Finite Group Databasebeta

← #72342835

Subgroups — #72342835

Order 3072.

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

normal subgroups only

This group has 343 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
1#72342697C211yesmaximal, characteristic
2A4×#1368C211yesmaximal, characteristic
3#72342686C211yesmaximal, characteristic
4#72224970—31nomaximal, Sylow
5#11711655C411yescharacteristic
6#11711655C411yescharacteristic
7#11711530—21no—
8A4×#346C2211yescharacteristic
9A4×#334—21no—
10#10682856—41nomaximal
11C22×#1368D311yescharacteristic
12#151891—31no—
13C2×#7457—31no—
14C2×#7457—31no—
15#134662—31no—
16#94030—31no—
17#94030—31no—
18C2×A4×C16C2×C411yescharacteristic
19C2×A4×C16Q811yescharacteristic
20A4×C4⋊C8—41no—
21C4×C8×A4D411yescharacteristic
22#73104—41no—
23A4⋊C32—21no—
24A4⋊C32C812yescomplemented
25C3×#1368—41no—
26#67519—41no—
27#67514—41no—
28C22×#346C2×D311yescharacteristic
29C22×#334—21no—
30#13553—31no—
31#13553—31no—
32C2×#1398—31no—
33C2×#1398—31no—
34#12788—61no—
35C2×#1369—31no—
36C2×#1369—31no—
37C2×#1368—31no—
38C2×#1368—31no—
39C2×#1368—61no—
40#7466—31no—
41#7466—31no—
42#7457—62no—
43#7457—61no—
44#7457—61no—
45#7161—61no—
46C2×C8×A4—81no—
47C2×C8×A4C4⋊C411yescharacteristic
48C2×C8×A4SD1611yescharacteristic
49C42×A4SD1611yescharacteristic
50A4×C16—21no—
51A4×C16C2×C811yescharacteristic
52A4×C16M1611yescharacteristic
53A4⋊C16—81no—
54C3×#346—41no—
55C3×#334—81no—
56#4499—41no—
57#4499—41no—
58#4463—81no—
59C23×C16Dic611yescharacteristic
60C23×C16C4×D311yescharacteristic
61C22×C4⋊C8—41no—
62C22×C4×C8#6711yescharacteristic
63C2×M64—31no—
64C2×M64—31no—
65C2×M64—31no—
66C2×M64—31no—
67C2×#364—61no—
68C2×#364—61no—
69#2112—121no—
70C2×#346—31no—
71C2×#346—31no—
72C2×#346—31no—
73C2×#346—31no—
74C2×#334—61no—
75C2×#334—61no—
76C2×#334—121no—
77C22⋊C32—32no—
78C22⋊C32—32no—
79C22⋊C32—61no—
80C22⋊C32—61no—
81#1398—61no—
82#1398—61no—
83#1398—61no—
84#1398—61no—
85#1376—121no—
86#1369—61no—
87#1369—61no—
88#1369—61no—
89#1369—61no—
90#1368—32no—
91#1368—31no—
92#1368—61no—
93#1368S411yescharacteristic
94C2×C4×A4—21no—
95C2×C4×A4#10611yescharacteristic
96C8×A4—21no—
97C8×A4—161no—
98C8×A4C4⋊C811yescharacteristic
99C8×A4#10811yescharacteristic
100C2×C3×C16—41no—
101C2×C3×C16—41no—
102C3×C4⋊C8—161no—
103C3×C4×C8—41no—
104#812—161no—
105C3⋊C32—42no—
106C3⋊C32—81no—
107C23×C8—81no—
108C23×C8Dic3⋊C411yescharacteristic
109C23×C8#21511yescharacteristic
110C22×C42#21411yescharacteristic
111C2×M32—121no—
112C2×M32—121no—
113C22×C16—21no—
114C22×C16—31no—
115C22×C16—31no—
116C22×C16—31no—
117C22×C16—31no—
118C22×C16—31no—
119C22×C16—31no—
120C22×C16—61no—
121C22×C16D3×C811yescharacteristic
122C22×C16ΓL(1,𝔽25)11yescharacteristic
123C2×C4⋊C8—121no—
124C2×C4⋊C8—121no—
125C2×C4⋊C8—241no—
126C2×C4×C8—31no—
127C2×C4×C8—31no—
128C2×C4×C8—31no—
129C2×C4×C8—31no—
130M64—61no—
131M64—61no—
132M64—61no—
133M64—61no—
134M64—61no—
135M64—61no—
136M64—61no—
137M64—61no—
138C2×C32—32no—
139C2×C32—32no—
140C2×C32—61no—
141C2×C32—61no—
142#364—121no—
143#364—121no—
144#364—121no—
145#364—121no—
146C22⋊C16—241no—
147C22⋊C16—241no—
148#346—31no—
149#346—31no—
150#346—31no—
151#346—61no—
152#346C2×S411yescharacteristic
153#334—21no—
154#334—61no—
155#334—121no—
156#334—121no—
157C22×A4#36511yescharacteristic
158C4×A4—41no—
159C4×A4#33411yescharacteristic
