Finite Group Databasebeta

← #13155052

Subgroups — #13155052

Order 1728.

420 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 29 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
1#11723728C211yesmaximal, characteristic
2#11723111C211yesmaximal, characteristic, complemented
3C3×#63581C211yesmaximal, characteristic, complemented
4#10590080—31nomaximal
5C2×C3×#638—21no—
6C2×C3×#638C2211yescharacteristic
7C2×#6295—21no—
8C2×#6295C411yescharacteristic, complemented
9#88994—21no—
10C32×#217—21no—
11#88813C411yescharacteristic, complemented
12#63979—31no—
13#63979—61no—
14#63786—31no—
15C3×#806—61no—
16#63581D311yescharacteristic, complemented
17C23×C33D411yescharacteristic
18C22×C3×C3⋊D3—21no—
19C22×C3×C3⋊D3D411yescharacteristic
20C2×C3⋊#157—41no—
21C3×#638—41no—
22C3×#638—42no—
23C2×C3×C3⋊Dic3—21no—
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C32×Dic3—41no—
26#6295—42no—
27#6295—41no—
28#4474—91nomaximal
29C2×#638—21no—
30C2×#638C2×D311yescharacteristic
31C2×C3×#67—61no—
32C2×C3×#67—61no—
33C2×C3×#67—61no—
34C2×C3×#67—61no—
35C2×C3×#67—121no—
36C2×#626—61no—
37C2×#626—61no—
38C2×#626—61no—
39C2×#625—31no—
40C2×#625—61no—
41C22⋊#157—61no—
42C32×C22⋊C4—61no—
43C3×#217—21no—
44C3×#217—21no—
45C3×#217—41no—
46Dic32—31no—
47Dic32—61no—
48C22×C33—21no—
49C22×C33—41no—
50C22×C33C22⋊C411yescharacteristic
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—41no—
53C2×C3×C3⋊D3—41no—
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3—41no—
56C3⋊#157—81no—
57C3×C3⋊Dic3—22no—
58C3×C3⋊Dic3—21no—
59C32×Dic3—81no—
60#940—91no—
61#940—181no—
62C3×#99—91no—
63#834—91no—
64#806—61no—
65C23×C32—21no—
66C23×C32—21no—
67C23×C32—41no—
68C23×C32—41no—
69C23×C32#6711yescharacteristic
70C22×C3⋊D3—21no—
71C22×C3⋊D3D1211yescharacteristic
72C22×C3×D3—61no—
73C22×C3×D3—61no—
74C22×C3×D3—61no—
75C22×C3×D3—61no—
76C22×C3×D3—121no—
77C2×#157—121no—
78C2×C32×C4—121no—
79#638—41no—
80#638—42no—
81C2×C3⋊Dic3—21no—
82C2×C3⋊Dic3C4×D311yescharacteristic
83C3×#67—122no—
84C3×#67—122no—
85C3×#67—122no—
86C3×#67—121no—
87C3×#67—121no—
88C3×#67—124no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—121no—
97#626—122no—
98#626—122no—
99#626—122no—
100#625—121no—
101#625—122no—
102#353—271noSylow
103C2×C33—21no—
104C2×C33—41no—
105C2×C33#9911yescharacteristic
106C3×C3⋊D3—41no—
107C3×C3⋊D3—81no—
108C2×C3×D4—91no—
109C2×C3×D4—181no—
110C2×#67—61no—
111C2×#67—61no—
112C2×#67—61no—
113C2×#67—61no—
114C2×#67—121no—
115C2×#67—181no—
116C2×#67—181no—
117C2×#67—181no—
118C2×D12—91no—
119C2×D12—181no—
120C3×C22⋊C4—61no—
121C3×C22⋊C4—61no—
122C3×C22⋊C4—121no—
123#217—21no—
124#217—181no—
125C4×Dic3—91no—
126C4×Dic3—181no—
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(C2×D3)⋊C411yescharacteristic
146C2×C3⋊D3—21no—
147C2×C3⋊D3—41no—
148C2×C3⋊D3—41no—
149C2×C3⋊D3—41no—
150C2×C3⋊D3—41no—
151C2×C3×D3—121no—
152C2×C3×D3—121no—
153C2×C3×D3—121no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C2×C3×D3—121no—
160C2×C3×D3—122no—
161C2×C3×D3—121no—
162C2×C3×D3—241no—
163C2×C3×D3—241no—
164C2×C3×D3—241no—
165#157—241no—
