Finite Group Databasebeta

← #480485

Subgroups — #480485

Order 512.

873 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#35619C211yesmaximal, characteristic
2#17370C211yesmaximal, characteristic
3#17351C211yesmaximal, characteristic
4#11555C214yesmaximal, complemented
5#11551C214yesmaximal, complemented
6#11447C212yesmaximal, complemented
7#11434C212yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2975—21no—
11#2972—21no—
12#2503C2212yes—
13#2502C2212yes—
14#2499C412yes—
15#2499C412yes—
16#2447—21no—
17#2447—21no—
18#2446—21no—
19#2446—21no—
20#2419—22no—
21#2419—22no—
22#2317C411yescharacteristic
23#2310C2211yescharacteristic
24#2297C411yescharacteristic
25C2×#391C411yescharacteristic
26#2285C2211yescharacteristic
27#1465—22no—
28#1463C2214yes—
29#1461—21no—
30#1461—21no—
31#1459—22no—
32#1458C2212yes—
33#1458C2212yes—
34C2×#324C2214yes—
35#1325C2214yes—
36#1324C2214yes—
37#1319C2218yes—
38C22×#41C2×C411yescharacteristic
39C23×Q8C2×C411yescharacteristic
40#536—44no—
41C2×C4⋊Q8—21no—
42C2×#124—21no—
43C2×#123—22no—
44C2×C22⋊Q8C2×C412yes—
45C2×C22⋊Q8C2×C412yes—
46#520—44no—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C412yes—
55C2×#117C2×C411yescharacteristic
56C2×#117C2311yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42C2×C411yescharacteristic
60#436—44no—
61C8×D4—44no—
62#431—22no—
63#431—22no—
64#407—24no—
65C2×C22⋊C8—24no—
66#400—22no—
67#400—22no—
68#398—22no—
69#398—22no—
70#397—22no—
71#397—22no—
72#397—22no—
73#397—22no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2312yes—
83#390—21no—
84#389—21no—
85#389—21no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#386—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#384C2×C412yes—
95#384C2×C411yescharacteristic
96#384C2311yescharacteristic
97#382C2×C412yes—
98#382C2×C411yescharacteristic
99#382C2311yescharacteristic
100#381—22no—
101#381—22no—
102#381—21no—
103#381—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C22⋊C4C2×C412yes—
109C4×C22⋊C4C2×C412yes—
110C2×#95—22no—
111#324—24no—
112#324—24no—
113#324C2314yes—
114#324C2314yes—
115#323—22no—
116#323—22no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—42no—
120C22×Q8—21no—
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C412yes—
126C22×D4C22⋊C412yes—
127C22×D4C22⋊C411yescharacteristic
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C22⋊C412yes—
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C2×M16—44no—
136C22×C8—44no—
137C4⋊Q8—44no—
138#124—44no—
139#123—48no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—44no—
147C4×D4—44no—
148#117—42no—
149#117—44no—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2411yescharacteristic
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C2×D411yescharacteristic
181C2×C22⋊C4C2×D411yescharacteristic
182C2×C42—24no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C4⋊C8—42no—
193C4⋊C8—42no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198C4×C8—42no—
199C4×C8—42no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—22no—
216C24C2×C22⋊C411yescharacteristic
217C24C2×C22⋊C412yes—
218#41—416no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—24no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—88no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4#9914yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4#13114yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268M16—164no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—164no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—82no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23—41no—
316C23—41no—
317C23#40812yes—
318C23C2×#9914yes—
319C23C22×C22⋊C411yescharacteristic
320Q8—44no—
321Q8—84no—
322Q8—84no—
323D4—28no—
324D4—48no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—28no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—48no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—162no—
367C2×C4#40812yes—
368C2×C4C2×#9912yes—
369C2×C4#41312yes—
370C2×C4C2×#13112yes—
371C8—84no—
372C8—84no—
373C22—21no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22—22no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—82no—
384C22—82no—
385C22C2×#40811yescharacteristic
386C22C22×#9911yescharacteristic
387C22C2×#41311yescharacteristic
388C4—22no—
389C4—42no—
390C4—41no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—41no—
395C4—41no—
396C4—42no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—82no—
402C4#146814yes—
403C2—21no—
404C2—21no—
405C2—44no—
406C2—42no—
407C2C2×#146811yescharacteristic
408C2#3336111yescharacteristic
409C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11555C2semidirect
5. #11551C2semidirect
6. #11447C2semidirect
7. #11434C2semidirect

Maximal subgroups

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

subgroupindex
#356192
#173702
#173512
#115552
#115512
#114472
#114342