Finite Group Databasebeta

← #164812

Subgroups — #164812

Order 512.

703 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36299C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#22022C211yesmaximal, characteristic, complemented
4#7733C212yesmaximal, complemented
5#7729C212yesmaximal, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7642C212yesmaximal
9#7635C211yesmaximal, characteristic
10#7594C212yesmaximal
11#7567C211yesmaximal, characteristic
12#2863C2212yes—
13#2859C2211yescharacteristic
14#2582C411yescharacteristic
15#2538C411yescharacteristic
16#2515—22no—
17#2509C412yes—
18#2495—21no—
19#2475—22no—
20#2447C412yes—
21#2421—21no—
22#2360—21no—
23#2355C411yescharacteristic
24D4×C4⋊C4C411yescharacteristic
25#2344—21no—
26C4×#124C2211yescharacteristic
27#2293C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1839C2212yes—
36#1839C2212yes—
37#1831C2212yes—
38#1831C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C2×#125—21no—
54C2×#124C2×C411yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C411yescharacteristic
62C2×#117—22no—
63C22×C4⋊C4—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×#97—21no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#400—22no—
73#400—22no—
74#398—22no—
75#397—22no—
76#397—22no—
77#396—22no—
78#391—21no—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2311yescharacteristic
82#390C2×C412yes—
83#390C2311yescharacteristic
84#389—21no—
85#389—22no—
86#389—21no—
87#389—21no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2312yes—
93#388—22no—
94#388—21no—
95#388—22no—
96#388—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#383—22no—
107#382—21no—
108#382—21no—
109#382C2×C411yescharacteristic
110#381—22no—
111#380—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D411yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—22no—
124C43C2312yes—
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8C22⋊C411yescharacteristic
141C22×Q8C22×C411yescharacteristic
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151#125—81no—
152#125—81no—
153#124—22no—
154#124—41no—
155C22⋊Q8—44no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—42no—
162C4×D4—42no—
163#117—82no—
164#117—82no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2411yescharacteristic
209C4⋊C8—42no—
210C4⋊C8—42no—
211C4⋊C8—82no—
212C4⋊C8—82no—
213#97—81no—
214#97—81no—
215C4×C8—42no—
216C4×C8—42no—
217#95—21no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24—21no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—21no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4#13112yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—82no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—81no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—82no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—22no—
327C42—21no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—41no—
337C42—42no—
338C42—82no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C22×C22⋊C411yescharacteristic
347Q8—44no—
348Q8—81no—
349Q8—81no—
350D4—42no—
351D4—42no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—88no—
395C2×C4—84no—
396C2×C4#41312yes—
397C2×C4#42112yes—
398C2×C4#42112yes—
399C2×C4C2×#13111yescharacteristic
400C2×C4#56811yescharacteristic
401C2×C4#57611yescharacteristic
402C2×C4#57711yescharacteristic
403C2×C4#57711yescharacteristic
404C2×C4#57811yescharacteristic
405C8—81no—
406C8—82no—
407C8—81no—
408C8—81no—
409C8—82no—
410C8—81no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22#288011yescharacteristic
419C22#288411yescharacteristic
420C22C2×#41311yescharacteristic
421C22#289711yescharacteristic
422C22#289811yescharacteristic
423C22C2×#42111yescharacteristic
424C22C2×#42111yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—41no—
432C4—42no—
433C4—41no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—81no—
439C4—81no—
440C4—82no—
441C4—82no—
442C4—81no—
443C4#153514yes—
444C2—82no—
445C2#1742311yescharacteristic
446C2#1752211yescharacteristic
447C2#1752311yescharacteristic
448C2#1760111yescharacteristic
449C2#1760211yescharacteristic
450C2C2×#153511yescharacteristic
451C2#1777211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
3. #22022C2semidirect
4. #7733C2semidirect
5. #7729C2semidirect
6. #7683C2semidirect
7. #7680C2semidirect

Maximal subgroups

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

subgroupindex
#362992
#220222
#77332
#77292
#76832
#76802
#76422
#76352
#75942
#75672