Finite Group Databasebeta

← #164872

Subgroups — #164872

Order 512.

692 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36312C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#22022C211yesmaximal, characteristic, complemented
4#7732C212yesmaximal
5#7726C212yesmaximal
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7644C212yesmaximal, complemented
9#7635C211yesmaximal, characteristic
10#7596C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic
12#2860C2212yes—
13#2859C2211yescharacteristic
14#2582C411yescharacteristic
15#2538C411yescharacteristic
16#2529—22no—
17#2519—22no—
18#2508C412yes—
19#2496—21no—
20#2448C412yes—
21#2423—21no—
22#2362—21no—
23#2354C411yescharacteristic
24Q8×C4⋊C4C411yescharacteristic
25#2346—21no—
26C4×#124C2211yescharacteristic
27#2295C2212yes—
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#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×#126—22no—
54C2×#125—21no—
55C2×#124C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×#117—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#399—22no—
74#398—22no—
75#398—22no—
76#397—22no—
77#396—22no—
78#391—21no—
79#391C2×C412yes—
80#391C2311yescharacteristic
81#390C2×C412yes—
82#390C2×C412yes—
83#390C2311yescharacteristic
84#389—21no—
85#389—21no—
86#389—21no—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2312yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—21no—
98#388—21no—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#384—21no—
105#384C2×C411yescharacteristic
106#384C2×C411yescharacteristic
107#382—21no—
108#382—22no—
109#382C2×C411yescharacteristic
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
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—
123#376—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—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150#126—82no—
151#126—82no—
152#125—81no—
153#125—81no—
154#124—22no—
155#124—41no—
156C22⋊Q8—44no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—42no—
164#117—44no—
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—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
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×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C412yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—82no—
214C4⋊C8—82no—
215#97—81no—
216#97—81no—
217C4×C8—42no—
218C4×C8—42no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8#13112yes—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—22no—
318C42—21no—
319C42—22no—
320C42—21no—
321C42—21no—
322C42—22no—
323C42—21no—
324C42—22no—
325C42—21no—
326C42—22no—
327C42—22no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—41no—
336C42—44no—
337C42—44no—
338C42—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C22×C22⋊C411yescharacteristic
347Q8—42no—
348Q8—42no—
349Q8—44no—
350D4—81no—
351D4—81no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—84no—
395C2×C4#41312yes—
396C2×C4#42112yes—
397C2×C4#42112yes—
398C2×C4C2×#13111yescharacteristic
399C2×C4#56811yescharacteristic
400C2×C4#57611yescharacteristic
401C2×C4#57711yescharacteristic
402C2×C4#57711yescharacteristic
403C2×C4#57811yescharacteristic
404C8—81no—
405C8—82no—
406C8—81no—
407C8—81no—
408C8—82no—
409C8—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#288011yescharacteristic
418C22#288411yescharacteristic
419C22C2×#41311yescharacteristic
420C22#289711yescharacteristic
421C22#289811yescharacteristic
422C22C2×#42111yescharacteristic
423C22C2×#42111yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—81no—
439C4—82no—
440C4—82no—
441C4—81no—
442C4#153514yes—
443C2—161no—
444C2#1742311yescharacteristic
445C2#1752511yescharacteristic
446C2#1752611yescharacteristic
447C2#1760111yescharacteristic
448C2#1760211yescharacteristic
449C2C2×#153511yescharacteristic
450C2#1778111yescharacteristic
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
6. #7683C2semidirect
7. #7680C2semidirect
8. #7644C2semidirect
10. #7596C2semidirect

Maximal subgroups

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

subgroupindex
#363122
#220222
#77322
#77262
#76832
#76802
#76442
#76352
#75962
#75672