Finite Group Databasebeta

← #72321268

Subgroups — #72321268

Order 2048.

490 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72321153C212yesmaximal, complemented
2#72321129C212yesmaximal, complemented
3#72286896C211yesmaximal, characteristic
4#72286396C211yesmaximal, characteristic
5#72234369C211yesmaximal, characteristic
6C22×#1397C2211yescharacteristic
7#483403—21no—
8#153427C412yes—
9C2×#7511C2212yes—
10#151054—24no—
11C2×#7464C2211yescharacteristic
12C2×#7464C2211yescharacteristic
13#151037—21no—
14#151037C2212yes—
15#134679—21no—
16#94756C411yescharacteristic
17#94756C411yescharacteristic
18#94754—24no—
19#94028—21no—
20C23×C32C2×C411yescharacteristic
21C23×C32Q811yescharacteristic
22C22×C4×C16D411yescharacteristic
23#17793—41no—
24C2×M128C2×C412yes—
25C2×M128C2×C412yes—
26C2×#1397—22no—
27C2×#1397—22no—
28C2×#1397C2×C411yescharacteristic
29C2×#1397D411yescharacteristic
30C2×#1397Q811yescharacteristic
31C2×#1397C2311yescharacteristic
32#12788—41no—
33#12024—22no—
34#12024—41no—
35#12023—22no—
36#12022—41no—
37C2×#1373—21no—
38C2×#1373—21no—
39#12008—22no—
40#12008—41no—
41C22⋊C64—24no—
42#7511—22no—
43#7511—22no—
44#7464—22no—
45#7464—22no—
46#7463—24no—
47#7463—22no—
48#7463D414yes—
49#7160—41no—
50C23×C16C4⋊C411yescharacteristic
51C23×C16Q1611yescharacteristic
52#2920—81no—
53C22×C4×C8D811yescharacteristic
54C22×C32—22no—
55C22×C32—22no—
56C22×C32—22no—
57C22×C32—22no—
58C22×C32—21no—
59C22×C32—21no—
60C22×C32—41no—
61C22×C32C4⋊C411yescharacteristic
62C22×C32C4⋊C411yescharacteristic
63C22×C32C22×C411yescharacteristic
64C22×C32C2×Q811yescharacteristic
65C2×#364—41no—
66C2×#364—41no—
67#2112—81no—
68C2×C4×C16—21no—
69C2×C4×C16—21no—
70C2×C4×C16—21no—
71C2×C4×C16—21no—
72C2×C4×C16C4⋊C411yescharacteristic
73C2×C4×C16D811yescharacteristic
74C2×C4×C16Q1611yescharacteristic
75C2×C4×C16C2×D411yescharacteristic
76#1568—42no—
77#1568—81no—
78#1567—42no—
79#1566—81no—
80#1564—42no—
81#1564—81no—
82M128—22no—
83M128—22no—
84M128—22no—
85M128—22no—
86C2×C64—22no—
87C2×C64M1614yescomplemented
88#1397—22no—
89#1397—22no—
90#1397—21no—
91#1397—21no—
92#1397C2×D412yes—
93#1397#4112yes—
94#1376—81no—
95#1373—41no—
96#1373—41no—
97#1373—41no—
98#1373—41no—
99#1372—24no—
100#1372—42no—
101#1371—42no—
102#1371—42no—
103C23×C8#10711yescharacteristic
104C23×C8SD3211yescharacteristic
105C22×C42SD3211yescharacteristic
106C2×M32—81no—
107C2×M32—81no—
108C22×C16—22no—
109C22×C16—21no—
110C22×C16—21no—
111C22×C16—21no—
112C22×C16—21no—
113C22×C16—21no—
114C22×C16—21no—
115C22×C16—21no—
116C22×C16—21no—
117C22×C16—41no—
118C22×C16#10711yescharacteristic
119C22×C16#10711yescharacteristic
120C22×C16C2×C4⋊C411yescharacteristic
121C22×C16C2×Q1611yescharacteristic
122#432—82no—
123#432—161no—
124#431—82no—
125#431—161no—
126#423—82no—
127#407—161no—
128C4×M16—161no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8#10711yescharacteristic
134C2×C4×C8SD3211yescharacteristic
135C2×C4×C8SD3211yescharacteristic
136C2×C4×C8C2×D811yescharacteristic
137C2×C32—21no—
138C2×C32—21no—
139C2×C32—21no—
140C2×C32—21no—
141C2×C32—21no—
142C2×C32—21no—
143C2×C32—21no—
144C2×C32—44no—
145C2×C32—42no—
146C2×C32—42no—
147C2×C32#10812yes—
148C2×C32C4×D412yes—
149C2×C32C22⋊Q812yes—
150C2×C32C2×M1612yes—
151C2×C32#13111yescharacteristic
152C2×C32#13111yescharacteristic
153#364—81no—
154#364—81no—
155#364—81no—
156#364—81no—
157C22⋊C16—162no—
158C4×C16—21no—
159C4×C16—21no—
160C4×C16—21no—
161C4×C16—21no—
162C4×C16—21no—
163C4×C16—21no—
164C4×C16#12011yescharacteristic
165C4×C16#12311yescharacteristic
166C4×C16AGL(1,ℤ/8)11yescharacteristic
167C4×C16#13711yescharacteristic
