Finite Group Databasebeta

← #232128

Subgroups — #232128

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21692C211yesmaximal, characteristic
2#9094C211yesmaximal, characteristic, complemented
3#9094C211yesmaximal, characteristic, complemented
4#9091C211yesmaximal, characteristic, complemented
5#9091C211yesmaximal, characteristic, complemented
6#9090C211yesmaximal, characteristic, complemented
7#9090C211yesmaximal, characteristic, complemented
8#9087C211yesmaximal, characteristic, complemented
9#9087C211yesmaximal, characteristic, complemented
10#8963C211yesmaximal, characteristic
11#8962C211yesmaximal, characteristic, complemented
12#8959C211yesmaximal, characteristic
13#8958C211yesmaximal, characteristic, complemented
14#8801C211yesmaximal, characteristic, complemented
15#8800C211yesmaximal, characteristic, complemented
16C2×#492C411yescharacteristic
17#2983C2211yescharacteristic
18#2982C2211yescharacteristic
19#2975C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#440C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×#440C2211yescharacteristic
24C2×#440C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27#1973C411yescharacteristic
28#1973C411yescharacteristic
29#1973C411yescharacteristic
30#1973C411yescharacteristic
31#1957C411yescharacteristic
32#1956C411yescharacteristic
33#1845C411yescharacteristic
34C4×#97C2211yescharacteristic
35#1479C2211yescharacteristic
36#1479C2211yescharacteristic
37#1479C2211yescharacteristic
38#1479C2211yescharacteristic
39#1478C2211yescharacteristic
40#1478C2211yescharacteristic
41#1478C2211yescharacteristic
42#1478C2211yescharacteristic
43#1461C2211yescharacteristic
44#1461C2211yescharacteristic
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×SD16C2×C411yescharacteristic
62C22×SD16C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—41no—
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70#492—24no—
71#446—41no—
72#446—41no—
73C8×Q8—41no—
74C8×Q8—41no—
75#440—41no—
76#440—41no—
77#440—41no—
78#440—41no—
79#440—41no—
80#440—41no—
81#440—41no—
82#440—41no—
83#436—41no—
84#436—41no—
85C8×D4—41no—
86C8×D4—41no—
87#432—41no—
88#431—41no—
89#431C2311yescharacteristic
90#431C2311yescharacteristic
91#431C2311yescharacteristic
92#431C2311yescharacteristic
93#431C2311yescharacteristic
94#431C2311yescharacteristic
95#431C2311yescharacteristic
96#431C2311yescharacteristic
97C2×#107C2×C411yescharacteristic
98C2×#107C2×C411yescharacteristic
99C2×#107C2×C411yescharacteristic
100C2×#107C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109#407—41no—
110C2×C22⋊C8—41no—
111C2×#97C2×C411yescharacteristic
112C2×#97C2×C411yescharacteristic
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115C2×C4×C8C2×C411yescharacteristic
116C2×C4×C8C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#390—41no—
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#389—41no—
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125C4×C4⋊C4—41no—
126C43C2311yescharacteristic
127Q8⋊C8—41no—
128Q8⋊C8—41no—
129Q8⋊C8—41no—
130Q8⋊C8—41no—
131D4⋊C8—41no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134D4⋊C8—41no—
135#324—41no—
136#324—41no—
137#323—41no—
138#323—41no—
139C22×Q8C22×C411yescharacteristic
140C22×Q8C22×C411yescharacteristic
141C22×D4—41no—
142C22×D4C22×C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C2×SD16—22no—
147C2×SD16—22no—
148C2×SD16—22no—
149C2×SD16—22no—
150C2×SD16—22no—
151C2×SD16—41no—
152C2×SD16—41no—
153C2×SD16—41no—
154C2×SD16—41no—
155C2×SD16—41no—
156C2×SD16—41no—
157C2×SD16M1614yescomplemented
158C2×M16—42no—
159C22×C8—42no—
160C22×C8C22×C411yescharacteristic
161C22×C8C22×C411yescharacteristic
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166#127—22no—
167#127—22no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—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—84no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C2411yescharacteristic
205#107—41no—
206#107—41no—
207#107—41no—
208#107—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—
218C4⋊C8—42no—
219#103—41no—
220#103—41no—
221#103—41no—
222#103—41no—
223#102—41no—
224#102—41no—
225#102—41no—
226#102—41no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237#97—22no—
238#97—21no—
239#97—21no—
240#97—21no—
241#97—21no—
242#97—41no—
243#97—42no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—42no—
255#95—41no—
256#95—41no—
257C24—21no—
258C24—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8C2×M1612yes—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—44no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4C2×M1612yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C23×C411yescharacteristic
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303SD16—44no—
304SD16—44no—
305SD16—44no—
306SD16—44no—
307M16—162no—
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×C8C2×M1612yes—
328C2×C8#13112yes—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—44no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42#13112yes—
360C42H(5,𝔽2)11yescharacteristic
361C42H(5,𝔽2)11yescharacteristic
362C42H(5,𝔽2)11yescharacteristic
363C42H(5,𝔽2)11yescharacteristic
364C23—22no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23#51811yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—41no—
378Q8—41no—
379Q8—41no—
380Q8—41no—
381Q8—41no—
382Q8—41no—
383D4—22no—
384D4—22no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—88no—
392C2×C4—22no—
393C2×C4—22no—
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—41no—
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#51811yescharacteristic
419C2×C4#51811yescharacteristic
420C2×C4#56011yescharacteristic
421C2×C4#56011yescharacteristic
422C2×C4#56011yescharacteristic
423C2×C4#56011yescharacteristic
424C2×C4C22×M1611yescharacteristic
425C2×C4C2×#13111yescharacteristic
426C2×C4C2×#13111yescharacteristic
427C2×C4#56811yescharacteristic
428C8—22no—
429C8—22no—
430C8—41no—
431C8—41no—
432C8—81no—
433C8—81no—
434C8—81no—
435C8—81no—
436C8—81no—
437C8—81no—
438C8—81no—
439C8—81no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—161no—
451C22—161no—
452C22#238011yescharacteristic
453C22#297411yescharacteristic
454C22#298111yescharacteristic
455C4—24no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4#297311yescharacteristic
468C4#297411yescharacteristic
469C4#297911yescharacteristic
470C4#298111yescharacteristic
471C2—21no—
472C2—21no—
473C2—82no—
474C2—161no—
475C2#2335611yescharacteristic
476C2#2336011yescharacteristic
477C2#2336111yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×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. #9094C2semidirect
3. #9094C2semidirect
4. #9091C2semidirect
5. #9091C2semidirect
6. #9090C2semidirect
7. #9090C2semidirect
8. #9087C2semidirect
9. #9087C2semidirect
11. #8962C2semidirect
13. #8958C2semidirect
14. #8801C2semidirect
15. #8800C2semidirect
157. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#216922
#90942
#90912
#90902
#90872
#89632
#89622
#89592
#89582
#88012
#88002