Finite Group Databasebeta

← #120595

Subgroups — #120595

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21370C211yesmaximal, characteristic
2C4×#428C211yesmaximal, characteristic
3#8097C211yesmaximal, characteristic
4C2×#1303C211yesmaximal, characteristic
5#7950C211yesmaximal, characteristic
6#7938C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1950C412yescomplemented
17#1950C412yescomplemented
18#1855—21no—
19#1855—21no—
20C4×#108—22no—
21#1776—22no—
22C4×#107—21no—
23C4×#106—21no—
24#1736—21no—
25#1303—22no—
26#1303—21no—
27#1303—21no—
28#1285—22no—
29#1283—22no—
30#1276—22no—
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C22×M16D411yescharacteristic
34C2×#117—21no—
35C22×C4⋊C4—21no—
36C22×C42—21no—
37C22×C42C2311yescharacteristic
38#432—21no—
39#432—21no—
40#432—21no—
41#432Q812yes—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431D412yes—
49#431Q812yes—
50C2×#108—22no—
51C2×#108—22no—
52#428—28no—
53C2×#107—22no—
54C2×#106—22no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423—21no—
59#423—21no—
60#423C2×C411yescharacteristic
61#423C2×C411yescharacteristic
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2×C412yes—
64C2×C4⋊C8Q812yes—
65#407—21no—
66#407—21no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16C2×C412yes—
73C4×M16D412yes—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4D412yes—
82C4×C4⋊C4D411yescharacteristic
83C4×C4⋊C4D411yescharacteristic
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86#376—21no—
87C2×#95—21no—
88C43—21no—
89#341—22no—
90#341—22no—
91#340—22no—
92#340—22no—
93C23×C4—41no—
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—21no—
98C2×M16—24no—
99C2×M16—21no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—42no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—81no—
107C2×M16—81no—
108C2×M16C22⋊C412yes—
109C2×M16C22⋊C412yes—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—21no—
113C22×C8—22no—
114#117—24no—
115#117—42no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—81no—
130C2×C42—81no—
131C2×C42C22×C411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×Q811yescharacteristic
137C2×C42C2×Q811yescharacteristic
138C2×C42C2×Q811yescharacteristic
139C2×C42#4111yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143#108—44no—
144#108—44no—
145#107—28no—
146#106—28no—
147C4⋊C8—22no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8C4⋊C414yes—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165#97—22no—
166#97—22no—
167#97—22no—
168#97—41no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—41no—
174C4×C8—41no—
175#95—22no—
176#95—21no—
177#95—21no—
178#95—41no—
179#95—41no—
180C24#12411yescharacteristic
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—44no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C4×Q811yescharacteristic
196C22×C4C4×Q811yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4#12311yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12511yescharacteristic
209C22×C4#12611yescharacteristic
210C22×C4#12611yescharacteristic
211C22×C4C4⋊Q811yescharacteristic
212M16—24no—
213M16—22no—
214M16—22no—
215M16—22no—
216M16—42no—
217M16—42no—
218M16—42no—
219M16—82no—
220M16—82no—
221M16—81no—
222M16—81no—
223M16#10414yes—
224C2×C8—24no—
225C2×C8—22no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—82no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—44no—
244C42—44no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—82no—
250C42—82no—
251C42—81no—
252C42—81no—
253C42C2×C4⋊C411yescharacteristic
254C42#11711yescharacteristic
255C42C4×D411yescharacteristic
256C42C4×D411yescharacteristic
257C42#12111yescharacteristic
258C42#12111yescharacteristic
259C42#12111yescharacteristic
260C42C22⋊Q811yescharacteristic
261C42C22⋊Q811yescharacteristic
262C42C22⋊Q811yescharacteristic
263C42C22⋊Q811yescharacteristic
264C42C22⋊Q811yescharacteristic
265C42#12511yescharacteristic
266C42#12511yescharacteristic
267C42C4⋊Q811yescharacteristic
268C42C4⋊Q811yescharacteristic
269C23—21no—
270C23—21no—
271C23—22no—
272C23—41no—
273C23#39111yescharacteristic
274C23#39411yescharacteristic
275C23#40011yescharacteristic
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—88no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4#38011yescharacteristic
310C2×C4#38611yescharacteristic
311C2×C4#38711yescharacteristic
312C2×C4#38711yescharacteristic
313C2×C4#38911yescharacteristic
314C2×C4#38911yescharacteristic
315C2×C4#39511yescharacteristic
316C2×C4#39611yescharacteristic
317C2×C4#39611yescharacteristic
318C2×C4#39811yescharacteristic
319C2×C4#39811yescharacteristic
320C2×C4#39811yescharacteristic
321C2×C4C2×#10412yes—
322C2×C4#42112yes—
323C2×C4#47111yescharacteristic
324C2×C4#47111yescharacteristic
325C2×C4#47211yescharacteristic
326C2×C4#47311yescharacteristic
327C8—24no—
328C8—44no—
329C8—42no—
330C8—82no—
331C8—82no—
332C8—81no—
333C8—81no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22#144111yescharacteristic
344C22#198612yes—
345C22#205111yescharacteristic
346C22#205211yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—82no—
356C4—82no—
357C4—84no—
358C4#180612yes—
359C4#189611yescharacteristic
360C4#189711yescharacteristic
361C4#195111yescharacteristic
362C4#195211yescharacteristic
363C4#206311yescharacteristic
364C4#206411yescharacteristic
365C2—21no—
366C2—21no—
367C2—22no—
368C2—41no—
369C2#1122611yescharacteristic
370C2#1124911yescharacteristic
371C2#1125111yescharacteristic
subgroupG/H
centerC2×C4#396
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1441
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1950C4semidirect
17. #1950C4semidirect

Maximal subgroups

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

subgroupindex
#213702
C4×#4282
#80972
C2×#13032
#79502
#79382
#78692