Finite Group Databasebeta

← #232174

Subgroups — #232174

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23806C211yesmaximal, characteristic
2#9093C211yesmaximal, characteristic, complemented
3#9091C211yesmaximal, characteristic, complemented
4#9090C211yesmaximal, characteristic, complemented
5#9090C211yesmaximal, characteristic, complemented
6#9089C211yesmaximal, characteristic, complemented
7#9089C211yesmaximal, characteristic, complemented
8#9089C211yesmaximal, characteristic, complemented
9#9087C211yesmaximal, characteristic, complemented
10#8963C211yesmaximal, characteristic, complemented
11#8960C211yesmaximal, characteristic, complemented
12#8959C211yesmaximal, characteristic, complemented
13#8957C211yesmaximal, characteristic, complemented
14#8802C211yesmaximal, characteristic, complemented
15#8802C211yesmaximal, characteristic, complemented
16#2982C2211yescharacteristic
17#2975C2211yescharacteristic
18#2972C2211yescharacteristic
19#2972C2211yescharacteristic
20C2×#442C2211yescharacteristic
21C2×#442C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×#440C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C4×#124C2211yescharacteristic
26#2096C411yescharacteristic
27#2093C411yescharacteristic
28#2092C411yescharacteristic
29#2092C411yescharacteristic
30#2014C411yescharacteristic
31#2012C411yescharacteristic
32#2012C411yescharacteristic
33#2011C411yescharacteristic
34#1745C2211yescharacteristic
35#1479C2211yescharacteristic
36#1479C2211yescharacteristic
37#1478C2211yescharacteristic
38#1478C2211yescharacteristic
39#1478C2211yescharacteristic
40#1478C2211yescharacteristic
41#1478C2211yescharacteristic
42#1478C2211yescharacteristic
43#1461C2211yescharacteristic
44#1460C2211yescharacteristic
45#1460C2211yescharacteristic
46#1460C2211yescharacteristic
47#1281C2211yescharacteristic
48#1281C2211yescharacteristic
49#1281C2211yescharacteristic
50#1281C2211yescharacteristic
51#1276C2211yescharacteristic
52#1276C2211yescharacteristic
53#1276C2211yescharacteristic
54#1276C2211yescharacteristic
55#1275C2211yescharacteristic
56#1275C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59C22×SD16C2×C411yescharacteristic
60C22×SD16C2×C411yescharacteristic
61C22×D8C2×C411yescharacteristic
62C22×D8C2×C411yescharacteristic
63C2×#127C2×C411yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—41no—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C8×Q8—41no—
71C8×Q8—41no—
72#442—41no—
73#442—41no—
74#442—41no—
75#442—41no—
76#440—41no—
77#440—41no—
78#440—41no—
79#440—41no—
80#436—41no—
81#436—41no—
82C8×D4—41no—
83C8×D4—41no—
84C8×D4—41no—
85C8×D4—41no—
86#431—41no—
87#431C2311yescharacteristic
88#431C2311yescharacteristic
89#431C2311yescharacteristic
90#431C2311yescharacteristic
91#431C2311yescharacteristic
92#431C2311yescharacteristic
93#431C2311yescharacteristic
94#431C2311yescharacteristic
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107#407—41no—
108C2×C22⋊C8—41no—
109C2×C22⋊C8—41no—
110C2×#97C2311yescharacteristic
111C2×#97C2311yescharacteristic
112#391C2×C411yescharacteristic
113#390—41no—
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#388—41no—
118#384C2×C411yescharacteristic
119#382C2×C411yescharacteristic
120C4×C22⋊C4—41no—
121C43C2311yescharacteristic
122#336C2×C411yescharacteristic
123#336C2×C411yescharacteristic
124#336C2×C411yescharacteristic
125#336C2×C411yescharacteristic
126Q8⋊C8—41no—
127Q8⋊C8—41no—
128D4⋊C8—41no—
129D4⋊C8—41no—
130D4⋊C8—41no—
131D4⋊C8—41no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134#324—41no—
135#323—41no—
136#323—41no—
137#323—41no—
138C22×Q8C22×C411yescharacteristic
139C22×D4—41no—
140C22×D4C22×C411yescharacteristic
