353
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723111C2semidirect
3. #11722083C2semidirect
8. C2×C3×#625C4semidirect
11. C3×Dic32C4semidirect
40. C2×#625Dic3semidirect
47. Dic32Dic3semidirect
180. C33#353semidirect
305. C32#4545semidirect
359. C3#10590080semidirect

Maximal subgroups

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

subgroupindex
C3×#639792
#117231112
#117220832
#105900803
#45459