168#335—44no—
169#335—82no—
170#334—82no—
171#334—82no—
172C64—24no—
173C64—42no—
174C23×C4—21no—
175C23×C4#36711yescharacteristic
176C2×M16—162no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—41no—
187C22×C8#36711yescharacteristic
188C22×C8#36711yescharacteristic
189C22×C8C2×#10711yescharacteristic
190C22×C8C2×SD3211yescharacteristic
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42#36711yescharacteristic
197C2×C42C2×SD3211yescharacteristic
198M32—161no—
199M32—161no—
200M32—161no—
201M32—161no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—42no—
210C2×C16—42no—
211C2×C16—41no—
212C2×C16—41no—
213C2×C16—41no—
214C2×C16—41no—
215C2×C16—161no—
216C2×C16—161no—
217C2×C16#36812yes—
218C2×C16#38511yescharacteristic
219C2×C16#42311yescharacteristic
220C2×C16#42411yescharacteristic
221C2×C16#42811yescharacteristic
222C2×C16C2×#10811yescharacteristic
223C2×C16#43011yescharacteristic
224C2×C16#45111yescharacteristic
225C2×C16#48411yescharacteristic
226C4⋊C8—84no—
227C4⋊C8—162no—
228C22⋊C8—162no—
229C22⋊C8—162no—
230#97—162no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—162no—
238C4×C8#44711yescharacteristic
239C4×C8#48011yescharacteristic
240C4×C8#50911yescharacteristic
241C4×C8#51011yescharacteristic
242C32—22no—
243C32—22no—
244C32—41no—
245C32#43512yes—
246C32#43612yes—
247C24#142611yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—82no—
261C22×C4#142611yescharacteristic
262C22×C4C2×#36711yescharacteristic
263M16—324no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—164no—
278C2×C8#142612yes—
279C2×C8#156711yescharacteristic
280C2×C8#157111yescharacteristic
281C2×C8#185211yescharacteristic
282C2×C8#215911yescharacteristic
283C2×C8C2×#36811yescharacteristic
284C2×C8#216111yescharacteristic
285C2×C8#218511yescharacteristic
286C2×C8#223311yescharacteristic
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42#142611yescharacteristic
294C42#142611yescharacteristic
295C42#218711yescharacteristic
296C42#223511yescharacteristic
297C16—22no—
298C16—21no—
299C16—21no—
300C16—41no—
301C16—321no—
302C16—321no—
303C16#156911yescharacteristic
304C16#157411yescharacteristic
305C16#185111yescharacteristic
306C16#185511yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23C2×#142611yescharacteristic
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4#1202311yescharacteristic
330C2×C4#1203311yescharacteristic
331C2×C4#1292011yescharacteristic
332C2×C4C2×#142611yescharacteristic
333C2×C4C2×#142611yescharacteristic
334C2×C4#1363811yescharacteristic
335C8—21no—
336C8—21no—
337C8—22no—
338C8—41no—
339C8—324no—
340C8#883211yescharacteristic
341C8#1202511yescharacteristic
342C8#1203611yescharacteristic
343C8#1292211yescharacteristic
344C22—22no—
345C22—22no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#15105411yescharacteristic
353C22#15106811yescharacteristic
354C22#15242711yescharacteristic
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4#13584011yescharacteristic
361C4#15105511yescharacteristic
362C4#15106711yescharacteristic
363C4#15242611yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2#7232108311yescharacteristic
subgroupG/H
centerC4#151055
commutatorC22×C32C22×C4
Frattini, cosocleC2×#1397C23
socleC2#72321083
prime radicalC24#1426
LCSG ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C4×C8 ◁ C22×C4×C16 ◁ C22×#1397 ◁ G
LECSG ▷ C2×#1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72321153C2semidirect
2. #72321129C2semidirect
87. C2×C64M16semidirect

Maximal subgroups

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

subgroupindex
#723211532
#723211292
#722868962
#722863962
#722343692