141C22×D4C22×C411yescharacteristic
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C2×SD16—22no—
147C2×SD16—22no—
148C2×SD16—22no—
149C2×SD16—41no—
150C2×SD16—41no—
151C2×SD16—41no—
152C2×D8—22no—
153C2×D8—22no—
154C2×D8—22no—
155C2×D8—41no—
156C2×D8—41no—
157C2×D8—41no—
158C2×M16—42no—
159C22×C8—42no—
160C22×C8—42no—
161C22×C8C22×C411yescharacteristic
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165#127—22no—
166#127—41no—
167#124—22no—
168#124—22no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—84no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C2411yescharacteristic
205#107—41no—
206#107—41no—
207#106—41no—
208#106—41no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—42no—
218#103—41no—
219#103—41no—
220#102—41no—
221#102—41no—
222#102—41no—
223#102—41no—
224#102—41no—
225#102—41no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—42no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237#97—21no—
238#97—21no—
239#97—21no—
240#97—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—42no—
250#95—41no—
251#95—41no—
252C24—21no—
253C24—21no—
254C24—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8#13112yes—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—44no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4#13112yes—
281C2×D4#13112yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4C23×C411yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301SD16—44no—
302SD16—44no—
303D8—44no—
304D8—44no—
305M16—162no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—162no—
327C2×C8—162no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—82no—
339C22⋊C4—44no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—42no—
354C42—41no—
355C42—41no—
356C42—81no—
357C42H(5,𝔽2)11yescharacteristic
358C42H(5,𝔽2)11yescharacteristic
359C42H(5,𝔽2)11yescharacteristic
360C42H(5,𝔽2)11yescharacteristic
361C23—22no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23#51811yescharacteristic
377Q8—22no—
378Q8—22no—
379Q8—41no—
380Q8—41no—
381D4—22no—
382D4—22no—
383D4—22no—
384D4—22no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—88no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—161no—
417C2×C4—161no—
418C2×C4—161no—
419C2×C4#51811yescharacteristic
420C2×C4#51811yescharacteristic
421C2×C4#56011yescharacteristic
422C2×C4#56011yescharacteristic
423C2×C4#56011yescharacteristic
424C2×C4#56011yescharacteristic
425C2×C4C2×#13111yescharacteristic
426C2×C4C2×#13111yescharacteristic
427C2×C4C2×#13111yescharacteristic
428C2×C4#56811yescharacteristic
429C8—41no—
430C8—41no—
431C8—41no—
432C8—41no—
433C8—81no—
434C8—81no—
435C8—81no—
436C8—81no—
437C8—81no—
438C8—81no—
439C8—81no—
440C8—81no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22#238011yescharacteristic
459C22#297611yescharacteristic
460C22#297711yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4#297611yescharacteristic
473C4#297611yescharacteristic
474C4#297711yescharacteristic
475C4#297711yescharacteristic
476C2—21no—
477C2—21no—
478C2—82no—
479C2—82no—
480C2—161no—
481C2#2335811yescharacteristic
482C2#2336111yescharacteristic
483C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2011C4
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. #9093C2semidirect
3. #9091C2semidirect
4. #9090C2semidirect
5. #9090C2semidirect
6. #9089C2semidirect
7. #9089C2semidirect
8. #9089C2semidirect
9. #9087C2semidirect
10. #8963C2semidirect
11. #8960C2semidirect
12. #8959C2semidirect
13. #8957C2semidirect
14. #8802C2semidirect
15. #8802C2semidirect

Maximal subgroups

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

subgroupindex
#238062
#90932
#90912
#90902
#90892
#90872
#89632
#89602
#89592
#89572
#88022