Finite Group Databasebeta

← #188773

Subgroups — #188773

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2925C211yesmaximal, characteristic
2#17806C212yesmaximal, complemented
3#17361C211yesmaximal, characteristic
4#17349C211yesmaximal, characteristic
5#8603C214yesmaximal, complemented
6#8602C212yesmaximal, complemented
7#8601C212yesmaximal, complemented
8#7863C212yesmaximal, complemented
9C2×C4×#41C2211yescharacteristic
10#2925—24no—
11#2925—22no—
12#2925—22no—
13#2920C2212yes—
14C2×#433C412yes—
15C2×#433C2212yes—
16C2×#432C411yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C411yescharacteristic
19C2×#431C2212yes—
20C2×#431C2211yescharacteristic
21C2×#423C411yescharacteristic
22C22×C4⋊C8C411yescharacteristic
23C2×#408C412yes—
24C2×#405C2212yes—
25#1574—22no—
26#1574—22no—
27#1573—24no—
28#1564C2214yes—
29#1564C2214yes—
30#1467—21no—
31#1466—21no—
32#1465—21no—
33#1465—21no—
34#1461—21no—
35#1460—21no—
36#1459—21no—
37#1459—21no—
38#1458C2211yescharacteristic
39#1458C2212yes—
40#1458C2211yescharacteristic
41#1458C2212yes—
42#1458C2212yes—
43#1275C2214yes—
44#1275C2214yes—
45C22×#41C2×C411yescharacteristic
46C22×M16C2×C412yes—
47C23×C8C2×C412yes—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41—22no—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8Q811yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4Q811yescharacteristic
58C2×C4×D4Q811yescharacteristic
59C2×C4×D4Q811yescharacteristic
60C2×#117C2×C411yescharacteristic
61C2×#117C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2311yescharacteristic
66#433—24no—
67#433—24no—
68#433—24no—
69#433—24no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#431—22no—
77#431—21no—
78#431—21no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431D414yes—
84#431D412yes—
85#431D412yes—
86#431C2312yes—
87#431C2312yes—
88#431C2314yes—
89#423—22no—
90#423—22no—
91#423—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2×C412yes—
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2×C412yes—
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8C2×C411yescharacteristic
102#408—24no—
103#408—24no—
104#407—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8C2×C412yes—
107C2×C22⋊C8C2×C412yes—
108C2×C22⋊C8C2×C412yes—
109C2×C22⋊C8C2×C412yes—
110#405—24no—
111#405—24no—
112C4×M16—22no—
113C2×#97C2×C412yes—
114C2×#97C2312yes—
115C2×C4×C8—22no—
116C2×C4×C8C2×C412yes—
117C2×C4×C8C2312yes—
118#335—24no—
119#335—24no—
120#334—24no—
121#334—24no—
122#324—21no—
123#324—21no—
124#324—22no—
125#324—22no—
126#324—22no—
127#323—21no—
128#323—21no—
129#323—22no—
130#323—22no—
131#323—22no—
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—41no—
137C2×#41—41no—
138C22×Q8C4⋊C411yescharacteristic
139C22×D4C4⋊C411yescharacteristic
140C22×D4C4⋊C411yescharacteristic
141C22×D4C4⋊C411yescharacteristic
142C23×C4C4⋊C411yescharacteristic
143C23×C4C4⋊C411yescharacteristic
144C23×C4C4⋊C411yescharacteristic
145C23×C4C4⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×Q811yescharacteristic
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—42no—
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8C22×C412yes—
161C22×C8C22×C412yes—
162C22×C8C22×C412yes—
163C22×C8C22×C412yes—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186C2×C4⋊C4C4⋊C411yescharacteristic
187C2×C4⋊C4C4⋊C411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×Q811yescharacteristic
191C2×C4⋊C4C2×Q811yescharacteristic
192C2×C22⋊C4C4⋊C411yescharacteristic
193C2×C22⋊C4C4⋊C411yescharacteristic
194C2×C22⋊C4C2×D411yescharacteristic
195C2×C22⋊C4C2×D411yescharacteristic
196C2×C22⋊C4C2×Q811yescharacteristic
197C2×C22⋊C4C2×Q811yescharacteristic
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C4⋊C411yescharacteristic
