Finite Group Databasebeta

← #480498

Subgroups — #480498

Order 512.

893 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35618C211yesmaximal, characteristic
2#17370C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11553C214yesmaximal, complemented
5#11551C214yesmaximal, complemented
6#11449C212yesmaximal, complemented
7#11431C212yesmaximal, complemented
8C2×#537C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2975—21no—
11#2972—21no—
12#2504C2212yes—
13#2503C2212yes—
14#2500C412yes—
15#2499C412yes—
16#2444—22no—
17#2444—22no—
18#2421—21no—
19#2421—21no—
20#2420—21no—
21#2420—21no—
22#2317C2211yescharacteristic
23#2311C411yescharacteristic
24#2297C411yescharacteristic
25#2297C2211yescharacteristic
26C2×#382C411yescharacteristic
27#1465—22no—
28#1463C2214yes—
29#1461—21no—
30#1461—21no—
31#1459—22no—
32#1458C2212yes—
33#1458C2212yes—
34C2×#324C2214yes—
35#1324C2218yes—
36#1319C2214yes—
37#1318C2214yes—
38C22×#41C2×C411yescharacteristic
39C22×#41C2×C411yescharacteristic
40#537—44no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8C2×C412yes—
43C2×#121—22no—
44C2×#121—21no—
45C2×#121C2×C412yes—
46#520—44no—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×#117C2×C412yes—
54C2×#117C2×C411yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
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#397—22no—
69#397—22no—
70#397—22no—
71#397—22no—
72#394—22no—
73#394—22no—
74#391—21no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#391C2311yescharacteristic
78#390—21no—
79#390C2×C412yes—
80#390C2×C411yescharacteristic
81#390C2311yescharacteristic
82#389—21no—
83#389—21no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#386—22no—
89#386—22no—
90#386—22no—
91#386—22no—
92#386—21no—
93#386—21no—
94#385—21no—
95#385—21no—
96#384C2×C412yes—
97#382—21no—
98#382—21no—
99#382C2×C412yes—
100#382C2×C412yes—
101#382C2312yes—
102#381—22no—
103#381—22no—
104#380—21no—
105#380—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊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—24no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—42no—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C411yescharacteristic
125C22×D4—21no—
126C22×D4—22no—
127C22×D4C22⋊C412yes—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C22⋊C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C2×M16—44no—
139C22×C8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142#121—44no—
143#121—48no—
144#121—44no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—44no—
150C4×D4—44no—
151#117—24no—
152#117—42no—
153#117—44no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2411yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C2×D411yescharacteristic
184C2×C22⋊C4C2×D411yescharacteristic
185C2×C42—21no—
186C2×C42—42no—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C4⋊C8—42no—
195C4⋊C8—42no—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—44no—
199C22⋊C8—44no—
200C4×C8—42no—
201C4×C8—42no—
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—
216C24—21no—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C412yes—
219#41—416no—
220#41—88no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—44no—
239C2×D4—82no—
240C2×D4—82no—
241C2×D4#9914yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4#13114yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—164no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—164no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—42no—
312C42—42no—
313C42—41no—
314C42—41no—
315C42—42no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23#40812yes—
332C23C2×#9914yes—
333C23C22×C22⋊C411yescharacteristic
334Q8—44no—
335Q8—82no—
336D4—28no—
337D4—48no—
338D4—82no—
339D4—84no—
340D4—164no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—28no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—48no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—162no—
380C2×C4—162no—
381C2×C4#40812yes—
382C2×C4C2×#9912yes—
383C2×C4#41312yes—
384C2×C4C2×#13112yes—
385C8—84no—
386C8—84no—
387C22—21no—
388C22—21no—
389C22—24no—
390C22—24no—
391C22—22no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22C2×#40811yescharacteristic
407C22C22×#9911yescharacteristic
408C22C2×#41311yescharacteristic
409C4—22no—
410C4—41no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4—82no—
421C4—81no—
422C4#146814yes—
423C2—21no—
424C2—21no—
425C2—44no—
426C2—42no—
427C2—161no—
428C2C2×#146811yescharacteristic
429C2#3336111yescharacteristic
430C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#537C4
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
2. #17370C2semidirect
3. #17351C2semidirect
4. #11553C2semidirect
5. #11551C2semidirect
6. #11449C2semidirect
7. #11431C2semidirect

Maximal subgroups

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

subgroupindex
#356182
#173702
#173512
#115532
#115512
#114492
#114312