Finite Group Databasebeta

← #219505

Subgroups — #219505

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21703C211yesmaximal, characteristic
2#17897C211yesmaximal, characteristic, complemented
3#17855C211yesmaximal, characteristic
4#9669C211yesmaximal, characteristic, complemented
5#9667C211yesmaximal, characteristic
6#9106C211yesmaximal, characteristic
7#9104C211yesmaximal, characteristic, complemented
8#9100C211yesmaximal, characteristic, complemented
9#9099C211yesmaximal, characteristic
10#9094C211yesmaximal, characteristic
11#9092C211yesmaximal, characteristic, complemented
12#9088C211yesmaximal, characteristic, complemented
13#9087C211yesmaximal, characteristic
14#8781C211yesmaximal, characteristic, complemented
15#8137C211yesmaximal, characteristic, complemented
16C2×#497C411yescharacteristic
17#2983C2211yescharacteristic
18#2983C2211yescharacteristic
19#2982C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#441C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25#2327C2211yescharacteristic
26#2326C2211yescharacteristic
27#1973C411yescharacteristic
28#1971C411yescharacteristic
29#1969C411yescharacteristic
30#1968C411yescharacteristic
31#1958C411yescharacteristic
32#1956C411yescharacteristic
33#1847C411yescharacteristic
34#1726C2211yescharacteristic
35#1593—22no—
36#1591—22no—
37SD16⋊C8—22no—
38C8×SD16—22no—
39#1564C2211yescharacteristic
40#1564C2211yescharacteristic
41#1479C2211yescharacteristic
42#1479C2211yescharacteristic
43#1479C2211yescharacteristic
44#1479C2211yescharacteristic
45#1479C2211yescharacteristic
46#1479C2211yescharacteristic
47#1478C2211yescharacteristic
48#1478C2211yescharacteristic
49#1461C2211yescharacteristic
50#1461C2211yescharacteristic
51#1461C2211yescharacteristic
52#1460C2211yescharacteristic
53#1450C2211yescharacteristic
54#1450C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59#1274C2211yescharacteristic
60#1274C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C22×Q16C2×C411yescharacteristic
64C22×Q16C2×C411yescharacteristic
65C22×SD16C2×C411yescharacteristic
66C22×SD16C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#124C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#497—24no—
74#446—22no—
75#446—22no—
76#446—41no—
77#446—41no—
78C8×Q8—22no—
79C8×Q8—41no—
80#441—41no—
81#441—41no—
82#440—41no—
83#440—41no—
84C4×Q16—21no—
85C4×Q16—21no—
86C4×Q16—21no—
87C4×Q16—21no—
88C4×SD16—21no—
89C4×SD16—21no—
90C4×SD16—21no—
91C4×SD16—21no—
92C8×D4—22no—
93C8×D4—41no—
94#432—41no—
95#432—41no—
96#431—41no—
97#431C2311yescharacteristic
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104#431C2311yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107C2×#106C2×C411yescharacteristic
108C2×#106C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×C22⋊C8—41no—
118C2×#97C2×C411yescharacteristic
119C2×#97C2×C411yescharacteristic
120C2×#97C2311yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122C2×C4×C8C2×C411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#388—41no—
129#387—41no—
130#387—41no—
131#385—41no—
132#384C2×C411yescharacteristic
133#382C2×C411yescharacteristic
134#376C2311yescharacteristic
135#335—21no—
136#335—21no—
137#334—21no—
138#334—21no—
139Q8⋊C8—22no—
140Q8⋊C8—22no—
141Q8⋊C8—22no—
142Q8⋊C8—41no—
143Q8⋊C8—41no—
144Q8⋊C8—41no—
145D4⋊C8—22no—
146D4⋊C8—41no—
147#324—41no—
148#324—41no—
149#324—41no—
150#323—41no—
151#322—21no—
152#322D412yes—
153C82—21no—
154C82D412yes—
155C22×Q8C22×C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C2×Q16—22no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—22no—
164C2×Q16—22no—
165C2×Q16—22no—
166C2×Q16—41no—
167C2×SD16—22no—
168C2×SD16—21no—
169C2×SD16—21no—
170C2×SD16—22no—
