Finite Group Databasebeta

← #120917

Subgroups — #120917

Order 512.

444 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21960C211yesmaximal, characteristic
2#10853C211yesmaximal, characteristic
3#8074C211yesmaximal, characteristic
4#7974C211yesmaximal, characteristic
5#7962C211yesmaximal, characteristic
6#7960C211yesmaximal, characteristic
7#7935C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—21no—
16#1949—21no—
17#1949—21no—
18#1949—21no—
19#1808—21no—
20#1807—21no—
21C2×#341—21no—
22C2×#341—21no—
23#1300—22no—
24#1288—22no—
25#1286—21no—
26#1286—21no—
27#1286—21no—
28#1286—21no—
29#1283C414yescomplemented
30C22×M16—21no—
31C22×M16—21no—
32C23×C8D411yescharacteristic
33C23×C8D411yescharacteristic
34C22×C42C2311yescharacteristic
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431C2×C412yes—
43C2×#107—21no—
44C2×#107—21no—
45C2×#106—21no—
46C2×#106—21no—
47#423—21no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423Q812yes—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2×C412yes—
60C2×C4⋊C8Q812yes—
61#407—21no—
62#407—21no—
63C4×M16—21no—
64C4×M16—21no—
65C4×M16—21no—
66C4×M16—21no—
67C4×M16C2×C412yes—
68C4×M16D412yes—
69C2×#97—21no—
70C2×C4×C8—21no—
71#383—21no—
72#383—21no—
73#382D411yescharacteristic
74#382D411yescharacteristic
75#382D411yescharacteristic
76#382D411yescharacteristic
77#381—21no—
78#381—21no—
79#380—21no—
80#380—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85#376—21no—
86#376—21no—
87#341—42no—
88#341—42no—
89#340—21no—
90#340—21no—
91#340—21no—
92#340—21no—
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8C4⋊C412yes—
121C22×C8C4⋊C412yes—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42C22×C411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×Q811yescharacteristic
135C2×C42C2×Q811yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42#4111yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142#107—81no—
143#107—81no—
144#106—81no—
145#106—81no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162#97—22no—
163#97—22no—
164#97—41no—
165#97—41no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—41no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—21no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182C24#12411yescharacteristic
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—81no—
193C22×C4C2×C4⋊C411yescharacteristic
194C22×C4#11711yescharacteristic
195C22×C4#11711yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12311yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4#12411yescharacteristic
205C22×C4#12511yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12511yescharacteristic
209C22×C4#12611yescharacteristic
210C22×C4#12611yescharacteristic
211M16—24no—
212M16—42no—
213M16—42no—
214M16—42no—
215M16—44no—
216M16—42no—
217M16—42no—
218M16—82no—
219M16—81no—
220M16—81no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—81no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—81no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C42—82no—
250C42—82no—
251C42C4×D411yescharacteristic
252C42C4×D411yescharacteristic
253C42C4×D411yescharacteristic
254C42C4×Q811yescharacteristic
255C42C22⋊Q811yescharacteristic
256C42C22⋊Q811yescharacteristic
257C42C22⋊Q811yescharacteristic
258C42C22⋊Q811yescharacteristic
259C42#12311yescharacteristic
260C42#12311yescharacteristic
261C42#12411yescharacteristic
262C42#12411yescharacteristic
263C42#12511yescharacteristic
264C42#12611yescharacteristic
265C42#12611yescharacteristic
266C42C4⋊Q811yescharacteristic
267C23—22no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23#38211yescharacteristic
272C23#39411yescharacteristic
273C23#40011yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#38511yescharacteristic
309C2×C4#38711yescharacteristic
310C2×C4#38711yescharacteristic
311C2×C4#38811yescharacteristic
312C2×C4#38811yescharacteristic
313C2×C4#38911yescharacteristic
314C2×C4#39311yescharacteristic
315C2×C4#39711yescharacteristic
316C2×C4#39811yescharacteristic
317C2×C4#39811yescharacteristic
318C2×C4#40011yescharacteristic
319C2×C4#40011yescharacteristic
320C2×C4#45311yescharacteristic
321C2×C4#45411yescharacteristic
322C2×C4#45511yescharacteristic
323C2×C4#45611yescharacteristic
324C8—44no—
325C8—42no—
326C8—42no—
327C8—82no—
328C8—82no—
329C8—82no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#144211yescharacteristic
340C22#183312yes—
341C22#203911yescharacteristic
342C22#204011yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—161no—
354C4—161no—
355C4—161no—
356C4—161no—
357C4#185611yescharacteristic
358C4#185711yescharacteristic
359C4#202711yescharacteristic
360C4#202811yescharacteristic
361C2—21no—
362C2—21no—
363C2—22no—
364C2—41no—
365C2#1126011yescharacteristic
366C2#1128411yescharacteristic
367C2#1129611yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1283C4semidirect

Maximal subgroups

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

subgroupindex
#219602
#108532
#80742
#79742
#79622
#79602
#79352