166C32×C4—241no—
167C3⋊Dic3—22no—
168C3⋊Dic3—21no—
169C3×Dic3—62no—
170C3×Dic3—62no—
171C3×Dic3—62no—
172C3×Dic3—81no—
173C3×Dic3—82no—
174C3×Dic3—81no—
175C3×Dic3—121no—
176C3×Dic3—122no—
177#127—271no—
178#99—91no—
179#99—271no—
180C33#35311yescharacteristic, complemented, Sylow
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—41no—
184C23×C3—41no—
185C23×C3—91no—
186C23×C3—181no—
187C23×C3GO+(4,𝔽2)11yescharacteristic
188C22×D3—61no—
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—61no—
192C22×D3—121no—
193C3×D4—361no—
194C3×D4—362no—
195C2×C3×C4—91no—
196C2×C3×C4—121no—
197C2×C3×C4—121no—
198C2×C3×C4—122no—
199C2×C3×C4—181no—
200#67—122no—
201#67—122no—
202#67—122no—
203#67—121no—
204#67—121no—
205#67—124no—
206#67—362no—
207#67—362no—
208#67—362no—
209C2×Dic3—41no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—121no—
215C2×Dic3—361no—
216D12—361no—
217D12—362no—
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#80611yescharacteristic
237C3⋊D3—41no—
238C3⋊D3—81no—
239C3×D3—241no—
240C3×D3—241no—
241C3×D3—241no—
242C3×D3—242no—
243C3×D3—241no—
244C2×D4—91no—
245C2×D4—181no—
246C2×D4—271no—
247C2×D4—541no—
248C22⋊C4—61no—
249C22⋊C4—541no—
250C42—271no—
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—181no—
270C22×C3—361no—
271C22×C3—361no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3D32⋊C411yescharacteristic
275C2×D3—121no—
276C2×D3—121no—
277C2×D3—121no—
278C2×D3—121no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—121no—
284C2×D3—122no—
285C2×D3—121no—
286C2×D3—241no—
287C2×D3—241no—
288C2×D3—241no—
289C3×C4—182no—
290C3×C4—181no—
291C3×C4—241no—
292C3×C4—242no—
293C3×C4—241no—
294Dic3—62no—
295Dic3—62no—
296Dic3—62no—
297Dic3—81no—
298Dic3—121no—
299Dic3—122no—
300Dic3—721no—
301C32—21no—
302C32—21no—
303C32—41no—
304C32—41no—
305C32#447411yescharacteristic, complemented
306C23—91no—
307C23—181no—
308C23D32⋊D311yescharacteristic
309D4—361no—
310D4—362no—
311D4—1081no—
312D4—1082no—
313C2×C4—91no—
314C2×C4—121no—
315C2×C4—181no—
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—361no—
336C2×C3—721no—
337C2×C3#6373811yescharacteristic
338D3—241no—
339D3—241no—
340D3—241no—
341D3—242no—
342D3—241no—
343C22—21no—
344C22—41no—
345C22—181no—
346C22—361no—
347C22—361no—
348C22—361no—
349C22—361no—
350C22#8893211yescharacteristic
351C4—182no—
352C4—181no—
353C4—241no—
354C4—2161no—
355C3—21no—
356C3—21no—
357C3—41no—
358C3—41no—
359C3#1059008011yescharacteristic, complemented
360C2—21no—
361C2—41no—
362C2—361no—
363C2—721no—
364C2#1172277011yescharacteristic
subgroupG/H
centerC2#11722770
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC23D32⋊D3
FittingC23×C33D4
socleC2×C33#99
prime radical#11723728C2
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. C3×#63581C2semidirect
8. C2×#6295C4semidirect
11. #88813C4semidirect
16. #63581D3semidirect
180. C33#353semidirect
305. C32#4474semidirect
359. C3#10590080semidirect

Maximal subgroups

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

subgroupindex
#117237282
#117231112
C3×#635812
#105900803
#44749