Finite Group Databasebeta

← #13166113

Subgroups — #13166113

Order 1728.

586 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 74 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×A4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×C3⋊SL(2,ℤ/4)C212yesmaximal
3Dic3⋊SL(2,ℤ/4)—32nomaximal
4#10591720—31nomaximal
5C22×C32×A4C2211yescharacteristic
6A4×C3⋊Dic3C412yescomplemented
7C3×C3⋊SL(2,ℤ/4)—22no—
8#88815—41nomaximal
9C23×C3⋊Dic3D311yescharacteristic
10C2×Dic3×A4—32no—
11C2×Dic3×A4—61no—
12C2×C3⋊SL(2,ℤ/4)D312yes—
13C2×C3×SL(2,ℤ/4)—32no—
14C2×C3×SL(2,ℤ/4)—62no—
15C2×C3×#217—32no—
16C2×#3724—31no—
17C2×#3724—31no—
18#63579—32no—
19C2×C32×A4C2×C411yescharacteristic
20C2×C32×A4D411yescharacteristic
21C2×C32×A4Q811yescharacteristic
22C2×C3×C3⋊Dic3—41no—
23C2×C3×C3⋊Dic3—42no—
24C4⋊SL(2,ℤ/4)—91no—
25#4942—92no—
26C24×C32C2×D311yescharacteristic
27C22×C3×A4—22no—
28C22×C3×A4—21no—
29C22×C3×A4—41no—
30C22×C3×A4C2×D312yes—
31C22×C3⋊Dic3—31no—
32C22×C3⋊Dic3—31no—
33C22×C3⋊Dic3—32no—
34C22×C3⋊Dic3Dic312yescomplemented
35C22×C3×Dic3—32no—
36C22×C3×Dic3—32no—
37Dic3×A4—34no—
38Dic3×A4—62no—
39C3⋊SL(2,ℤ/4)—22no—
40C3×SL(2,ℤ/4)—64no—
41C3×SL(2,ℤ/4)—62no—
42C3×#217—62no—
43C3×#217—62no—
44#3724—62no—
45#3724—61no—
46#3724—61no—
47Dic3⋊Dic3—122no—
48C22×C33—41no—
49C32×A4C4⋊C411yescharacteristic, complemented
50C3×C3⋊Dic3—42no—
51C3×C3⋊Dic3—82no—
52C23×Dic3—32no—
53C23×Dic3—61no—
54C2×C4×A4—91no—
55C2×SL(2,ℤ/4)—32no—
56C2×SL(2,ℤ/4)—62no—
57C2×C3×C22⋊C4—92no—
58C2×#217—32no—
59C2×Dic3⋊C4—92no—
60C2×Dic3⋊C4—92no—
61C3⋊#95—92no—
62C3⋊#95—92no—
63C23×C32—31no—
64C23×C32—31no—
65C23×C32—31no—
66C23×C32—31no—
67C23×C32Dic611yescharacteristic
68C23×C32D1211yescharacteristic
69C23×C32C2×Dic311yescharacteristic
70C2×C3×A4—22no—
71C2×C3×A4—22no—
72C2×C3×A4—22no—
73C2×C3×A4—21no—
74C2×C3×A4—21no—
75C2×C3×A4—21no—
76C2×C3×A4—41no—
77C2×C3×A4—41no—
78C2×C3×A4—41no—
79C2×C3×A4Dic612yes—
80C2×C3×A4C4×D312yes—
81C2×C3×A4#6712yes—
82C2×C3⋊Dic3—31no—
83C2×C3⋊Dic3—32no—
84C2×C3⋊Dic3—32no—
85C2×C3⋊Dic3—32no—
86C2×C3⋊Dic3—42no—
87C2×C3⋊Dic3—61no—
88C2×C3⋊Dic3S411yescharacteristic
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—122no—
96C2×C3×Dic3—122no—
97C2×C3×Dic3—241no—
98C2×C3×Dic3—242no—
99#380—271noSylow
100C2×C33—41no—
101C2×C33—41no—
102C2×C33—41no—
103C24×C3—21no—
104C24×C3D3212yes—
105C22×A4—22no—
106C22×A4—41no—
107C22×A4D3211yescharacteristic
108C22×C3×C4—92no—
109C22×C3×C4—92no—
110C22×Dic3—32no—
111C22×Dic3—32no—
112C22×Dic3—34no—
113C22×Dic3—62no—
114C22×Dic3—92no—
115C22×Dic3—92no—
116C22×Dic3—94no—
117C22×Dic3—181no—
118C22×Dic3—181no—
119C22×Dic3—182no—
120C4×A4—92no—
121SL(2,ℤ/4)—64no—
122SL(2,ℤ/4)—62no—
123C3×C22⋊C4—182no—
124C3×C22⋊C4—182no—
125#217—62no—
126#217—62no—
127C4⋊Dic3—361no—
128Dic3⋊C4—182no—
129Dic3⋊C4—184no—
130Dic3⋊C4—182no—
131C22×C32—31no—
132C22×C32—31no—
133C22×C32—31no—
134C22×C32—31no—
135C22×C32—31no—
136C22×C32—31no—
137C22×C32—31no—
138C22×C32—31no—
139C22×C32—31no—
140C22×C32—42no—
141C22×C32—61no—
142C22×C32—81no—
143C22×C32—82no—
144C22×C32—161no—
145C22×C32C4⋊Dic311yescharacteristic, complemented
146C22×C32C2×S411yescharacteristic
147C3×A4—22no—
148C3×A4—21no—
149C3×A4—41no—
150C3×A4Dic3⋊C412yescomplemented
151C3⋊Dic3—32no—
152C3⋊Dic3—82no—
153C3⋊Dic3SL(2,ℤ/4)12yescomplemented
154C3×Dic3—124no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157C3×Dic3—242no—
158C3×Dic3—242no—
159C3×Dic3—244no—
160C23×C4—91no—
161C2×C4⋊C4—271no—
162C2×C4⋊C4—271no—
163C2×C22⋊C4—92no—
164#95—272no—
165C33—41noSylow
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—32no—
170C23×C3—32no—
171C23×C3—32no—
172C23×C3—32no—
173C23×C3—61no—
174C23×C3—61no—
175C23×C3—61no—
