Finite Group Databasebeta

← #391731

Subgroups — #391731

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2#20626C212yesmaximal, complemented
3#11204C214yesmaximal, complemented
4#11198C212yesmaximal, complemented
5#11044C214yesmaximal, complemented
6#10967C212yesmaximal
7C22×#123C412yes—
8C22×#121C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#423C2214yes—
11C22×#100C2214yes—
12C2×#407C2214yes—
13#2854—22no—
14#2854—22no—
15#2854—22no—
16#2854—22no—
17C2×#390C2212yes—
18C2×#389C2212yes—
19C2×#388C414yes—
20C2×#386C412yes—
21C2×#380C2211yescharacteristic
22#1797—22no—
23#1797—22no—
24#1797—22no—
25#1797—22no—
26C2×#343C2218yes—
27C2×#343C2214yes—
28#1728C2214yes—
29C23×D4C2×C412yes—
30C23×D4C2311yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16C2318yes—
35C2×#123—24no—
36C2×#123—22no—
37C2×#123—22no—
38C2×#123—48no—
39C2×#121—22no—
40C2×#121—22no—
41C2×#121—44no—
42C22×C4⋊C4C2×C414yes—
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2312yes—
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C414yes—
48C22×C22⋊C4C2×C414yes—
49C22×C42—44no—
50C22×C42C2×C412yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42C2312yes—
56#423—48no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—24no—
59C2×#100—48no—
60C2×#100—44no—
61#407—48no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64#390—22no—
65#390—22no—
66#390—22no—
67#390—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#388—24no—
73#388—24no—
74#388—24no—
75#388—24no—
76#386—22no—
77#386—24no—
78#386—22no—
79#380—22no—
80#380—22no—
81C2×#95C2×C412yes—
82C2×#95C2×C414yes—
83#343—44no—
84#343—48no—
85#336—24no—
86#336—24no—
87C25C22×C411yescharacteristic
88C25C22×C412yes—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—24no—
93C22×D4—24no—
94C22×D4—24no—
95C22×D4—44no—
96C22×D4—44no—
97C22×D4—44no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—44no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C22×C414yes—
116C23×C4C22×C412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2411yescharacteristic
120C2×M16—28no—
121C2×M16—28no—
122C2×M16—28no—
123C2×M16—28no—
124C2×M16—28no—
125C2×M16—28no—
126C22×C8—28no—
127#123—48no—
128#123—416no—
129#123—48no—
130#121—48no—
131#121—48no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—48no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—48no—
150C2×C22⋊C4—48no—
151C2×C22⋊C4—44no—
152C2×C42—28no—
153C2×C42—44no—
154C2×C42—44no—
155C2×C42—44no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42C22⋊C412yes—
159C2×C42C22⋊C412yes—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C4⋊C8—48no—
167C4⋊C8—48no—
168#100—416no—
169C22⋊C8—48no—
170C22⋊C8—48no—
171#95—24no—
172#95—24no—
173#95—24no—
174C24—22no—
175C24—22no—
176C24—22no—
177C24—24no—
178C24—22no—
179C24—24no—
180C24—28no—
181C24—42no—
182C24—42no—
183C24—44no—
184C24—42no—
185C24C2×C22⋊C411yescharacteristic
186C24C2×C22⋊C411yescharacteristic
187C24C23×C411yescharacteristic
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—48no—
200C2×D4—44no—
201C2×D4—816no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—416no—
238C22×C4—416no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4#10014yescomplemented
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C22×D412yes—
253C22×C4H(5,𝔽2)14yes—
254C22×C4H(5,𝔽2)14yes—
255C22×C42−1+414yes—
256M16—48no—
257M16—48no—
258M16—48no—
259M16—48no—
260C2×C8—48no—
261C2×C8—48no—
262C2×C8—48no—
263C4⋊C4—48no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—48no—
271C22⋊C4—48no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C42—24no—
277C42—24no—
278C42—24no—
279C42—24no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C23—21no—
285C23—22no—
286C23—21no—
287C23—28no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—21no—
296C23—48no—
297C23—48no—
298C23—44no—
299C23—44no—
300C23—42no—
301C23—44no—
302C23—42no—
303C23—44no—
304C23—42no—
305C23—48no—
306C23—84no—
307C23—88no—
308C23—88no—
309C23—84no—
310C23C2×#10012yes—
311C23#41312yes—
312C23C22×C22⋊C411yescharacteristic
313C23#51812yes—
314C23#52014yes—
315D4—88no—
316D4—88no—
317D4—84no—
318D4—84no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—88no—
352C2×C4—84no—
353C2×C4C2×#10014yes—
354C2×C4#41314yes—
355C2×C4#53414yes—
356C2×C4#53512yes—
357C2×C4#53612yes—
358C2×C4#53714yes—
359C2×C4#56014yes—
360C2×C4#56218yes—
361C2×C4#56314yes—
362C8—88no—
363C22—24no—
364C22—24no—
365C22—22no—
366C22—22no—
367C22—24no—
368C22—24no—
369C22—22no—
370C22—22no—
371C22—416no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—82no—
379C22—84no—
380C22—84no—
381C22—82no—
382C22—82no—
383C22—168no—
384C22#230811yescharacteristic
385C22#231112yes—
386C22#237614yes—
387C22C22×#10011yescharacteristic
388C22C2×#41312yes—
389C22C2×#41311yescharacteristic
390C4—24no—
391C4—24no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—84no—
396C4—82no—
397C4#281314yes—
398C4#281414yes—
399C2—24no—
400C2—42no—
401C2—42no—
402C2—162no—
403C2#1465811yescharacteristic
404C2#2175211yescharacteristic
405C2#2175411yescharacteristic
406C2#2187812yes—
407C2#2188012yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. #11204C2semidirect
4. #11198C2semidirect
5. #11044C2semidirect
246. C22×C4#100semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#206262
#112042
#111982
#110442
#109672