Finite Group Databasebeta

← #164183

Subgroups — #164183

Order 512.

804 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36459C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20368C211yesmaximal, characteristic, complemented
4#7692C212yesmaximal, complemented
5#7688C211yesmaximal, characteristic, complemented
6#7666C212yesmaximal, complemented
7#7665C211yesmaximal, characteristic, complemented
8#7619C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7568C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12C2×#405—21no—
13#2860C2212yes—
14#2857C2211yescharacteristic
15#2563C411yescharacteristic
16#2560—22no—
17#2552C411yescharacteristic
18#2532—22no—
19#2496—21no—
20#2495C412yes—
21#2490C412yes—
22#2485—21no—
23#2369C411yescharacteristic
24#2354—21no—
25#2354C411yescharacteristic
26#2347—21no—
27C4×#125C2212yes—
28C4×#121C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1923—22no—
32#1831C2212yes—
33#1797C2211yescharacteristic
34C8×C4⋊C4—21no—
35C4×C22⋊C8C2211yescharacteristic
36#1746—22no—
37#1745C2212yes—
38#1744C2212yes—
39#1744C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#125C2×C412yes—
58C2×C22⋊Q8—21no—
59C2×#121C2×C411yescharacteristic
60C2×C4×D4—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70#405—28no—
71C2×#97—22no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75#400—22no—
76#398—22no—
77#397—22no—
78#397—22no—
79#394—22no—
80#393—22no—
81#391—21no—
82#391—22no—
83#390—22no—
84#390C2×C412yes—
85#390C2×C412yes—
86#390C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2311yescharacteristic
93#389C2312yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—21no—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—22no—
103#386—21no—
104#384—21no—
105#384C2×C411yescharacteristic
106#383—22no—
107#382C2×C412yes—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380C2×C412yes—
113#380C2×C411yescharacteristic
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
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×Q8—21no—
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#126—82no—
162#126—82no—
163#125—24no—
164#125—42no—
165C22⋊Q8—81no—
166C22⋊Q8—81no—
167#121—22no—
168#121—41no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172#117—22no—
173#117—22no—
174#117—24no—
175#117—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—42no—
221C4⋊C8—82no—
222C22⋊C8—42no—
223C22⋊C8—81no—
224#97—28no—
225C4×C8—28no—
226C4×C8—42no—
227C4×C8—44no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—22no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243C24—21no—
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—42no—
246C2×Q8—81no—
247C2×Q8—81no—
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⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
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—42no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—41no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4#10414yes—
326C4⋊C4#13112yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—44no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—42no—
351C42—41no—
352C42—41no—
353C42—42no—
354C42—42no—
355C42—44no—
356C42—82no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#41612yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—162no—
373D4—81no—
374D4—81no—
375D4—84no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—84no—
421C2×C4#41012yes—
422C2×C4#41612yes—
423C2×C4C2×#10414yes—
424C2×C4C2×#10412yes—
425C2×C4C2×#13111yescharacteristic
426C2×C4#56811yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57711yescharacteristic
431C8—42no—
432C8—42no—
433C8—42no—
434C8—44no—
435C8—81no—
436C8—82no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22#288011yescharacteristic
452C22C2×#41011yescharacteristic
453C22#288911yescharacteristic
454C22C2×#41611yescharacteristic
455C22#289711yescharacteristic
456C22C22×#10411yescharacteristic
457C22#290111yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—24no—
465C4—41no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—41no—
471C4—81no—
472C4—82no—
473C4—81no—
474C4—82no—
475C4#150014yes—
476C2—42no—
477C2—161no—
478C2#1742011yescharacteristic
479C2#1750811yescharacteristic
480C2#1752011yescharacteristic
481C2C2×#150011yescharacteristic
482C2#1761011yescharacteristic
483C2#1765311yescharacteristic
484C2#1772711yescharacteristic
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. #22632C2semidirect
3. #20368C2semidirect
4. #7692C2semidirect
5. #7688C2semidirect
6. #7666C2semidirect
7. #7665C2semidirect
8. #7619C2semidirect
10. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#364592
#226322
#203682
#76922
#76882
#76662
#76652
#76192
#76162
#75682
#75652