Finite Group Databasebeta

← #480496

Subgroups — #480496

Order 512.

917 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35530C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic, complemented
3#17353C211yesmaximal, characteristic, complemented
4#11556C214yesmaximal, complemented
5#11550C214yesmaximal, complemented
6#11449C212yesmaximal, complemented
7#11431C212yesmaximal, complemented
8C2×#537C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2983—21no—
11#2982—21no—
12#2502C412yes—
13#2501C412yes—
14#2500C2212yes—
15#2499C2212yes—
16#2443—22no—
17#2443—22no—
18#2423—21no—
19#2423—21no—
20#2422—21no—
21#2422—21no—
22#2317C2211yescharacteristic
23#2311C411yescharacteristic
24#2297C411yescharacteristic
25#2297C2211yescharacteristic
26C2×#384C411yescharacteristic
27#1466—22no—
28#1464C2214yes—
29#1460—21no—
30#1460—22no—
31#1460—21no—
32#1458C2214yes—
33#1458C2212yes—
34#1458C2212yes—
35#1324C2218yes—
36#1319C2214yes—
37#1318C2214yes—
38C22×#41C2×C411yescharacteristic
39C22×#41C2×C411yescharacteristic
40#537—44no—
41C2×#123—22no—
42C2×#123—22no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8C2×C412yes—
45C2×#121—21no—
46C2×#121—22no—
47C2×#121C2×C412yes—
48#520—44no—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×#117C2×C411yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2312yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C42C2×C411yescharacteristic
63#446—44no—
64C8×Q8—44no—
65#432—24no—
66#431—22no—
67#431—24no—
68#431—22no—
69#399—22no—
70#399—22no—
71#398—22no—
72#398—22no—
73#397—22no—
74#397—22no—
75#394—22no—
76#394—22no—
77#391—21no—
78#391C2×C412yes—
79#391C2×C411yescharacteristic
80#391C2311yescharacteristic
81#390—21no—
82#390C2×C412yes—
83#390C2×C411yescharacteristic
84#390C2311yescharacteristic
85#389—21no—
86#389—21no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#386—22no—
92#386—22no—
93#386—21no—
94#386—21no—
95#385—22no—
96#385—22no—
97#385—21no—
98#385—21no—
99#384—21no—
100#384—21no—
101#384C2×C412yes—
102#384C2×C412yes—
103#384C2×C412yes—
104#382C2312yes—
105#380—21no—
106#380—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4C2×C412yes—
111#324C2314yes—
112#324C2314yes—
113#323—22no—
114#323—24no—
115#323—22no—
116#323—24no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—42no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C411yescharacteristic
125C22×D4—21no—
126C22×D4—22no—
127C22×D4C22⋊C411yescharacteristic
128C22×D4C22⋊C412yes—
129C22×D4C22×C411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138#123—84no—
139#123—84no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142#121—44no—
143#121—48no—
144#121—44no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—44no—
150C4×D4—44no—
151#117—24no—
152#117—42no—
153#117—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C2411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4C2×D411yescharacteristic
187C2×C22⋊C4C2×D411yescharacteristic
188C2×C22⋊C4C2×D412yes—
189C2×C22⋊C4C2×D412yes—
190C2×C42—21no—
191C2×C42—42no—
192C2×C42C22×C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C4⋊C8—44no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—44no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205#97—44no—
206C4×C8—44no—
207C4×C8—42no—
208C4×C8—42no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95—22no—
219C24—21no—
220C24C2×C22⋊C412yes—
221C24C2×C22⋊C411yescharacteristic
222#41—416no—
223#41—88no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8#9914yes—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—82no—
244C2×D4—82no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4#13114yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—44no—
280C2×C8—44no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—24no—
314C42—22no—
315C42—22no—
316C42—24no—
317C42—24no—
318C42—41no—
319C42—41no—
320C42—42no—
321C23—21no—
322C23—21no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#40812yes—
338C23C22×C22⋊C411yescharacteristic
339Q8—28no—
340Q8—82no—
341D4—48no—
342D4—44no—
343D4—82no—
344D4—84no—
345D4—164no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—48no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—48no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—84no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4—162no—
384C2×C4—162no—
385C2×C4#40812yes—
386C2×C4C2×#9914yes—
387C2×C4C2×#9912yes—
388C2×C4#41312yes—
389C2×C4C2×#13112yes—
390C8—84no—
391C8—84no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22—82no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22C2×#40811yescharacteristic
412C22C22×#9911yescharacteristic
413C22C2×#41311yescharacteristic
414C4—22no—
415C4—24no—
416C4—42no—
417C4—42no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—42no—
423C4—42no—
424C4—82no—
425C4—82no—
426C4—81no—
427C4#146814yes—
428C2—21no—
429C2—21no—
430C2—42no—
431C2—82no—
432C2—161no—
433C2C2×#146811yescharacteristic
434C2#3336011yescharacteristic
435C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17372C2semidirect
3. #17353C2semidirect
4. #11556C2semidirect
5. #11550C2semidirect
6. #11449C2semidirect
7. #11431C2semidirect

Maximal subgroups

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

subgroupindex
#355302
#173722
#173532
#115562
#115502
#114492
#114312