171C2×SD16—22no—
172C2×SD16—41no—
173C2×SD16M1614yes—
174C22×C8—42no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181#124—22no—
182#124—22no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2411yescharacteristic
217#107—21no—
218#107—21no—
219#107—41no—
220#106—21no—
221#106—21no—
222#106—41no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234#103—21no—
235#103—21no—
236#103—21no—
237#103—21no—
238#103—21no—
239#103—21no—
240#103—41no—
241#103—41no—
242#103—41no—
243#102—21no—
244#102—21no—
245#102—41no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—42no—
255#97—22no—
256#97—22no—
257#97—21no—
258#97—21no—
259#97—42no—
260#97—42no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270C4×C8C2×D412yes—
271C4×C8C2×D412yes—
272C4×C8C2×D411yescharacteristic
273C4×C8C2×D411yescharacteristic
274C4×C8#4111yescharacteristic
275C4×C8#4111yescharacteristic
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8C2×M1612yes—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C2×M1612yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—161no—
319C22×C4C23×C411yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C22×C4H(5,𝔽2)11yescharacteristic
322Q16—44no—
323Q16—42no—
324Q16—42no—
325SD16—24no—
326SD16—24no—
327SD16—44no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—162no—
344C2×C8C4×D411yescharacteristic
345C2×C8C4×D411yescharacteristic
346C2×C8C4×D411yescharacteristic
347C2×C8C4×D411yescharacteristic
348C2×C8C2×M1612yes—
349C2×C8#13112yes—
350C2×C8#13112yes—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—44no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—161no—
367C22⋊C4—44no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—161no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42#13112yes—
381C42C22×D411yescharacteristic
382C42C2×#4111yescharacteristic
383C42H(5,𝔽2)11yescharacteristic
384C42H(5,𝔽2)11yescharacteristic
385C23—22no—
386C23—41no—
387C23—41no—
388C23—81no—
389C23—81no—
390C23#51811yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—41no—
394Q8—41no—
395Q8—41no—
396Q8—41no—
397Q8—41no—
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—41no—
403D4—22no—
404D4—22no—
405D4—41no—
406D4—41no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—161no—
434C2×C4C2×C4×D411yescharacteristic
435C2×C4#51811yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4#53511yescharacteristic
438C2×C4#54011yescharacteristic
439C2×C4#54011yescharacteristic
440C2×C4C22×M1611yescharacteristic
441C2×C4C2×#13111yescharacteristic
442C2×C4C2×#13111yescharacteristic
443C2×C4C2×#13111yescharacteristic
444C8—21no—
445C8—21no—
446C8—22no—
447C8—42no—
448C8—42no—
449C8—41no—
450C8—81no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8#43512yes—
457C8#43512yes—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—82no—
462C22—81no—
463C22—81no—
464C22—161no—
465C22#232811yescharacteristic
466C22#297411yescharacteristic
467C22#297911yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—42no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4C2×#43511yescharacteristic
482C4#293111yescharacteristic
483C4#297411yescharacteristic
484C4#297911yescharacteristic
485C2—21no—
486C2—21no—
487C2—82no—
488C2#2226211yescharacteristic
489C2#2228411yescharacteristic
490C2#2228511yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17897C2semidirect
4. #9669C2semidirect
7. #9104C2semidirect
8. #9100C2semidirect
11. #9092C2semidirect
12. #9088C2semidirect
14. #8781C2semidirect
15. #8137C2semidirect

Maximal subgroups

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

subgroupindex
#217032
#178972
#178552
#96692
#96672
#91062
#91042
#91002
#90992
#90942
#90922
#90882
#90872
#87812
#81372