Finite Group Databasebeta

← #219712

Subgroups — #219712

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23809C211yesmaximal, characteristic
2#17880C211yesmaximal, characteristic, complemented
3#17879C211yesmaximal, characteristic
4#9669C211yesmaximal, characteristic, complemented
5#9667C211yesmaximal, characteristic
6#9104C211yesmaximal, characteristic, complemented
7#9104C211yesmaximal, characteristic, complemented
8#9102C211yesmaximal, characteristic
9#9099C211yesmaximal, characteristic
10#9094C211yesmaximal, characteristic
11#9092C211yesmaximal, characteristic, complemented
12#9088C211yesmaximal, characteristic, complemented
13#9087C211yesmaximal, characteristic
14#8782C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#2983C2211yescharacteristic
17#2983C2211yescharacteristic
18#2982C2211yescharacteristic
19#2972C2211yescharacteristic
20C2×#441C2211yescharacteristic
21C2×#440C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24#2327C2211yescharacteristic
25#2326C2211yescharacteristic
26#2097C411yescharacteristic
27#2096C411yescharacteristic
28#2095C411yescharacteristic
29#2092C411yescharacteristic
30#2016C411yescharacteristic
31#2015C411yescharacteristic
32#2015C411yescharacteristic
33#2013C411yescharacteristic
34#1746C2211yescharacteristic
35#1587—22no—
36#1585—22no—
37#1584—22no—
38#1582—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#1452C2211yescharacteristic
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#446—22no—
74#446—22no—
75#446—41no—
76#446—41no—
77C8×Q8—22no—
78C8×Q8—41no—
79#441—41no—
80#441—41no—
81#440—41no—
82#440—41no—
83C4×Q16—21no—
84C4×Q16—21no—
85C4×Q16—21no—
86C4×Q16—21no—
87C4×SD16—21no—
88C4×SD16—21no—
89C4×SD16—21no—
90C4×SD16—21no—
91C8×D4—22no—
92C8×D4—41no—
93#432—41no—
94#432—41no—
95#431—41no—
96#431C2311yescharacteristic
97#431C2311yescharacteristic
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104C2×#107C2×C411yescharacteristic
105C2×#107C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×#106C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×C22⋊C8—41no—
117C2×#97C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#388—41no—
124#387—41no—
125#387—41no—
126#385—41no—
127#384C2×C411yescharacteristic
128#382C2×C411yescharacteristic
129#376C2311yescharacteristic
130#336C2×C411yescharacteristic
131#336C2×C411yescharacteristic
132#336C2×C411yescharacteristic
133#336C2×C411yescharacteristic
134#335—21no—
135#335—21no—
136#334—21no—
137#334—21no—
138Q8⋊C8—22no—
139Q8⋊C8—22no—
140Q8⋊C8—22no—
141Q8⋊C8—41no—
142Q8⋊C8—41no—
143Q8⋊C8—41no—
144D4⋊C8—22no—
145D4⋊C8—41no—
146#324—41no—
147#324—41no—
148#324—41no—
149#323—41no—
150#322—21no—
151#322D412yes—
152#322D412yes—
153C82—21no—
154C22×Q8C22×C411yescharacteristic
155C22×Q8C22×C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C2×Q16—22no—
160C2×Q16—22no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—41no—
164C2×Q16—41no—
165C2×Q16—41no—
166C2×SD16—22no—
167C2×SD16—22no—
168C2×SD16—21no—
169C2×SD16—21no—
170C2×SD16—41no—
171C2×SD16—41no—
172C2×SD16—41no—
173C22×C8—42no—
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180#124—22no—
181#124—22no—
182C4×Q8—21no—
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×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C2411yescharacteristic
216#107—21no—
217#107—21no—
218#107—41no—
219#106—21no—
220#106—21no—
221#106—41no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233#103—21no—
234#103—21no—
235#103—21no—
236#103—21no—
237#103—21no—
238#103—21no—
239#103—41no—
240#103—41no—
241#103—41no—
242#102—21no—
243#102—21no—
244#102—41no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—42no—
254#97—21no—
255#97—21no—
256#97—42no—
257#97—42no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—42no—
265C4×C8C2×D412yes—
266C4×C8C2×D412yes—
267C4×C8C2×D411yescharacteristic
268C4×C8C2×D411yescharacteristic
269C4×C8#4111yescharacteristic
270C4×C8#4111yescharacteristic
271#95—21no—
272#95—21no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—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#13112yes—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4#13112yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—161no—
314C22×C4C23×C411yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317Q16—42no—
318Q16—42no—
319Q16—44no—
320SD16—42no—
321SD16—42no—
322SD16—44no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—162no—
342C2×C8C4×D411yescharacteristic
343C2×C8C4×D411yescharacteristic
344C2×C8C4×D411yescharacteristic
345C2×C8C4×D411yescharacteristic
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—44no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—161no—
362C22⋊C4—44no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—161no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42C22×D411yescharacteristic
377C42C2×#4111yescharacteristic
378C42H(5,𝔽2)11yescharacteristic
379C42H(5,𝔽2)11yescharacteristic
380C23—22no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23—81no—
385C23#51811yescharacteristic
386Q8—22no—
387Q8—22no—
388Q8—41no—
389Q8—41no—
390Q8—41no—
391Q8—41no—
392Q8—41no—
393Q8—41no—
394Q8—41no—
395Q8—41no—
396Q8—41no—
397Q8—41no—
398D4—22no—
399D4—22no—
400D4—41no—
401D4—41no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
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—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—161no—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4#51811yescharacteristic
431C2×C4#53511yescharacteristic
432C2×C4#53511yescharacteristic
433C2×C4#54011yescharacteristic
434C2×C4#54011yescharacteristic
435C2×C4C2×#13111yescharacteristic
436C2×C4C2×#13111yescharacteristic
437C2×C4#56811yescharacteristic
438C2×C4#56811yescharacteristic
439C8—21no—
440C8—21no—
441C8—21no—
442C8—21no—
443C8—42no—
444C8—42no—
445C8—41no—
446C8—41no—
447C8—81no—
448C8—81no—
449C8—81no—
450C8—81no—
451C8—81no—
452C8—81no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—82no—
457C22—81no—
458C22—81no—
459C22—161no—
460C22#232811yescharacteristic
461C22#297711yescharacteristic
462C22#297811yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4#293311yescharacteristic
477C4#293311yescharacteristic
478C4#297611yescharacteristic
479C4#297611yescharacteristic
480C2—21no—
481C2—21no—
482C2—82no—
483C2#2226711yescharacteristic
484C2#2228411yescharacteristic
485C2#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. #17880C2semidirect
4. #9669C2semidirect
6. #9104C2semidirect
7. #9104C2semidirect
11. #9092C2semidirect
12. #9088C2semidirect
14. #8782C2semidirect
15. #8141C2semidirect

Maximal subgroups

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

subgroupindex
#238092
#178802
#178792
#96692
#96672
#91042
#91022
#90992
#90942
#90922
#90882
#90872
#87822
#81412