Finite Group Databasebeta

← #198487

Subgroups — #198487

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic, complemented
3#21811C211yesmaximal, characteristic, complemented
4#21794C211yesmaximal, characteristic
5#17595C211yesmaximal, characteristic
6#9262C218yesmaximal, complemented
7#7980C211yesmaximal, characteristic, complemented
8#7928C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3206—22no—
12#3206—22no—
13#3206—22no—
14#3206—22no—
15C2×#492C412yescomplemented
16C2×#487C412yescomplemented
17C2×#461C414yescomplemented
18C2×#448C2214yes—
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2381—22no—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4C2214yes—
25#2331—22no—
26#2327—22no—
27#2313—22no—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30#2283C2211yescharacteristic
31#1981—22no—
32#1964—22no—
33#1964C2214yes—
34#1960—22no—
35#1787C2214yes—
36#1782—22no—
37#1519C2214yes—
38#1514—22no—
39#1505—22no—
40#1289C2214yes—
41#1282C2214yes—
42C22×#41—21no—
43C23×Q8C2312yes—
44C22×SD16C2×C418yes—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8D412yes—
66C2×#127C2×C411yescharacteristic
67C2×#124—21no—
68C2×#121—21no—
69C2×#121C2312yes—
70C2×C4×Q8—24no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#492—24no—
77#492—24no—
78#487—24no—
79#487—24no—
80#461—28no—
81#461—28no—
82#448—28no—
83#448—28no—
84#432—21no—
85#432—21no—
86#432—21no—
87#432—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C412yes—
94C2×#103—28no—
95C2×#103C2×C418yes—
96C2×#99—28no—
97C2×C22⋊C8C2312yes—
98C2×C22⋊C8C2312yes—
99C2×#97—21no—
100C2×C4×C8C2×C412yes—
101#391—24no—
102#391—28no—
103#389—24no—
104#387—28no—
105#387—28no—
106#387—28no—
107#383—21no—
108#383—21no—
109#383D412yes—
110#382—21no—
111#382—21no—
112#382—21no—
113#381—21no—
114#381—21no—
115#381D412yes—
116#380—24no—
117#380D412yes—
118#380C2312yes—
119#380C2312yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2312yes—
128#376—21no—
129#376—21no—
130#376D412yes—
131#327—28no—
132C2×#41—41no—
133C2×#41—41no—
134C2×#41—41no—
135C2×#41—41no—
136C2×#41—84no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—84no—
142C22×Q8C22×C414yes—
143C22×Q8C2×D414yes—
144C22×Q8C2×D414yes—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×SD16—216no—
155C2×SD16—88no—
156C22×C8—22no—
157C22×C8C22×C414yes—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8C22⋊C414yes—
175C4⋊Q8C22⋊C414yes—
176C4⋊Q8C22⋊C414yes—
177C4⋊Q8C22⋊C414yes—
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#124—41no—
183#124—41no—
184#124—41no—
185#124—41no—
186#121—22no—
187#121—22no—
188#121—22no—
189#121—22no—
190#121—41no—
191#121—41no—
192#121—41no—
193#121—41no—
194C4×Q8—88no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4C22×C412yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4C2×D414yes—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—28no—
229C2×C42—41no—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246#103—216no—
247#103—88no—
248#99—88no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253#97—41no—
254#97—41no—
255C4×C8—22no—
256C4×C8—22no—
257#95—28no—
258#95—28no—
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—41no—
268#95—41no—
269#95—41no—
270#95C2×D414yes—
271C24—22no—
272C24C22×D411yescharacteristic
273#41—84no—
274#41—84no—
275#41—84no—
276#41—84no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—48no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—82no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×Q8C2×C22⋊C418yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—48no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—82no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334SD16—816no—
335C2×C8—24no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—88no—
355C22⋊C4—28no—
356C22⋊C4—44no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C42—44no—
363C42—44no—
364C42—82no—
365C42—88no—
366C42C2×C22⋊C411yescharacteristic
367C42C2×C22⋊C411yescharacteristic
368C42C2×C22⋊C411yescharacteristic
369C42C2×C22⋊C411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12311yescharacteristic
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C42#12311yescharacteristic
382C23—41no—
383C23—41no—
384C23—42no—
385C23—44no—
386C23—82no—
387C23—82no—
388C23C2×C4×D411yescharacteristic
389C23C2×#12011yescharacteristic
390C23C2×#12111yescharacteristic
391C23#57714yes—
392Q8—216no—
393Q8—48no—
394Q8—84no—
395Q8—84no—
396Q8—82no—
397Q8—82no—
398D4—44no—
399D4—82no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—84no—
404D4—84no—
405D4—82no—
406C2×C4—22no—
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—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—28no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4#38611yescharacteristic
442C2×C4#38611yescharacteristic
443C2×C4#38611yescharacteristic
444C2×C4#38611yescharacteristic
445C2×C4#38611yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4C22×C22⋊C411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×#12111yescharacteristic
452C2×C4C2×#12311yescharacteristic
453C8—44no—
454C8—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—44no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—164no—
466C22C2×#38611yescharacteristic
467C22#294912yes—
468C22#301012yes—
469C22#304312yes—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—48no—
481C4—84no—
482C4—84no—
483C4—82no—
484C4—162no—
485C4#190512yes—
486C4#288312yes—
487C2—22no—
488C2—22no—
489C2—22no—
490C2—162no—
491C2#2093611yescharacteristic
492C2#2098511yescharacteristic
493C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34371C2semidirect
3. #21811C2semidirect
6. #9262C2semidirect
7. #7980C2semidirect
8. #7928C2semidirect
15. C2×#492C4semidirect
16. C2×#487C4semidirect
17. C2×#461C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
#343712
#218112
#217942
#175952
#92622
#79802
#79282