Finite Group Databasebeta

← #189785

Subgroups — #189785

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34060C211yesmaximal, characteristic
2#21853C211yesmaximal, characteristic
3#21374C211yesmaximal, characteristic
4#8711C212yesmaximal, complemented
5#8711C212yesmaximal, complemented
6#8710C212yesmaximal, complemented
7#8709C212yesmaximal, complemented
8#7904C211yesmaximal, characteristic, complemented
9#7903C211yesmaximal, characteristic, complemented
10#7902C211yesmaximal, characteristic, complemented
11#7901C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2320—21no—
18#2319—21no—
19#2318—22no—
20#2317C412yes—
21#2314C412yes—
22#2314C412yes—
23#2311C412yes—
24#2295—21no—
25#2293—21no—
26#2290—22no—
27#2285C2212yes—
28#2283C2212yes—
29C4×#117C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1985—21no—
32#1985—21no—
33#1955—21no—
34#1955—21no—
35#1870—22no—
36#1870—22no—
37#1797C2212yes—
38#1797C2212yes—
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1279C2211yescharacteristic
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2211yescharacteristic
49#1279C2211yescharacteristic
50#1279C2212yes—
51C22×#41C2×C411yescharacteristic
52C22×M16—21no—
53C22×M16—21no—
54C4×#41—22no—
55C4×#41—22no—
56C4×#41—22no—
57C4×#41—22no—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D411yescharacteristic
64C2×#117—22no—
65C2×#117C2×C412yes—
66C2×#117C2×C412yes—
67C2×#117C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74#431—22no—
75#431—22no—
76#423—22no—
77#423—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C4×M16—22no—
90C4×M16—22no—
91C2×#97—21no—
92C2×#97—21no—
93C2×C4×C8C2312yes—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—22no—
97#390—22no—
98#389—22no—
99#389—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#386—22no—
109#386—22no—
110#385—22no—
111#385—22no—
112#384C2×C412yes—
113#384C2×C412yes—
114#383—22no—
115#382—22no—
116#382C2×C412yes—
117#382C2×C412yes—
118#381—22no—
119#380—22no—
120#380C2×C412yes—
121#380C2×C412yes—
122#380C2×C412yes—
123#380C2×C412yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2312yes—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136#376—21no—
137#376—22no—
138C2×#95—21no—
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×C4—22no—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C2×M16—44no—
158C2×M16—42no—
159C2×M16—42no—
160C2×M16—81no—
161C2×M16—81no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180#117—22no—
181#117—22no—
182#117—24no—
183#117—24no—
184#117—44no—
185#117—82no—
186#117—82no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—81no—
221C2×C42—81no—
222C2×C42C22×C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C22⋊C8—82no—
246C22⋊C8—82no—
247#97—42no—
248#97—42no—
249C4×C8—22no—
250C4×C8—22no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—21no—
269C24C2×C22⋊C412yes—
270C24C2×C22⋊C411yescharacteristic
271#41—48no—
272#41—48no—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
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—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#11712yes—
314C22×C4#11712yes—
315C22×C4#12312yes—
316C22×C4#12312yes—
317C22×C4#12312yes—
318C22×C4#12312yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322M16—48no—
323M16—84no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—44no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—44no—
368C42—42no—
369C42—41no—
370C42—41no—
371C42—42no—
372C42—41no—
373C42—41no—
374C42—82no—
375C42—81no—
376C42—81no—
377C42#11712yes—
378C42#11712yes—
379C42#12312yes—
380C42#12312yes—
381C42#12312yes—
382C42#12312yes—
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C42C2×#4111yescharacteristic
387C23—22no—
388C23—21no—
389C23—21no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23#38112yes—
399C23C22×C22⋊C411yescharacteristic
400Q8—44no—
401D4—44no—
402D4—48no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—88no—
446C2×C4#38112yes—
447C2×C4#38112yes—
448C2×C4#38112yes—
449C2×C4C2×#11711yescharacteristic
450C2×C4C2×#11711yescharacteristic
451C2×C4C2×#12311yescharacteristic
452C2×C4C2×#12311yescharacteristic
453C2×C4C2×#12311yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4#56814yes—
456C2×C4#57611yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57711yescharacteristic
459C2×C4#57811yescharacteristic
460C8—44no—
461C8—81no—
462C8—81no—
463C8—82no—
464C8—81no—
465C8—81no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22C2×#38111yescharacteristic
479C22#287912yes—
480C22#289711yescharacteristic
481C22#289811yescharacteristic
482C22#290112yes—
483C4—22no—
484C4—22no—
485C4—21no—
486C4—21no—
487C4—22no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4#292112yes—
496C4#292112yes—
497C4#309611yescharacteristic
498C4#309711yescharacteristic
499C4#309811yescharacteristic
500C4#309911yescharacteristic
501C2—42no—
502C2—82no—
503C2#1750512yes—
504C2#1750612yes—
505C2#2068311yescharacteristic
506C2#2070311yescharacteristic
507C2#2070611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
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. #8711C2semidirect
5. #8711C2semidirect
6. #8710C2semidirect
7. #8709C2semidirect
8. #7904C2semidirect
9. #7903C2semidirect
10. #7902C2semidirect
11. #7901C2semidirect

Maximal subgroups

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

subgroupindex
#340602
#218532
#213742
#87112
#87102
#87092
#79042
#79032
#79022
#79012