207C2×C42C4⋊C411yescharacteristic
208C2×C42C4⋊C411yescharacteristic
209C2×C42C4⋊C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q811yescharacteristic
217C2×C42#4112yes—
218C2×C42#4112yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—24no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—24no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8C22⋊C412yes—
233C4⋊C8C22⋊C411yescharacteristic
234C4⋊C8C22⋊C411yescharacteristic
235C4⋊C8C22⋊C412yes—
236C4⋊C8C22⋊C412yes—
237C22⋊C8—24no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—24no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—24no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8C22⋊C414yes—
247C22⋊C8C22⋊C414yes—
248#97—24no—
249#97—24no—
250#97—42no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8—22no—
255C4×C8—24no—
256C4×C8—42no—
257C4×C8C2×D414yes—
258C4×C8C2×D414yes—
259C24C2×C4⋊C411yescharacteristic
260C24C2×C4⋊C411yescharacteristic
261C24C2×C4⋊C411yescharacteristic
262#41—44no—
263#41—44no—
264#41—44no—
265#41—44no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
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—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4C2×C4⋊C411yescharacteristic
309C22×C4C2×C4⋊C411yescharacteristic
310C22×C4C2×C4⋊C411yescharacteristic
311C22×C4C2×C4⋊C411yescharacteristic
312C22×C4C2×C4⋊C411yescharacteristic
313C22×C4C2×C4⋊C411yescharacteristic
314C22×C4C2×C4⋊C411yescharacteristic
315C22×C4C2×C4⋊C411yescharacteristic
316C22×C4#11712yes—
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4C22⋊Q812yes—
320C22×C4C22⋊Q812yes—
321C22×C4#13112yes—
322C22×C4#13112yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q811yescharacteristic
326M16—48no—
327M16—84no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—44no—
343C2×C8—84no—
344C2×C8C2×C22⋊C412yes—
345C2×C8C2×C22⋊C412yes—
346C2×C8C2×C22⋊C414yes—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42C2×C22⋊C411yescharacteristic
372C42C2×C22⋊C411yescharacteristic
373C42#11711yescharacteristic
374C42#11711yescharacteristic
375C42#12112yes—
376C42#12112yes—
377C42C22⋊Q812yes—
378C42C22⋊Q812yes—
379C42C22×D412yes—
380C42C2×#4112yes—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#38012yes—
395C23#42412yes—
396C23C22×C4⋊C411yescharacteristic
397Q8—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4#38011yescharacteristic
445C2×C4#38011yescharacteristic
446C2×C4#38011yescharacteristic
447C2×C4#38012yes—
448C2×C4#38011yescharacteristic
449C2×C4#42412yes—
450C2×C4C22×C22⋊C411yescharacteristic
451C2×C4C2×#11711yescharacteristic
452C2×C4C2×#12112yes—
453C2×C4C2×C22⋊Q812yes—
454C2×C4C2×#13111yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4#56811yescharacteristic
457C2×C4#56811yescharacteristic
458C8—22no—
459C8—22no—
460C8—24no—
461C8—42no—
462C8—42no—
463C8—82no—
464C8—82no—
465C8—84no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22C2×#38011yescharacteristic
484C22C2×#42411yescharacteristic
485C22#290511yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—22no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4#181312yes—
499C4#288011yescharacteristic
500C4#288011yescharacteristic
501C4#292411yescharacteristic
502C4#292411yescharacteristic
503C2—21no—
504C2—21no—
505C2—42no—
506C2—42no—
507C2—81no—
508C2#2062811yescharacteristic
509C2#2067111yescharacteristic
510C2C2×#181311yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#380
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#432 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17806C2semidirect
5. #8603C2semidirect
6. #8602C2semidirect
7. #8601C2semidirect
8. #7863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
#178062
#173612
#173492
#86032
#86022
#86012
#78632