Finite Group Databasebeta

← #13146392

Subgroups — #13146392

Order 1728.

447 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 61 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×Dic3×C22⋊C9C211yesmaximal, characteristic
2C2×C3⋊((C22⋊C9)⋊C4)C211yesmaximal, characteristic
3C2×C3×(C22⋊C9)⋊C4C211yesmaximal, characteristic
4#10591693—31nomaximal
5((C22⋊C9)⋊C4)⋊C4—31nomaximal
6C22×C3×C22⋊C9C2211yescharacteristic
7Dic3×C22⋊C9—21no—
8C3⋊((C22⋊C9)⋊C4)C412yescomplemented
9C3×(C22⋊C9)⋊C4—21no—
10#88442—41nomaximal
11C23×C3×Dic3D311yescharacteristic
12C2×#3757—31no—
13C2×C3×#217—31no—
14C2×#3724—31no—
15C2×#3724—31no—
16C2×C4×C22⋊C9—31no—
17C2×(C22⋊C9)⋊C4—31no—
18C2×(C22⋊C9)⋊C4—61no—
19C2×(C22⋊C9)⋊C4D311yescharacteristic
20#63579—31no—
21#63579—31no—
22C2×C3×C22⋊C9C2×C411yescharacteristic
23C2×C3×C22⋊C9D411yescharacteristic
24C2×C3×C22⋊C9Q811yescharacteristic
25C2×C3⋊Dic9—41no—
26C2×C9×Dic3—41no—
27C2×C3×Dic9—41no—
28#5210—91no—
29#4946—91no—
30C24×C32C2×D311yescharacteristic
31C22×C3⋊Dic3—31no—
32C22×C3⋊Dic3—31no—
33C22×C3×Dic3—21no—
34C22×C3×Dic3—31no—
35C22×C3×Dic3—31no—
36C22×C3×Dic3—31no—
37C22×C3×Dic3—31no—
38C22×C3×Dic3—61no—
39C22×C22⋊C9—21no—
40C22×C22⋊C9C2×D311yescharacteristic
41#3757—32no—
42#3757—32no—
43C3×#217—61no—
44C3×#217—61no—
45#3724—62no—
46#3724—61no—
47#3724—61no—
48C4×C22⋊C9—61no—
49(C22⋊C9)⋊C4—21no—
50(C22⋊C9)⋊C4—32no—
51(C22⋊C9)⋊C4—62no—
52Dic9⋊C4—121no—
53C22×C3×C9—41no—
54C3×C22⋊C9C4⋊C411yescharacteristic, complemented
55C3⋊Dic9—42no—
56C9×Dic3—81no—
57C3×Dic9—81no—
58C23×C3×C4—31no—
59C23×Dic3D911yescharacteristic
60C2×C3×C22⋊C4—91no—
61C2×#217—31no—
62C2×#217—91no—
63C2×#217—91no—
64C2×#217—181no—
65C2×Dic3⋊C4—91no—
66C2×Dic3⋊C4—91no—
67C2×Dic3⋊C4—91no—
68C2×Dic3⋊C4—91no—
69C3⋊#95—91no—
70C3⋊#95—91no—
71C3⋊#95—91no—
72C3⋊#95—91no—
73C23×C32—31no—
74C23×C32—31no—
75C23×C32—31no—
76C23×C32—31no—
77C23×C32Dic611yescharacteristic
78C23×C32C4×D311yescharacteristic
79C23×C32#6711yescharacteristic
80C2×C3⋊Dic3—32no—
81C2×C3⋊Dic3—32no—
82C2×C3⋊Dic3—61no—
83C2×C3⋊Dic3—61no—
84C2×C3⋊Dic3—61no—
85C2×C3⋊Dic3—61no—
86C2×C3×Dic3—32no—
87C2×C3×Dic3—31no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3S411yescharacteristic
98C2×C22⋊C9—21no—
99C2×C22⋊C9—21no—
100C2×C22⋊C9—21no—
101C2×C22⋊C9Dic611yescharacteristic
102C2×C22⋊C9C4×D311yescharacteristic
103C2×C22⋊C9#6711yescharacteristic
104C2×C4×C9—121no—
105C2×Dic9—41no—
106C2×Dic9—121no—
107C2×Dic9—241no—
108#385—271noSylow
109C2×C3×C9—41no—
110C2×C3×C9—41no—
111C2×C3×C9—41no—
112C24×C3—21no—
113C24×C3C2×D911yescharacteristic
