Finite Group Databasebeta

← #192362

Subgroups — #192362

Order 512.

1136 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21394C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5#13856C211yesmaximal, characteristic
6#8871C214yesmaximal
7#8868C214yesmaximal
8#8022C211yesmaximal, characteristic
9#7912C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2370—22no—
14#2369—21no—
15#2365—21no—
16#2355—21no—
17#2355—22no—
18Q8×C4⋊C4—21no—
19#2329C2214yes—
20#2317—21no—
21C4×C22⋊Q8—22no—
22#2285C2211yescharacteristic
23#2274—21no—
24C2×C4×C4⋊C4—22no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#1988—21no—
29#1981—21no—
30#1948—21no—
31#1947—21no—
32#1872C2212yes—
33#1872C2212yes—
34#1853C2212yes—
35#1852C2212yes—
36#1776—22no—
37#1763—22no—
38#1506C418yescomplemented
39C2×#328C2214yes—
40#1444—21no—
41#1444—21no—
42#1444C2212yes—
43#1444C2212yes—
44#1294C2214yes—
45#1280C2214yes—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8D412yes—
58C2×#117—21no—
59C2×#117—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70#432—22no—
71#432—22no—
72#432—22no—
73#432C2×C414yes—
74C2×#107—22no—
75C2×#106—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×#103—22no—
79C2×#103—22no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×#97—21no—
83C2×#97—21no—
84#391—22no—
85#391—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#385—24no—
94#384—22no—
95#384D411yescharacteristic
96#384D411yescharacteristic
97#382—22no—
98#382—21no—
99#380—21no—
100#380C2312yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—48no—
117C4×C4⋊C4D412yes—
118C4×C22⋊C4—24no—
119#376—22no—
120#376C2×C414yes—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#328C2×C4116yes—
132C22×Q8—21no—
133C22×Q8C2×D412yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8C4⋊C414yescomplemented
147C4⋊Q8C4⋊C414yescomplemented
148#125—24no—
149#125—24no—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8C22×C414yes—
155C22⋊Q8C2×Q814yes—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—84no—
160C4×Q8C4⋊C418yescomplemented
161#117—24no—
162#117—24no—
163#117—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—44no—
208C2×C42—44no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42C22×C412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q812yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219#107—44no—
220#106—44no—
221C4⋊C8—24no—
222C4⋊C8—42no—
223#103—44no—
224#103—44no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8C22×C414yes—
229#97—24no—
230#97—42no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—44no—
242#95—44no—
243#95—41no—
244#95—41no—
245#95C22×C414yes—
246C24C22×D411yescharacteristic
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8C2×C4⋊C414yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q812yes—
298C22×C4#13714yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×Q811yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C2×C8—24no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—28no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—416no—
335C4⋊C4—416no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4C2×C4⋊C414yes—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—44no—
344C42—216no—
345C42—28no—
346C42—44no—
347C42—44no—
348C42—82no—
349C42—82no—
350C42—84no—
351C42—84no—
352C42—82no—
353C42C2×C4⋊C412yes—
354C42C2×C4⋊C412yes—
355C42C4×D412yes—
356C42C4×D412yes—
357C42C22⋊Q812yes—
358C42C22⋊Q812yes—
359C42#12312yes—
360C42#12312yes—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23C2×C4×D411yescharacteristic
366C23C2×#12011yescharacteristic
367C23C2×C22⋊Q811yescharacteristic
368C23C2×#13712yes—
369C23#57512yes—
370Q8—44no—
371Q8—48no—
372Q8—82no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4#38512yes—
431C2×C4#38512yes—
432C2×C4#38512yes—
433C2×C4#38512yes—
434C2×C4#45312yes—
435C2×C4#45612yes—
436C2×C4C22×C4⋊C411yescharacteristic
437C2×C4C2×C4×D411yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C8—44no—
441C8—82no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22C2×#38511yescharacteristic
451C22C4×#13712yes—
452C22#300411yescharacteristic
453C22#300511yescharacteristic
454C22C2×#45311yescharacteristic
455C22C2×#45611yescharacteristic
456C22#302511yescharacteristic
457C22#302511yescharacteristic
458C22#307812yes—
459C4—22no—
460C4—22no—
461C4—22no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—48no—
467C4—44no—
468C4—44no—
469C4—44no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4#185914yes—
476C2—22no—
477C2—22no—
478C2#1818011yescharacteristic
479C2#1818311yescharacteristic
480C2#1818911yescharacteristic
481C2#1819011yescharacteristic
482C2C2×#185911yescharacteristic
483C2#2087411yescharacteristic
484C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
38. #1506C4semidirect
146. C4⋊Q8C4⋊C4semidirect
147. C4⋊Q8C4⋊C4semidirect
160. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#213942
#213652
C2×#15062
#138562
#88712
#88682
#80222
#79122