Finite Group Databasebeta

← #164539

Subgroups — #164539

Order 512.

771 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#36449C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic, complemented
3#20754C211yesmaximal, characteristic, complemented
4#7733C212yesmaximal, complemented
5#7709C212yesmaximal, complemented
6#7687C211yesmaximal, characteristic, complemented
7#7667C211yesmaximal, characteristic, complemented
8#7618C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7601C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12C2×#424—21no—
13#2863C2212yes—
14#2857C2211yescharacteristic
15#2578—22no—
16#2571C411yescharacteristic
17#2547C411yescharacteristic
18#2530—22no—
19#2495—21no—
20#2485—21no—
21#2427C412yes—
22#2421C412yes—
23#2365C411yescharacteristic
24#2352—21no—
25D4×C4⋊C4C411yescharacteristic
26#2345—21no—
27C4×#124C2212yes—
28C4×#121C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1924—22no—
32#1832—22no—
33#1831C2212yes—
34#1831C2212yes—
35#1831C2212yes—
36#1797C2211yescharacteristic
37#1770—21no—
38C4×C22⋊C8C2211yescharacteristic
39#1745C2212yes—
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×#124C2×C412yes—
57C2×C22⋊Q8C2×C412yes—
58C2×#121C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C2×C4×D4—21no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C412yes—
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C42C2311yescharacteristic
67#424—44no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×#97—22no—
75C2×C4×C8—21no—
76#400—22no—
77#400—22no—
78#397—22no—
79#397—22no—
80#397—22no—
81#394—22no—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2312yes—
85#390C2311yescharacteristic
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2311yescharacteristic
93#388—22no—
94#388—22no—
95#388—22no—
96#388—21no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#386—21no—
103#386—22no—
104#385—22no—
105#384C2×C411yescharacteristic
106#383—22no—
107#382—21no—
108#382C2×C412yes—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—22no—
115#380C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C22⋊C4—22no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127#376D412yes—
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8C22×C412yes—
142C22×D4—21no—
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4—22no—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8—42no—
161C4⋊Q8—42no—
162#124—24no—
163#124—42no—
164C22⋊Q8—82no—
165#121—22no—
166#121—41no—
167C4×Q8—82no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—82no—
172#117—22no—
173#117—22no—
174#117—42no—
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—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—42no—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2411yescharacteristic
217C4⋊C8—42no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220C4⋊C8—82no—
221C22⋊C8—42no—
222C22⋊C8—81no—
223#97—44no—
224C4×C8—42no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
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—41no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—81no—
254C2×D4#13112yes—
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—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4AGL(1,ℤ/8)12yes—
284C22×C4#13712yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—84no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—84no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—82no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—24no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—82no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23#41712yes—
369C23C22×C22⋊C411yescharacteristic
370Q8—82no—
371Q8—82no—
372D4—42no—
373D4—42no—
374D4—84no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—84no—
421C2×C4—164no—
422C2×C4#41012yes—
423C2×C4#41712yes—
424C2×C4#42112yes—
425C2×C4C2×#13111yescharacteristic
426C2×C4#56811yescharacteristic
427C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
428C2×C4C2×#13711yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57811yescharacteristic
431C8—42no—
432C8—81no—
433C8—82no—
434C8—81no—
435C8—81no—
436C8—82no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22#288011yescharacteristic
452C22C2×#41011yescharacteristic
453C22#288811yescharacteristic
454C22C2×#41711yescharacteristic
455C22#289811yescharacteristic
456C22C2×#42111yescharacteristic
457C22#290111yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—41no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—82no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4#151214yes—
476C2—42no—
477C2—82no—
478C2#1741511yescharacteristic
479C2#1751211yescharacteristic
480C2#1752311yescharacteristic
481C2C2×#151211yescharacteristic
482C2#1760511yescharacteristic
483C2#1768111yescharacteristic
484C2#1776411yescharacteristic
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. #22636C2semidirect
3. #20754C2semidirect
4. #7733C2semidirect
5. #7709C2semidirect
6. #7687C2semidirect
7. #7667C2semidirect
8. #7618C2semidirect
10. #7601C2semidirect

Maximal subgroups

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

subgroupindex
#364492
#226362
#207542
#77332
#77092
#76872
#76672
#76182
#76162
#76012
#75652