Finite Group Databasebeta

← #188528

Subgroups — #188528

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic
6#8578C214yesmaximal
7#8572C214yesmaximal
8#8026C211yesmaximal, characteristic
9#8014C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2370—22no—
14#2369C412yescomplemented
15#2365C412yescomplemented
16#2355—22no—
17#2355C412yescomplemented
18Q8×C4⋊C4C412yescomplemented
19#2329—22no—
20#2317—21no—
21C4×C22⋊Q8C2214yes—
22C2×#389—21no—
23C2×#389—21no—
24#2285C2211yescharacteristic
25C2×#382C2211yescharacteristic
26#2274—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—22no—
30#1939—22no—
31#1797C2214yes—
32#1518—24no—
33#1503—24no—
34C2×#328C2212yes—
35C2×#328C2214yes—
36C2×#328C2212yes—
37#1444—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2214yes—
43#533—22no—
44#533—22no—
45#533D414yes—
46#533Q814yes—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2312yes—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C414yes—
55C2×#117—21no—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—24no—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#391—24no—
78#391C2×C414yes—
79#389—22no—
80#389—22no—
81#389—24no—
82#389—44no—
83#389—44no—
84#389C2×C414yes—
85#387—24no—
86#387—24no—
87#387—24no—
88#385—24no—
89#384—22no—
90#384C2×C411yescharacteristic
91#384C2×C411yescharacteristic
92#382—22no—
93#382—21no—
94#382—22no—
95#382—22no—
96#380—21no—
97#380C2312yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4C2×C414yes—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—24no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95D412yes—
117C2×#95C2311yescharacteristic
118C2×#95C2311yescharacteristic
119C2×#95C2312yes—
120#336—24no—
121#328—24no—
122#328—24no—
123#328—28no—
124C22×Q8—21no—
125C22×Q8C22×C412yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8C4⋊C414yescomplemented
139C4⋊Q8C4⋊C414yescomplemented
140#125—24no—
141#125—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8C2×D414yes—
147C22⋊Q8C2×Q814yes—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—84no—
152C4×Q8C4⋊C418yescomplemented
153#117—24no—
154#117—24no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4SD1614yes—
187C2×C4⋊C4Q1614yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—82no—
201C2×C42C22×C414yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D414yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×Q812yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—84no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C4×C8—42no—
219C4×C8—42no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—24no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—44no—
233#95—42no—
234#95—44no—
235#95C22⋊C414yes—
236#95C22⋊C414yes—
237C24C22×D411yescharacteristic
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—81no—
243C2×Q8—81no—
244C2×Q8C2×C4⋊C414yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4C2×C22⋊C414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4C22⋊Q812yes—
289C22×C4C22⋊Q812yes—
290C22×C4C2×SD1614yes—
291C22×C4C2×SD1612yes—
292C22×C4C2×Q1614yes—
293C22×C4C2×Q1612yes—
294C22×C4AGL(1,ℤ/8)12yes—
295C22×C4#13712yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×Q811yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—88no—
327C4⋊C4—88no—
328C4⋊C4—84no—
329C4⋊C4#10318yes—
330C4⋊C4#10318yes—
331C4⋊C4C2×C4⋊C414yes—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C42—28no—
337C42—44no—
338C42—44no—
339C42—44no—
340C42—84no—
341C42—84no—
342C42—84no—
343C42C2×C4⋊C412yes—
344C42C2×C4⋊C412yes—
345C42#11712yes—
346C42#11712yes—
347C42#12112yes—
348C42#12112yes—
349C42C22⋊Q812yes—
350C42C22⋊Q812yes—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23C22×C22⋊C411yescharacteristic
356C23C2×#12111yescharacteristic
357C23C2×C22⋊Q811yescharacteristic
358C23C22×SD1611yescharacteristic
359C23C22×Q1611yescharacteristic
360C23C2×AGL(1,ℤ/8)11yescharacteristic
361C23C2×#13711yescharacteristic
362Q8—44no—
363Q8—44no—
364Q8—84no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4#38012yes—
422C2×C4#38012yes—
423C2×C4#38012yes—
424C2×C4#38012yes—
425C2×C4C2×#10318yes—
426C2×C4C2×#10312yes—
427C2×C4C2×#10312yes—
428C2×C4#41712yes—
429C2×C4#41712yes—
430C2×C4#46014yes—
431C2×C4#46214yes—
432C2×C4#47514yes—
433C2×C4#47714yes—
434C2×C4C22×C4⋊C411yescharacteristic
435C2×C4C2×#11711yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×C22⋊Q811yescharacteristic
438C8—82no—
439C8—82no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22C2×#38011yescharacteristic
453C22#288311yescharacteristic
454C22#288411yescharacteristic
455C22C22×#10311yescharacteristic
456C22C2×#41711yescharacteristic
457C22#290111yescharacteristic
458C22#290111yescharacteristic
459C22C2×#46011yescharacteristic
460C22C2×#46211yescharacteristic
461C22C2×#47511yescharacteristic
462C22C2×#47711yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—22no—
469C4—28no—
470C4—44no—
471C4—44no—
472C4—44no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—162no—
478C4#180314yes—
479C4#180314yes—
480C2—21no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2#1761411yescharacteristic
485C2#1761911yescharacteristic
486C2#1762811yescharacteristic
487C2#1763011yescharacteristic
488C2#2063411yescharacteristic
489C2C2×#180311yescharacteristic
490C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2369C4semidirect
15. #2365C4semidirect
17. #2355C4semidirect
18. Q8×C4⋊C4C4semidirect
138. C4⋊Q8C4⋊C4semidirect
139. C4⋊Q8C4⋊C4semidirect
152. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219512
C2×#15182
C2×#15032
#139542
#85782
#85722
#80262
#80142