Finite Group Databasebeta

← #120236

Subgroups — #120236

Order 512.

559 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#21949C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic
3#8107C211yesmaximal, characteristic
4C2×#1305C211yesmaximal, characteristic
5#8057C211yesmaximal, characteristic
6#8009C211yesmaximal, characteristic
7#7915C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1855—22no—
18#1855C414yescomplemented
19#1811—22no—
20#1810—22no—
21#1305—22no—
22#1305—22no—
23#1298—22no—
24#1293—22no—
25#1280—22no—
26C24×C4C2311yescharacteristic
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16C2×C412yes—
31C22×M16D412yes—
32C23×C8—21no—
33C22×C4⋊C4—21no—
34C22×C22⋊C4Q812yes—
35C2×#108—24no—
36C2×#108—24no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407—21no—
50#407—21no—
51#407—21no—
52#407C2×C412yes—
53#407D412yes—
54#407D412yes—
55#407D412yes—
56#407Q812yes—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8C2×C412yes—
71C2×C22⋊C8D412yes—
72C2×C22⋊C8Q812yes—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77C2×#95—21no—
78#341—24no—
79#341—24no—
80C25C2×Q811yescharacteristic
81C23×C4—22no—
82C23×C4—82no—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4C2×Q811yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—82no—
107C2×M16—82no—
108C2×M16—82no—
109C2×M16—82no—
110C2×M16—82no—
111C2×M16C22⋊C414yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—82no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—82no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—82no—
127#108—84no—
128#108—84no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8C4⋊C414yes—
153#95—41no—
154#95—41no—
155C24—41no—
156C24—41no—
157C24—42no—
158C24—81no—
159C24C4×Q811yescharacteristic
160C24C22⋊Q811yescharacteristic
161C24C22⋊Q811yescharacteristic
162C24C22⋊Q811yescharacteristic
163C24#12511yescharacteristic
164C24C4⋊Q811yescharacteristic
165C24C4⋊Q811yescharacteristic
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4C2×C22⋊C411yescharacteristic
185C22×C4C2×C4⋊C411yescharacteristic
186C22×C4#11711yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×Q811yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4C22⋊Q811yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4#12311yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12411yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12511yescharacteristic
209C22×C4#12611yescharacteristic
210C22×C4#12611yescharacteristic
211C22×C4C4⋊Q811yescharacteristic
212M16—44no—
213M16—44no—
214M16—44no—
215M16—44no—
216M16—82no—
217M16—82no—
218M16—82no—
219M16—82no—
220M16—82no—
221M16—82no—
222C2×C8—24no—
223C2×C8—24no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—84no—
265C23—84no—
266C23#38711yescharacteristic
267C23#38911yescharacteristic
268C23#39111yescharacteristic
269C23#39511yescharacteristic
270C23#39611yescharacteristic
271C23#39811yescharacteristic
272C23#39811yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—164no—
305C2×C4#38011yescharacteristic
306C2×C4#38611yescharacteristic
307C2×C4#38711yescharacteristic
308C2×C4#38911yescharacteristic
309C2×C4#39411yescharacteristic
310C2×C4#39611yescharacteristic
311C2×C4#39811yescharacteristic
312C2×C4#40011yescharacteristic
313C8—44no—
314C8—44no—
315C8—82no—
316C8—82no—
317C8—82no—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—81no—
343C22—81no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22#144111yescharacteristic
349C22#195312yes—
350C22#206911yescharacteristic
351C22#207011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—162no—
360C4#191312yes—
361C2—21no—
362C2—21no—
363C2—22no—
364C2—41no—
365C2—41no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2#1120511yescharacteristic
370C2#1125011yescharacteristic
371C2#1125211yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×Q8
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1855C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#219482
#81072
C2×#13052
#80572
#80092
#79152