114C24×C3D3211yescharacteristic
115C22×C3×C4—61no—
116C22×C3×C4—91no—
117C22×C3×C4—91no—
118C22×C3×C4—91no—
119C22×C3×C4—91no—
120C22×C3×C4—181no—
121C22×Dic3—21no—
122C22×Dic3—31no—
123C22×Dic3—31no—
124C22×Dic3—31no—
125C22×Dic3—31no—
126C22×Dic3—61no—
127C22×Dic3—91no—
128C22×Dic3—91no—
129C22×Dic3—91no—
130C22×Dic3—91no—
131C22×Dic3—181no—
132C22×Dic3—181no—
133C3×C22⋊C4—181no—
134C3×C22⋊C4—181no—
135#217—61no—
136#217—61no—
137#217—92no—
138#217—92no—
139#217—92no—
140#217—92no—
141#217—182no—
142#217—182no—
143Dic3⋊C4—182no—
144Dic3⋊C4—181no—
145Dic3⋊C4—181no—
146Dic3⋊C4—182no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149C22×C32—31no—
150C22×C32—31no—
151C22×C32—31no—
152C22×C32—31no—
153C22×C32—31no—
154C22×C32—31no—
155C22×C32—31no—
156C22×C32—31no—
157C22×C32—31no—
158C22×C32—61no—
159C22×C32Dic3⋊C411yescharacteristic
160C22×C32C2×S411yescharacteristic
161C3⋊Dic3—62no—
162C3⋊Dic3—62no—
163C3×Dic3—21no—
164C3×Dic3—61no—
165C3×Dic3—121no—
166C3×Dic3—121no—
167C22×C9—41no—
168C22×C9—81no—
169C22⋊C9—21no—
170C22⋊C9Dic3⋊C411yescharacteristic, complemented
171C4×C9—241no—
172Dic9—81no—
173Dic9—122no—
174Dic9—242no—
175C23×C4—31no—
176C2×C4⋊C4—271no—
177C2×C4⋊C4—271no—
178C2×C22⋊C4—91no—
179C2×C22⋊C4—271no—
180#95—271no—
181#95—271no—
182C3×C9—41noSylow
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—31no—
187C23×C3—31no—
188C23×C3—31no—
189C23×C3—31no—
190C23×C3—31no—
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—61no—
195C23×C3—61no—
196C23×C3—61no—
197C23×C3—61no—
198C23×C3Dic1811yescharacteristic
199C23×C3C4×D911yescharacteristic
200C23×C3#61111yescharacteristic
201C23×C3#62411yescharacteristic
202C23×C3#62511yescharacteristic
203C23×C3#62711yescharacteristic
204C2×C3×C4—31no—
205C2×C3×C4—92no—
206C2×C3×C4—91no—
207C2×C3×C4—181no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×Dic3—32no—
217C2×Dic3—31no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—92no—
228C2×Dic3—92no—
229C2×Dic3—92no—
230C2×Dic3—92no—
231C2×Dic3—182no—
232C2×Dic3—182no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235C2×Dic3—181no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238C2×Dic3—181no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241C2×Dic3—361no—
242C2×Dic3—361no—
243C2×Dic3—361no—
244C2×Dic3—361no—
245C2×Dic3(C22⋊C9)⋊C211yescharacteristic
246C2×C32—31no—
247C2×C32—31no—
248C2×C32—31no—
249C2×C32—31no—
250C2×C32A4⋊Q811yescharacteristic
251C2×C32C4×S411yescharacteristic
252C2×C32GL(2,ℤ/4)11yescharacteristic
253C2×C9—41no—
254C2×C9—41no—
255C2×C9—41no—
256C2×C9—81no—
257C2×C9—81no—
