Finite Group Databasebeta

← #112436

Subgroups — #112436

Order 512.

902 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×#3229C211yesmaximal, characteristic
2#21337C211yesmaximal, characteristic
3#21202C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#8095C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3229—22no—
11#3229—22no—
12#3229—22no—
13#3229—22no—
14C2×#498—21no—
15C2×#487—21no—
16C2×#483—22no—
17C2×#475C414yescomplemented
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#419—22no—
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24#1998—22no—
25#1997—21no—
26#1997—21no—
27#1960—22no—
28#1956—22no—
29#1742—22no—
30#1546—22no—
31#1546—22no—
32C2×#332—22no—
33#1520—22no—
34#1518—22no—
35#1503—22no—
36#1303—22no—
37#1278—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C22×M16D412yes—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C42C2311yescharacteristic
73#498—24no—
74#498—24no—
75#487—24no—
76#487—24no—
77#483—28no—
78#483—28no—
79#475—28no—
80#475—28no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431D412yes—
87#431D412yes—
88#431D412yes—
89C2×#106—24no—
90#423—21no—
91#423—21no—
92#423—21no—
93#423D412yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C412yes—
98#419—44no—
99#419—44no—
100C2×#103—24no—
101C2×#103—28no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C4×M16—21no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112#384—21no—
113#382D412yes—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118#343—24no—
119#343—24no—
120#332—28no—
121#328—28no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8C22⋊C412yes—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—41no—
145C2×M16—42no—
146C2×M16—42no—
147C2×M16SD1614yes—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—41no—
151C22×C8—41no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—22no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8C22⋊C414yes—
176#125—41no—
177#125—41no—
178#125—41no—
179#125—41no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196#117—41no—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—42no—
201#117—42no—
202C2×C4⋊C4—22no—
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—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C22⋊C412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—41no—
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239#106—216no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8C22⋊C414yes—
248#103—216no—
249#103—416no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—41no—
254#97—41no—
255#97—41no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—41no—
260C4×C8—41no—
261C4×C8—41no—
262#95—22no—
263#95—22no—
264C24#12111yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—44no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—81no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×Q8—81no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12411yescharacteristic
316C22×C4#12711yescharacteristic
317C22×C4C2×SD1612yes—
318C22×C4#13512yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4#13712yes—
321M16—24no—
322M16—24no—
323M16—42no—
324M16—42no—
325M16—44no—
326M16—81no—
327M16—81no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—44no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—82no—
336C2×C8—82no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—44no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—82no—
363C42—42no—
364C42—42no—
365C42—81no—
366C42—81no—
367C42C2×C22⋊C411yescharacteristic
368C42C2×C22⋊C411yescharacteristic
369C42C4×D411yescharacteristic
370C42C4×D411yescharacteristic
371C42#12011yescharacteristic
372C42#12011yescharacteristic
373C42#12011yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12311yescharacteristic
381C42#12411yescharacteristic
382C42#12711yescharacteristic
383C23—41no—
384C23—41no—
385C23#38611yescharacteristic
386C23#39211yescharacteristic
387C23#39411yescharacteristic
388C23#46012yes—
389C23#46412yes—
390Q8—44no—
391Q8—48no—
392Q8—82no—
393Q8—82no—
394Q8—82no—
395Q8—82no—
396Q8—82no—
397Q8—84no—
398Q8—82no—
399Q8—82no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#37911yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#39011yescharacteristic
429C2×C4#39011yescharacteristic
430C2×C4#39211yescharacteristic
431C2×C4#39211yescharacteristic
432C2×C4#39211yescharacteristic
433C2×C4#39411yescharacteristic
434C2×C4#39411yescharacteristic
435C2×C4#39411yescharacteristic
436C2×C4C4×SD1612yes—
437C2×C4#44012yes—
438C2×C4#44812yes—
439C2×C4#44912yes—
440C2×C4#45012yes—
441C2×C4#45212yes—
442C2×C4#46112yes—
443C2×C4#46312yes—
444C2×C4#46512yes—
445C2×C4#46812yes—
446C2×C4#49211yescharacteristic
447C2×C4#49511yescharacteristic
448C2×C4#49611yescharacteristic
449C2×C4#49711yescharacteristic
450C8—44no—
451C8—44no—
452C8—82no—
453C8—82no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22#143811yescharacteristic
463C22#188012yes—
464C22#201411yescharacteristic
465C22#201511yescharacteristic
466C22#205512yes—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—84no—
474C4—82no—
475C4—82no—
476C4—162no—
477C4#179612yes—
478C4#187712yes—
479C4#196811yescharacteristic
480C4#197311yescharacteristic
481C4#200012yes—
482C4#201211yescharacteristic
483C4#201311yescharacteristic
484C4#204912yes—
485C2—21no—
486C2—22no—
487C2—22no—
488C2—21no—
489C2#1098511yescharacteristic
490C2#1102311yescharacteristic
491C2#1103511yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#475C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#213372
#212022
C2×#15462
#175752
#80952
#78822