Finite Group Databasebeta

← #480289

Subgroups — #480289

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C2×#1464C212yesmaximal
3#11516C212yesmaximal, complemented
4#11515C212yesmaximal, complemented
5#11513C214yesmaximal, complemented
6#11448C214yesmaximal, complemented
7C22×#117C411yescharacteristic
8C22×#117C411yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#432—22no—
11#2317C411yescharacteristic
12#2317C411yescharacteristic
13#2317C412yes—
14#2317C2211yescharacteristic
15#2317C2211yescharacteristic
16#2317C2212yes—
17#2316—22no—
18#2315—22no—
19#2296—22no—
20#2296—22no—
21#2285C412yes—
22#2285C2212yes—
23#1464—22no—
24#1464—22no—
25#1464C2214yes—
26#1464C2214yes—
27C2×#324C2214yes—
28C2×#323—22no—
29#1325C2218yescomplemented
30#1324C2218yescomplemented
31C24×C4C2×C411yescharacteristic
32C2×#117—21no—
33C2×#117—21no—
34C2×#117—28no—
35C2×#117—21no—
36C2×#117—21no—
37C2×#117—42no—
38C2×#117—42no—
39C2×#117—42no—
40C2×#117—42no—
41C2×#117C2×C412yes—
42C2×#117C2×C412yes—
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C412yes—
48C2×#117D412yes—
49C2×#117D412yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4D412yes—
60C22×C42—21no—
61C22×C42C2×C412yes—
62C22×C42D412yes—
63#432—24no—
64#432—24no—
65#432—42no—
66#432—42no—
67C2×C4⋊C8—42no—
68C2×C22⋊C8—24no—
69C2×#97—42no—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#385—24no—
77#385—24no—
78#385—22no—
79#385—22no—
80#385—22no—
81#385—22no—
82#384C2×C412yes—
83#384C2×C412yes—
84#384C2×C412yes—
85#384C2312yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#382C2×C412yes—
89#382C2312yes—
90#380—24no—
91#380—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94#324—24no—
95#324C2318yes—
96#323—24no—
97#323—24no—
98C25C22⋊C411yescharacteristic
99C23×C4—24no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D412yes—
114C22×C8—44no—
115#117—28no—
116#117—28no—
117#117—24no—
118#117—24no—
119#117—24no—
120#117—42no—
121#117—41no—
122#117—41no—
123#117—42no—
124#117—42no—
125#117—42no—
126#117—42no—
127#117—41no—
128#117—42no—
129#117—41no—
130#117—44no—
131#117—42no—
132#117—42no—
133#117—44no—
134#117—42no—
135#117—42no—
136#117M1618yes—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2411yescharacteristic
159C2×C22⋊C4—28no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C2×D412yes—
171C2×C22⋊C4C2×D412yes—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C4⋊C8—44no—
191C22⋊C8—28no—
192C22⋊C8—44no—
193#97—44no—
194#97—44no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199C24—22no—
200C24—22no—
201C24—81no—
202C24—81no—
203C24#10114yes—
204C24C2×C22⋊C411yescharacteristic
205C24C2×C22⋊C411yescharacteristic
206C24C2×C22⋊C411yescharacteristic
207C22×C4—216no—
208C22×C4—216no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—28no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4#9912yes—
245C22×C4#9912yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×M1614yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D412yes—
255C2×C8—48no—
256C2×C8—88no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—28no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—84no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—28no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C42—28no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—44no—
295C42—42no—
296C42—42no—
297C42C2×M1614yes—
298C42C2×M1614yes—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—28no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—82no—
311C23—82no—
312C23—84no—
313C23—84no—
314C23#40712yes—
315C23C2×#9912yes—
316C23C2×#10112yes—
317C23C2×#10112yes—
318C23C2×#10112yes—
319C23C22×C22⋊C411yescharacteristic
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—88no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4#40712yes—
363C2×C4C2×#9911yescharacteristic
364C2×C4C2×#9911yescharacteristic
365C2×C4C2×#9912yes—
366C2×C4C22×M1612yes—
367C8—88no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—88no—
385C22#146314yes—
386C22C2×#40711yescharacteristic
387C22C22×#9911yescharacteristic
388C22C22×#10111yescharacteristic
389C4—22no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—44no—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—82no—
401C4—82no—
402C2—21no—
403C2—21no—
404C2—24no—
405C2—42no—
406C2—42no—
407C2C2×#146311yescharacteristic
408C2#3335811yescharacteristic
409C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11516C2semidirect
4. #11515C2semidirect
5. #11513C2semidirect
6. #11448C2semidirect
29. #1325C22semidirect
30. #1324C22semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#14642
#115162
#115152
#115132
#114482