160C4×A4#36511yescharacteristic
161C2×C3×C8—41no—
162C2×C3×C8—41no—
163C2×C3×C8—321no—
164C3×C42—41no—
165C3×C16—41no—
166C3×C16—41no—
167C3×C16—81no—
168C3⋊C16—321no—
169C23×C4—21no—
170C23×C4#80811yescharacteristic
171C22×C8—21no—
172C22×C8—31no—
173C22×C8—31no—
174C22×C8—31no—
175C22×C8—31no—
176C22×C8—31no—
177C22×C8—31no—
178C22×C8—61no—
179C22×C8—161no—
180C22×C8—241no—
181C22×C8—241no—
182C22×C8—481no—
183C22×C8Dic3⋊C811yescharacteristic
184C22×C8#82211yescharacteristic
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—61no—
188M32—241no—
189M32—241no—
190M32—241no—
191M32—241no—
192C2×C16—31no—
193C2×C16—31no—
194C2×C16—31no—
195C2×C16—31no—
196C2×C16—31no—
197C2×C16—31no—
198C2×C16—31no—
199C2×C16—31no—
200C2×C16—61no—
201C2×C16—61no—
202C2×C16—61no—
203C2×C16—61no—
204C2×C16—61no—
205C2×C16—241no—
206C2×C16—241no—
207C2×C16A4⋊Q811yescharacteristic
208C2×C16C4×S411yescharacteristic
209C4⋊C8—41no—
210C4⋊C8—121no—
211C4⋊C8—241no—
212C4⋊C8—241no—
213C4×C8—31no—
214C4×C8—31no—
215C4×C8—31no—
216C4×C8—61no—
217C4×C8GL(2,ℤ/4)11yescharacteristic
218C32—62no—
219C32—62no—
220C32—121no—
221C32—121no—
222C2×A4—21no—
223C2×A4#136811yescharacteristic
224C2×C3×C4—41no—
225C2×C3×C4—81no—
226C3×C8—41no—
227C3×C8—41no—
228C3×C8—81no—
229C3×C8—641no—
230C24#448711yescharacteristic
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—41no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—121no—
239C22×C4#448011yescharacteristic
240C22×C4#449411yescharacteristic
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—81no—
255C2×C8—241no—
256C2×C8—481no—
257C2×C8—481no—
258C2×C8—481no—
259C2×C8—481no—
260C2×C8SL(2,ℤ/4)⋊C411yescharacteristic
261C2×C8#541711yescharacteristic
262C42—32no—
263C42—31no—
264C42—61no—
265C42#541411yescharacteristic
266C16—21no—
267C16—31no—
268C16—31no—
269C16—61no—
270C16—481no—
271C16—481no—
272C16C8×S411yescharacteristic
273C16#540011yescharacteristic
274C22×C3—41no—
275A4#745711yescharacteristic, complemented
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—161no—
279C23—21no—
280C23—31no—
281C23—31no—
282C23—61no—
283C23#6746411yescharacteristic
284C2×C4—21no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4#7305111yescharacteristic
299C8—21no—
300C8—31no—
301C8—31no—
302C8—61no—
303C8—161no—
304C8—481no—
305C8SL(2,ℤ/4)⋊C811yescharacteristic
306C8#7311411yescharacteristic
307C2×C3—41no—
308C2×C3—81no—
309C22—31no—
310C22—31no—
311C22—32no—
312C22—31no—
313C22—61no—
314C22—61no—
315C22—61no—
316C22#1068285611yescharacteristic, complemented
317C22#1171155311yescharacteristic
318C4—31no—
319C4—31no—
320C4—41no—
321C4—121no—
322C4#1171108111yescharacteristic
323C4#1171155211yescharacteristic
324C3—41noSylow
325C2—21no—
326C2—31no—
327C2—31no—
328C2—61no—
329C2#7234274411yescharacteristic
subgroupG/H
centerC2#72342744
commutatorA4×C16C2×C8
Frattini#346C2×S4
FittingC22×#1368D3
socleC23#67464
prime radicalC22×A4#365
cosocleA4×#346C22
hypercenter#1368S4
hypocenterA4#7457
LCSG ▷ A4×C16 ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ #1368

Complements

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

NH ≅ G/Ntype
24. A4⋊C32C8semidirect
275. A4#7457semidirect
316. C22#10682856semidirect

Maximal subgroups

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

subgroupindex
#723426972
A4×#13682
#723426862
#722249703
#106828564