Finite Group Databasebeta

← #196456

Subgroups — #196456

Order 512.

1168 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3C2×#1515C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5#13961C211yesmaximal, characteristic
6#9116C214yesmaximal
7#9112C214yesmaximal
8#8022C211yesmaximal, characteristic
9#8018C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2983—24no—
13#2983—24no—
14C2×#446—22no—
15C2×#446—22no—
16C2×#432C2212yes—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2212yes—
20#2855C2212yes—
21#2855C2212yes—
22#2627—21no—
23#2627—21no—
24#2582C414yescomplemented
25#2581—22no—
26#2568C412yescomplemented
27#2567C412yescomplemented
28#2565—22no—
29#2317—21no—
30#2310—21no—
31#2285C2211yescharacteristic
32C2×#383C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1515—24no—
35#1506—24no—
36C2×#328C2214yes—
37C2×#328C2214yes—
38#1446—21no—
39#1445—21no—
40#1436—21no—
41#1436—21no—
42#1434C2212yes—
43#1434C2212yes—
44#1294C2214yes—
45#1294C2214yes—
46C23×Q8D411yescharacteristic
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2312yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8D412yes—
65C2×C4×Q8D412yes—
66C2×#117—21no—
67C2×#117—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#446—216no—
76#446—216no—
77#432—22no—
78#432—22no—
79#432—24no—
80#432—24no—
81#432D414yes—
82#432D414yes—
83#431—24no—
84#431—24no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×#97—22no—
92C2×#97—21no—
93C2×#97—21no—
94C2×C4×C8—22no—
95#400—22no—
96#400—22no—
97#398—24no—
98#398—24no—
99#395—22no—
100#395—22no—
101#391—22no—
102#391—22no—
103#391—24no—
104#391C2×C414yes—
105#391C2×C414yes—
106#389—24no—
107#389C2×C414yes—
108#389C2×C414yes—
109#387—24no—
110#387—24no—
111#385—22no—
112#385—22no—
113#384—22no—
114#384C2×C412yes—
115#383—22no—
116#383—22no—
117#382—22no—
118#382—21no—
119#381—22no—
120#381—22no—
121#380—21no—
122#380C2312yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C412yes—
125#376—22no—
126#376—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95C2312yes—
135C2×#95C2312yes—
136#328—28no—
137#328—28no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—42no—
143C22×Q8C22⋊C412yes—
144C22×Q8C22⋊C412yes—
145C22×Q8C22×C412yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161#125—42no—
162#125—42no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—42no—
180C4×Q8—42no—
181#117—44no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C414yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42—42no—
221C2×C42C22×C412yes—
222C2×C42C22×C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233C4⋊C8—28no—
234C4⋊C8—28no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8C2×D414yes—
240C22⋊C8C2×D414yes—
241#97—28no—
242#97—24no—
243#97—24no—
244C4×C8—28no—
245#95—22no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—44no—
264C24C22×D411yescharacteristic
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—44no—
276C2×Q8—42no—
277C2×Q8—82no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#13714yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—44no—
334C2×C8—44no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—44no—
366C42—48no—
367C42—44no—
368C42C4×D412yes—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C4×D412yes—
372C42#12012yes—
373C42#12012yes—
374C42#12312yes—
375C42#12312yes—
376C23—22no—
377C23—22no—
378C23C22×C22⋊C411yescharacteristic
379C23C2×#12111yescharacteristic
380C23C2×#12111yescharacteristic
381C23C2×#13712yes—
382C23#57512yes—
383Q8—48no—
384Q8—48no—
385Q8—82no—
386Q8—82no—
387Q8—84no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#38612yes—
443C2×C4#38612yes—
444C2×C4#38612yes—
445C2×C4#38612yes—
446C2×C4#41914yes—
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C8—44no—
452C8—44no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—21no—
461C22C2×#38611yescharacteristic
462C22#288412yes—
463C22C2×#41911yescharacteristic
464C22#289611yescharacteristic
465C22#290112yes—
466C22#305612yes—
467C22#305612yes—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—24no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—84no—
481C4—82no—
482C4—84no—
483C4—82no—
484C4—162no—
485C2—22no—
486C2—22no—
487C2#1762412yes—
488C2#1763212yes—
489C2#2090311yescharacteristic
490C2#2097211yescharacteristic
491C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #2582C4semidirect
26. #2568C4semidirect
27. #2567C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#351792
C2×#15152
C2×#15062
#139612
#91162
#91122
#80222
#80182