Finite Group Databasebeta

← #162437

Subgroups — #162437

Order 512.

810 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#35197C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7730C212yesmaximal, complemented
5#7718C212yesmaximal, complemented
6#7670C212yesmaximal, complemented
7#7658C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12#2854C2212yes—
13#2850C2212yes—
14#2648—22no—
15#2571C411yescharacteristic
16#2548C411yescharacteristic
17#2541C411yescharacteristic
18#2538C411yescharacteristic
19#2513—21no—
20#2457—22no—
21#2445—21no—
22#2427C412yes—
23#2420C412yes—
24#2328—21no—
25#2319—21no—
26C4×#127C2211yescharacteristic
27#2295C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1923—22no—
33#1923—22no—
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C2×#127—21no—
54C2×#127C2×C411yescharacteristic
55C2×#124C2×C412yes—
56C2×#123—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×#121C2×C412yes—
59C2×#121C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C22×C42C2312yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—22no—
80#400—22no—
81#397—22no—
82#397—22no—
83#396—22no—
84#394—22no—
85#394—22no—
86#394—22no—
87#392—22no—
88#391C2×C411yescharacteristic
89#391C2311yescharacteristic
90#390—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2312yes—
98#389—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C411yescharacteristic
102#389C2311yescharacteristic
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—21no—
108#387—21no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#384C2×C411yescharacteristic
115#382—21no—
116#382C2×C412yes—
117#381—22no—
118#380—22no—
119#380C2×C412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128#376—21no—
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140C22×Q8C22×C411yescharacteristic
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22⋊C411yescharacteristic
146C22×D4C22×C412yes—
147C23×C4—22no—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157#127—24no—
158#127—44no—
159#124—82no—
160#123—82no—
161#123—82no—
162C22⋊Q8—82no—
163#121—82no—
164#121—82no—
165C4×Q8—24no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—81no—
173C4×D4—81no—
174C4×D4—82no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—82no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—82no—
220C22⋊C8—44no—
221C22⋊C8—82no—
222C22⋊C8—82no—
223C22⋊C8—82no—
224C4×C8—42no—
225C4×C8—42no—
226C4×C8—81no—
227C4×C8—82no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C24—22no—
243C24C2×C22⋊C412yes—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—81no—
260C2×D4—81no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—21no—
344C42—21no—
345C42—22no—
346C42—42no—
347C42—41no—
348C42—42no—
349C42—41no—
350C42—42no—
351C42—41no—
352C42—42no—
353C42—42no—
354C42#13114yes—
355C42#13512yes—
356C42#13512yes—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23C22×C22⋊C411yescharacteristic
372Q8—82no—
373D4—44no—
374D4—84no—
375D4—84no—
376D4—84no—
377D4—164no—
378D4—164no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—84no—
421C2×C4—164no—
422C2×C4#40812yes—
423C2×C4#40812yes—
424C2×C4#41612yes—
425C2×C4#41612yes—
426C2×C4#42112yes—
427C2×C4C2×#13112yes—
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4#57611yescharacteristic
431C2×C4#57711yescharacteristic
432C8—42no—
433C8—81no—
434C8—82no—
435C8—81no—
436C8—81no—
437C8—82no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22C2×#40811yescharacteristic
453C22#288311yescharacteristic
454C22#288811yescharacteristic
455C22C2×#41611yescharacteristic
456C22#289711yescharacteristic
457C22C2×#42111yescharacteristic
458C22#290111yescharacteristic
459C4—22no—
460C4—22no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—41no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—81no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4#148214yes—
476C2—82no—
477C2—162no—
478C2#1738911yescharacteristic
479C2C2×#148211yescharacteristic
480C2#1748311yescharacteristic
481C2#1759911yescharacteristic
482C2#1761711yescharacteristic
483C2#1768111yescharacteristic
484C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22835C2semidirect
3. #22008C2semidirect
4. #7730C2semidirect
5. #7718C2semidirect
6. #7670C2semidirect
7. #7658C2semidirect
8. #7587C2semidirect
9. #7577C2semidirect
10. #7569C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#351972
#228352
#220082
#77302
#77182
#76702
#76582
#75872
#75772
#75692
#75622