258C2×C9—81no—
259C24D3×D911yescharacteristic
260C22×C4—61no—
261C22×C4—91no—
262C22×C4—91no—
263C22×C4—91no—
264C22×C4—91no—
265C22×C4—181no—
266C22×C4—271no—
267C22×C4—271no—
268C4⋊C4—541no—
269C4⋊C4—542no—
270C4⋊C4—541no—
271C22⋊C4—181no—
272C22⋊C4—181no—
273C22⋊C4—272no—
274C22⋊C4—272no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—121no—
307C22×C3Dic9⋊C411yescharacteristic, complemented
308C22×C3#372411yescharacteristic
309C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
310C22×C3D3×S411yescharacteristic
311C3×C4—61no—
312C3×C4—181no—
313C3×C4—361no—
314C3×C4—361no—
315Dic3—21no—
316Dic3—61no—
317Dic3—121no—
318Dic3—121no—
319Dic3—182no—
320Dic3—182no—
321Dic3—182no—
322Dic3—182no—
323Dic3—362no—
324Dic3—362no—
325C32SL(2,ℤ/4)⋊C411yescharacteristic
326C9—41no—
327C9—81no—
328C23—31no—
329C23—31no—
330C23—31no—
331C23—31no—
332C23#618911yescharacteristic
333C23(C3⋊D9)⋊C411yescharacteristic
334C23#619411yescharacteristic
335C2×C4—31no—
336C2×C4—92no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—272no—
348C2×C4—272no—
349C2×C4—541no—
350C2×C4—541no—
351C2×C4—541no—
352C2×C4—541no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—61no—
365C2×C3—61no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3(C22⋊C9)⋊Q811yescharacteristic
369C2×C3C4×(C22⋊C9)⋊C211yescharacteristic
370C2×C3#6369411yescharacteristic
371C2×C3#6420411yescharacteristic
372C2×C3(C3⋊S4)⋊C411yescharacteristic
373C2×C3#6420911yescharacteristic
374C22—31no—
375C22—31no—
376C22—31no—
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—31no—
383C22—61no—
384C22#8844211yescharacteristic, complemented
385C22D3×(C22⋊C9)⋊C211yescharacteristic
386C4—61no—
387C4—181no—
388C4—361no—
389C4—361no—
390C4—542no—
391C4—542no—
392C3—21no—
393C3((C22⋊C9)⋊C4)⋊C411yescharacteristic, complemented
394C3#1059323811yescharacteristic
395C2—31no—
396C2—31no—
397C2—31no—
398C2—31no—
399C2#1172226111yescharacteristic
400C2(C3⋊((C22⋊C9)⋊C2))⋊C411yescharacteristic
401C2#1172226611yescharacteristic
subgroupG/H
center, hypercenterC22D3×(C22⋊C9)⋊C2
commutatorC2×C3×C22⋊C9C2×C4
FrattiniC22×C3D3×S4
Fitting, socle, prime radicalC24×C32C2×D3
cosocleC22×C3×C22⋊C9C22
hypocenterC3×C22⋊C9C4⋊C4
LCSG ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
8. C3⋊((C22⋊C9)⋊C4)C4semidirect
54. C3×C22⋊C9C4⋊C4semidirect
170. C22⋊C9Dic3⋊C4semidirect
307. C22×C3Dic9⋊C4semidirect
384. C22#88442semidirect
393. C3((C22⋊C9)⋊C4)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×C22⋊C92
C2×C3⋊((C22⋊C9)⋊C4)2
C2×C3×(C22⋊C9)⋊C42
#105916933
((C22⋊C9)⋊C4)⋊C43
#884424