176C23×C3—61no—
177C23×C3D3×Dic312yes—
178C23×C3#62612yes—
179C23×C3#62712yes—
180C2×A4—22no—
181C2×A4—22no—
182C2×A4—22no—
183C2×A4—41no—
184C2×A4—41no—
185C2×A4—41no—
186C2×A4#62411yescharacteristic
187C2×A4#62511yescharacteristic
188C2×A4#62711yescharacteristic
189C2×C3×C4—182no—
190C2×C3×C4—182no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4—361no—
196C2×Dic3—32no—
197C2×Dic3—61no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—94no—
205C2×Dic3—94no—
206C2×Dic3—92no—
207C2×Dic3—94no—
208C2×Dic3—122no—
209C2×Dic3—182no—
210C2×Dic3—182no—
211C2×Dic3—181no—
212C2×Dic3—182no—
213C2×Dic3—182no—
214C2×Dic3—182no—
215C2×Dic3—242no—
216C2×C32—31no—
217C2×C32—31no—
218C2×C32—31no—
219C2×C32—31no—
220C2×C32—42no—
221C2×C32—42no—
222C2×C32—42no—
223C2×C32—81no—
224C2×C32—81no—
225C2×C32—81no—
226C2×C32—82no—
227C2×C32—82no—
228C2×C32—82no—
229C2×C32—161no—
230C2×C32—161no—
231C2×C32—161no—
232C2×C32A4⋊Q811yescharacteristic
233C2×C32C4⋊S411yescharacteristic
234C2×C32C2×SL(2,ℤ/4)11yescharacteristic
235C24#111311yescharacteristic
236C22×C4—92no—
237C22×C4—92no—
238C22×C4—92no—
239C22×C4—271no—
240C22×C4—271no—
241C22×C4—272no—
242C4⋊C4—541no—
243C4⋊C4—542no—
244C4⋊C4—541no—
245C22⋊C4—182no—
246C22⋊C4—182no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—32no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—32no—
256C22×C3—32no—
257C22×C3—32no—
258C22×C3—41no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—62no—
269C22×C3—62no—
270C22×C3—82no—
271C22×C3—161no—
272C22×C3Dic3⋊Dic312yescomplemented
273C22×C3D3×S412yes—
274A4—22no—
275A4—41no—
276A4#372411yescharacteristic, complemented
277C3×C4—362no—
278C3×C4—362no—
279C3×C4—362no—
280Dic3—34no—
281Dic3—62no—
282Dic3—94no—
283Dic3—122no—
284Dic3—122no—
285Dic3—182no—
286Dic3—242no—
287Dic3—244no—
288C32—42no—
289C32—81no—
290C32—82no—
291C32—161no—
292C32C4⋊SL(2,ℤ/4)11yescharacteristic, complemented
293C23—31no—
294C23—31no—
295C23—31no—
296C23—31no—
297C23#629411yescharacteristic
298C23#629511yescharacteristic
299C23#629611yescharacteristic
300C2×C4—91no—
301C2×C4—182no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4—182no—
305C2×C4—182no—
306C2×C4—182no—
307C2×C4—272no—
308C2×C4—272no—
309C2×C4—271no—
310C2×C4—272no—
311C2×C4—541no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—32no—
316C2×C3—32no—
317C2×C3—32no—
318C2×C3—32no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3—82no—
327C2×C3—82no—
328C2×C3—82no—
329C2×C3—161no—
330C2×C3—161no—
331C2×C3—161no—
332C2×C3#6420412yes—
333C2×C3#6420712yes—
334C2×C3D3×SL(2,ℤ/4)12yes—
335C22—31no—
336C22—31no—
337C22—31no—
338C22—31no—
339C22—31no—
340C22—31no—
341C22—31no—
342C22—31no—
343C22—31no—
344C22—61no—
345C22#8881511yescharacteristic, complemented
346C22#8910111yescharacteristic
347C4—92no—
348C4—272no—
349C4—362no—
350C4—362no—
351C3—21no—
352C3—41no—
353C3—82no—
354C3—161no—
355C3Dic3⋊SL(2,ℤ/4)12yescomplemented
356C2—31no—
357C2—31no—
358C2—31no—
359C2—31no—
360C2#1172421211yescharacteristic
361C2#1172421411yescharacteristic
362C2#1172421511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutatorC2×C32×A4C2×C4
Fitting, socleC24×C32C2×D3
prime radical, cosocleC22×C32×A4C22
hypocenterC32×A4C4⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
6. A4×C3⋊Dic3C4semidirect
34. C22×C3⋊Dic3Dic3semidirect
49. C32×A4C4⋊C4semidirect
145. C22×C32C4⋊Dic3semidirect
150. C3×A4Dic3⋊C4semidirect
153. C3⋊Dic3SL(2,ℤ/4)semidirect
272. C22×C3Dic3⋊Dic3semidirect
276. A4#3724semidirect
292. C32C4⋊SL(2,ℤ/4)semidirect
345. C22#88815semidirect
355. C3Dic3⋊SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C3⋊Dic32
C2×C3×C3⋊SL(2,ℤ/4)2
Dic3⋊SL(2,ℤ/4)3
#105917203
#888154