Finite Group Databasebeta

← #164174

Subgroups — #164174

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36315C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic, complemented
3#20368C211yesmaximal, characteristic, complemented
4#7687C211yesmaximal, characteristic, complemented
5#7676C212yesmaximal, complemented
6#7667C211yesmaximal, characteristic, complemented
7#7656C212yesmaximal, complemented
8#7616C211yesmaximal, characteristic
9#7611C212yesmaximal, complemented
10#7565C211yesmaximal, characteristic
11#7562C212yesmaximal, complemented
12C2×#405—21no—
13#2861C2212yes—
14#2857C2211yescharacteristic
15#2552C411yescharacteristic
16#2547C411yescharacteristic
17#2531—22no—
18#2529—22no—
19#2497—21no—
20#2489C412yes—
21#2485—21no—
22#2426C412yes—
23#2365C411yescharacteristic
24#2354—21no—
25#2354C411yescharacteristic
26#2346—21no—
27C4×#121C2211yescharacteristic
28#2295C2212yes—
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1923—22no—
32#1839C2212yes—
33#1797C2211yescharacteristic
34C8×C4⋊C4—21no—
35C4×C4⋊C8C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37#1746—22no—
38#1745C2212yes—
39#1744C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×#124C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×#121C2×C411yescharacteristic
60C2×C4×Q8C2×C412yes—
61C2×C4×D4—21no—
62C2×C4×D4C2×C411yescharacteristic
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71#405—28no—
72C2×#97—22no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76#400—22no—
77#398—22no—
78#397—22no—
79#397—22no—
80#396—22no—
81#394—22no—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2312yes—
85#390—22no—
86#390C2×C412yes—
87#390C2311yescharacteristic
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389C2×C412yes—
93#389C2311yescharacteristic
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—21no—
100#388—22no—
101#388—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#386—21no—
106#384—21no—
107#384C2×C411yescharacteristic
108#382—22no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380C2×C411yescharacteristic
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—21no—
124C4×C22⋊C4C2×C412yes—
125C4×C22⋊C4D412yes—
126#376—22no—
127#376D412yes—
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—21no—
133#336—22no—
134#336—22no—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141C22×Q8C22×C412yes—
142C22×D4—21no—
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C2×M16—42no—
150C2×M16—42no—
151C22×C8—24no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—42no—
159C22×C8—42no—
160C4⋊Q8—42no—
161#125—82no—
162#124—82no—
163C22⋊Q8—82no—
164#121—22no—
165#121—41no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—42no—
171#117—22no—
172#117—22no—
173#117—24no—
174#117—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2411yescharacteristic
219C4⋊C8—42no—
220C4⋊C8—44no—
221C22⋊C8—42no—
222C22⋊C8—81no—
223#97—28no—
224C4×C8—28no—
225C4×C8—42no—
226C4×C8—82no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C24—21no—
243C24C2×C22⋊C411yescharacteristic
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—81no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4#13512yes—
281C22×C4#13512yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—48no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—48no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—44no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—82no—
319C4⋊C4—81no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4#10414yes—
324C4⋊C4#13112yes—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—44no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—24no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—44no—
353C42—42no—
354C42—44no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23#41612yes—
369C23C22×C22⋊C411yescharacteristic
370Q8—82no—
371Q8—82no—
372D4—81no—
373D4—81no—
374D4—84no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—84no—
420C2×C4#41012yes—
421C2×C4#41612yes—
422C2×C4C2×#10414yes—
423C2×C4C2×#10412yes—
424C2×C4C2×#13111yescharacteristic
425C2×C4#56811yescharacteristic
426C2×C4C2×#13511yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57811yescharacteristic
430C8—42no—
431C8—42no—
432C8—42no—
433C8—44no—
434C8—82no—
435C8—81no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22#288011yescharacteristic
451C22C2×#41011yescharacteristic
452C22#288811yescharacteristic
453C22C2×#41611yescharacteristic
454C22#289811yescharacteristic
455C22C22×#10411yescharacteristic
456C22#290111yescharacteristic
457C4—21no—
458C4—21no—
459C4—24no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—41no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—41no—
469C4—42no—
470C4—82no—
471C4—81no—
472C4—81no—
473C4—82no—
474C4#150014yes—
475C2—42no—
476C2—161no—
477C2#1741811yescharacteristic
478C2#1750811yescharacteristic
479C2#1751911yescharacteristic
480C2C2×#150011yescharacteristic
481C2#1761211yescharacteristic
482C2#1765511yescharacteristic
483C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22639C2semidirect
3. #20368C2semidirect
4. #7687C2semidirect
5. #7676C2semidirect
6. #7667C2semidirect
7. #7656C2semidirect
9. #7611C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#363152
#226392
#203682
#76872
#76762
#76672
#76562
#76162
#76112
#75652
#75622