Finite Group Databasebeta

← #219498

Subgroups — #219498

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21694C211yesmaximal, characteristic
2#17896C211yesmaximal, characteristic, complemented
3#17855C211yesmaximal, characteristic, complemented
4#9667C211yesmaximal, characteristic, complemented
5#9667C211yesmaximal, characteristic, complemented
6#9106C211yesmaximal, characteristic, complemented
7#9103C211yesmaximal, characteristic, complemented
8#9102C211yesmaximal, characteristic, complemented
9#9099C211yesmaximal, characteristic, complemented
10#9093C211yesmaximal, characteristic
11#9092C211yesmaximal, characteristic, complemented
12#9089C211yesmaximal, characteristic
13#9088C211yesmaximal, characteristic, complemented
14#8780C211yesmaximal, characteristic, complemented
15#8138C211yesmaximal, characteristic, complemented
16C2×#495C411yescharacteristic
17#2983C2211yescharacteristic
18#2982C2211yescharacteristic
19#2975C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#442C2211yescharacteristic
22C2×#441C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25#2326C2211yescharacteristic
26#2326C2211yescharacteristic
27#1972C411yescharacteristic
28#1971C411yescharacteristic
29#1968C411yescharacteristic
30#1968C411yescharacteristic
31#1959C411yescharacteristic
32#1959C411yescharacteristic
33#1844C411yescharacteristic
34#1726C2211yescharacteristic
35#1592—22no—
36#1590—22no—
37SD16⋊C8—22no—
38C8×SD16—22no—
39#1564C2211yescharacteristic
40#1564C2211yescharacteristic
41#1479C2211yescharacteristic
42#1479C2211yescharacteristic
43#1479C2211yescharacteristic
44#1479C2211yescharacteristic
45#1478C2211yescharacteristic
46#1478C2211yescharacteristic
47#1478C2211yescharacteristic
48#1478C2211yescharacteristic
49#1461C2211yescharacteristic
50#1461C2211yescharacteristic
51#1460C2211yescharacteristic
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×SD16C2×C411yescharacteristic
65C22×SD16C2×C411yescharacteristic
66C22×D8C2×C411yescharacteristic
67C2×#124C2×C411yescharacteristic
68C2×#124C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#495—24no—
74#446—22no—
75#446—41no—
76C8×Q8—22no—
77C8×Q8—41no—
78#442—41no—
79#442—41no—
80#441—41no—
81#441—41no—
82C4×SD16—21no—
83C4×SD16—21no—
84C4×SD16—21no—
85C4×SD16—21no—
86C4×SD16—21no—
87C4×SD16—21no—
88C4×SD16—21no—
89C4×SD16—21no—
90#436—22no—
91#436—41no—
92C8×D4—22no—
93C8×D4—41no—
94#432—41no—
95#431—41no—
96#431C2311yescharacteristic
97#431C2311yescharacteristic
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×#106C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116#407—41no—
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#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#388—41no—
129#388—41no—
130#385—41no—
131#385—41no—
132#384C2×C411yescharacteristic
133#384C2×C411yescharacteristic
134#376C2311yescharacteristic
135#335—21no—
136#335—21no—
137#334—21no—
138#334—21no—
139Q8⋊C8—22no—
140Q8⋊C8—22no—
141Q8⋊C8—41no—
142Q8⋊C8—41no—
143D4⋊C8—22no—
144D4⋊C8—22no—
145D4⋊C8—41no—
146D4⋊C8—41no—
147#324—41no—
148#324—41no—
149#323—41no—
150#323—41no—
151#322—21no—
152#322D412yes—
153C82—21no—
154C82D412yes—
155C22×Q8C22×C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C2×Q16—22no—
162C2×Q16—22no—
163C2×Q16—41no—
164C2×SD16—21no—
165C2×SD16—21no—
166C2×SD16—21no—
167C2×SD16—21no—
168C2×SD16—22no—
169C2×SD16—22no—
170C2×SD16—22no—
171C2×SD16M1614yescomplemented
172C2×D8—22no—
173C2×D8—22no—
174C2×D8—41no—
175C2×M16—42no—
176C22×C8—42no—
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181#124—22no—
182#124—22no—
183#124—22no—
184#124—22no—
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×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2411yescharacteristic
220#106—21no—
221#106—21no—
222#106—21no—
223#106—21no—
224#106—41no—
225#106—41no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236#103—21no—
237#103—21no—
238#103—21no—
239#103—21no—
240#103—41no—
241#103—41no—
242#102—21no—
243#102—21no—
244#102—21no—
245#102—21no—
246#102—41no—
247#102—41no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#97—22no—
259#97—22no—
260#97—21no—
261#97—21no—
262#97—42no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—42no—
272C4×C8C2×D412yes—
273C4×C8C2×D412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D411yescharacteristic
276C4×C8#4111yescharacteristic
277C4×C8#4111yescharacteristic
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282C24—21no—
283C24—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×Q8C2×M1612yes—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4C2×M1612yes—
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—41no—
319C22×C4—41no—
320C22×C4—161no—
321C22×C4—161no—
322C22×C4C23×C411yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325Q16—44no—
326SD16—24no—
327SD16—24no—
328SD16—42no—
329SD16—42no—
330D8—44no—
331M16—162no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—162no—
348C2×C8C4×D411yescharacteristic
349C2×C8C4×D411yescharacteristic
350C2×C8C4×D411yescharacteristic
351C2×C8C4×D411yescharacteristic
352C2×C8C2×M1612yes—
353C2×C8#13112yes—
354C2×C8#13112yes—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—161no—
368C4⋊C4—161no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—161no—
376C22⋊C4—161no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42#13112yes—
385C42C22×D411yescharacteristic
386C42C2×#4111yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C42H(5,𝔽2)11yescharacteristic
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23#51811yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416C2×C4—22no—
417C2×C4—22no—
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—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—161no—
441C2×C4—161no—
442C2×C4C2×C4×D411yescharacteristic
443C2×C4#51811yescharacteristic
444C2×C4#53511yescharacteristic
445C2×C4#53511yescharacteristic
446C2×C4#54011yescharacteristic
447C2×C4#54011yescharacteristic
448C2×C4C22×M1611yescharacteristic
449C2×C4C2×#13111yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4C2×#13111yescharacteristic
452C8—21no—
453C8—21no—
454C8—22no—
455C8—42no—
456C8—42no—
457C8—41no—
458C8—81no—
459C8—81no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C8#43512yes—
465C8#43512yes—
466C22—21no—
467C22—21no—
468C22—42no—
469C22—82no—
470C22—82no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—161no—
477C22—161no—
478C22#232811yescharacteristic
479C22#297411yescharacteristic
480C22#297911yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4C2×#43511yescharacteristic
494C4#293111yescharacteristic
495C4#297411yescharacteristic
496C4#297911yescharacteristic
497C2—21no—
498C2—21no—
499C2—82no—
500C2—161no—
501C2#2226211yescharacteristic
502C2#2228411yescharacteristic
503C2#2228411yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8C2×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. #17896C2semidirect
3. #17855C2semidirect
4. #9667C2semidirect
5. #9667C2semidirect
6. #9106C2semidirect
7. #9103C2semidirect
8. #9102C2semidirect
9. #9099C2semidirect
11. #9092C2semidirect
13. #9088C2semidirect
14. #8780C2semidirect
15. #8138C2semidirect
171. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#216942
#178962
#178552
#96672
#91062
#91032
#91022
#90992
#90932
#90922
#90892
#90882
#87802
#81382