Finite Group Databasebeta

← #189801

Subgroups — #189801

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34129C211yesmaximal, characteristic
2#21851C211yesmaximal, characteristic
3#20718C211yesmaximal, characteristic
4#8710C212yesmaximal, complemented
5#8709C212yesmaximal, complemented
6#8708C212yesmaximal, complemented
7#8708C212yesmaximal, complemented
8#7902C211yesmaximal, characteristic, complemented
9#7900C211yesmaximal, characteristic, complemented
10#7893C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#2354C412yes—
18#2354C412yes—
19#2354C412yes—
20Q8×C4⋊C4C412yes—
21#2346—22no—
22#2346—21no—
23#2341—21no—
24#2293—22no—
25#2293—21no—
26#2285C2212yes—
27#2285C2211yescharacteristic
28C42×Q8—21no—
29C4×#117C2212yes—
30C4×#117C2211yescharacteristic
31#1985—21no—
32#1955—21no—
33#1869—22no—
34#1818—21no—
35#1798—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1780—21no—
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1279C2212yes—
42#1279C2211yescharacteristic
43#1279C2211yescharacteristic
44#1279C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2211yescharacteristic
49#1279C2212yes—
50#1279C2211yescharacteristic
51C22×#41D411yescharacteristic
52C23×C8—21no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C4×#41—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D411yescharacteristic
64C2×#117—22no—
65C2×#117—21no—
66C2×#117C2×C412yes—
67C2×#117C2×C412yes—
68C2×#117C2×C412yes—
69C2×#117D412yes—
70C2×#117D411yescharacteristic
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×#97—21no—
95C2×C4×C8C2312yes—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#390—22no—
99#390—22no—
100#390—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#384C2×C412yes—
111#384C2×C412yes—
112#384C2×C412yes—
113#382—22no—
114#382—21no—
115#380—22no—
116#380—21no—
117#380C2×C412yes—
118#380C2×C412yes—
119#380C2×C411yescharacteristic
120#380C2×C411yescharacteristic
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2312yes—
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136#376—22no—
137#376—21no—
138C43—22no—
139C43—21no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—44no—
147C22×Q8C22⋊C411yescharacteristic
148C22×D4C22⋊C412yes—
149C22×D4C22⋊C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—42no—
163C22×C8—81no—
164C22×C8—81no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—48no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178#117—22no—
179#117—22no—
180#117—24no—
181#117—24no—
182#117—24no—
183#117—24no—
184#117—24no—
185#117—42no—
186#117—42no—
187#117—42no—
188#117—81no—
189#117—82no—
190#117—81no—
191#117—82no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42C22×C412yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42C2411yescharacteristic
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C22⋊C8—82no—
255C22⋊C8—82no—
256#97—42no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—21no—
262C4×C8—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—21no—
273C24C2×C22⋊C412yes—
274C24C2×C22⋊C411yescharacteristic
275#41—48no—
276#41—48no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—42no—
283C2×D4—42no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4#11712yes—
318C22×C4#11712yes—
319C22×C4#12312yes—
320C22×C4#12312yes—
321C22×C4#12312yes—
322C22×C4#12312yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—84no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4#13114yes—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—81no—
368C22⋊C4—81no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—44no—
378C42—44no—
379C42—44no—
380C42—44no—
381C42—44no—
382C42—42no—
383C42—41no—
384C42—41no—
385C42—42no—
386C42—41no—
387C42—41no—
388C42—44no—
389C42#11712yes—
390C42#11712yes—
391C42#12312yes—
392C42#12312yes—
393C42#12312yes—
394C42#12312yes—
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C23—22no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23#38112yes—
411C23C22×C22⋊C411yescharacteristic
412Q8—28no—
413D4—48no—
414D4—44no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4#38112yes—
460C2×C4#38112yes—
461C2×C4#38112yes—
462C2×C4#42114yes—
463C2×C4C2×#11711yescharacteristic
464C2×C4C2×#11711yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C2×C4C2×#12311yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4C2×#13112yes—
470C2×C4#56812yes—
471C2×C4#57611yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57811yescharacteristic
475C8—81no—
476C8—81no—
477C8—82no—
478C8—82no—
479C8—81no—
480C8—81no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22C2×#38111yescharacteristic
494C22#288012yes—
495C22#289711yescharacteristic
496C22#289811yescharacteristic
497C22C2×#42112yes—
498C4—22no—
499C4—22no—
500C4—21no—
501C4—21no—
502C4—22no—
503C4—24no—
504C4—44no—
505C4—42no—
506C4—42no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4#292211yescharacteristic
511C4#292311yescharacteristic
512C4#292512yes—
513C4#309611yescharacteristic
514C4#309711yescharacteristic
515C4#309811yescharacteristic
516C4#309911yescharacteristic
517C2—42no—
518C2—82no—
519C2#1752312yes—
520C2#1752612yes—
521C2#2068411yescharacteristic
522C2#2070311yescharacteristic
523C2#2070511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8710C2semidirect
5. #8709C2semidirect
6. #8708C2semidirect
7. #8708C2semidirect
8. #7902C2semidirect
9. #7900C2semidirect
10. #7893C2semidirect
11. #7891C2semidirect

Maximal subgroups

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

subgroupindex
#341292
#218512
#207182
#87102
#87092
#87082
#79022
#79002
#78932
#78912