Finite Group Databasebeta

← #114218

Subgroups — #114218

Order 512.

892 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#10848C211yesmaximal, characteristic
7#7860C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3165—22no—
11#3165—22no—
12#3165—22no—
13#3165—22no—
14C2×#501—22no—
15C2×#500C412yescomplemented
16C2×#498C412yescomplemented
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428—22no—
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#1981—22no—
24#1956—22no—
25#1787—22no—
26C2×#344—22no—
27#1549—22no—
28#1549—22no—
29C2×#332—22no—
30#1520—22no—
31#1518—22no—
32#1503—22no—
33#1493—22no—
34#1487—22no—
35#1481—22no—
36#1275—22no—
37C23×Q8—21no—
38C22×M16—21no—
39C22×M16D412yes—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533D412yes—
47#533D412yes—
48C2×C4⋊Q8—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⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D412yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#501—28no—
73#501—28no—
74#500—24no—
75#500—24no—
76#498—24no—
77#498—24no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431D412yes—
86#428—28no—
87#428—28no—
88C2×#107—24no—
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⋊C8—21no—
98C2×#103—28no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C4×M16—21no—
102C4×M16—21no—
103C4×M16—21no—
104C4×M16—21no—
105C2×#97D412yes—
106C2×C4×C8—21no—
107C2×C4×C8C2×C412yes—
108C2×C4×C8D412yes—
109#384—21no—
110#382—21no—
111#380—21no—
112#380D412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115#344—44no—
116#332—28no—
117#328—28no—
118Q8⋊C8—28no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—84no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—41no—
138C2×M16—42no—
139C2×M16—42no—
140C2×M16SD1614yes—
141C2×M16Q1614yes—
142C22×C8—22no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8C22⋊C414yes—
146C4⋊Q8—22no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8C22⋊C414yes—
168C4⋊Q8C22⋊C412yes—
169C4⋊Q8C22⋊C412yes—
170#125—22no—
171#125—22no—
172#125—41no—
173#125—41no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190#117—22no—
191#117—22no—
192#117—41no—
193#117—41no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233#107—216no—
234#106—216no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8—41no—
243#103—88no—
244C22⋊C8—22no—
245C22⋊C8—41no—
246C22⋊C8—41no—
247C22⋊C8—41no—
248#97—22no—
249#97—22no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8—22no—
255C4×C8—41no—
256#95—22no—
257#95—41no—
258C24#12111yescharacteristic
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—48no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—82no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8—82no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8—84no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—84no—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12411yescharacteristic
299C22×C4#12411yescharacteristic
300C22×C4#12711yescharacteristic
301C22×C4#12711yescharacteristic
302C22×C4C2×SD1612yes—
303C22×C4C2×Q1612yes—
304C22×C4#13712yes—
305C22×C4#13712yes—
306M16—24no—
307M16—24no—
308M16—24no—
309M16—24no—
310M16—44no—
311M16—81no—
312M16—81no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—44no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—82no—
320C2×C8—82no—
321C2×C8#10318yescomplemented
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—82no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C42—42no—
350C42—42no—
351C42—82no—
352C42C2×C22⋊C411yescharacteristic
353C42C2×C22⋊C411yescharacteristic
354C42C4×D411yescharacteristic
355C42C4×D411yescharacteristic
356C42#12011yescharacteristic
357C42#12011yescharacteristic
358C42#12011yescharacteristic
359C42#12011yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42#12111yescharacteristic
364C42#12111yescharacteristic
365C42#12111yescharacteristic
366C42#12311yescharacteristic
367C42#12311yescharacteristic
368C23—41no—
369C23—41no—
370C23#39011yescharacteristic
371C23#39211yescharacteristic
372C23#39411yescharacteristic
373C23#46112yes—
374C23#46212yes—
375Q8—44no—
376Q8—48no—
377Q8—82no—
378Q8—84no—
379Q8—82no—
380Q8—84no—
381Q8—84no—
382Q8—82no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#37911yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38611yescharacteristic
409C2×C4#38611yescharacteristic
410C2×C4#38611yescharacteristic
411C2×C4#39011yescharacteristic
412C2×C4#39211yescharacteristic
413C2×C4#39211yescharacteristic
414C2×C4#39211yescharacteristic
415C2×C4#39411yescharacteristic
416C2×C4#39411yescharacteristic
417C2×C4#39411yescharacteristic
418C2×C4C2×#10312yes—
419C2×C4#41912yes—
420C2×C4#44812yes—
421C2×C4#44812yes—
422C2×C4#45112yes—
423C2×C4#45112yes—
424C2×C4#46112yes—
425C2×C4#46212yes—
426C2×C4#48712yes—
427C2×C4#49112yes—
428C2×C4#49211yescharacteristic
429C2×C4#49411yescharacteristic
430C2×C4#49711yescharacteristic
431C2×C4#49711yescharacteristic
432C8—28no—
433C8—82no—
434C8—82no—
435C8—82no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22#143811yescharacteristic
445C22#198112yes—
446C22#201311yescharacteristic
447C22#201611yescharacteristic
448C22#204312yes—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—84no—
456C4—82no—
457C4—82no—
458C4—162no—
459C4#178712yes—
460C4#190212yes—
461C4#190412yes—
462C4#195611yescharacteristic
463C4#196011yescharacteristic
464C4#201311yescharacteristic
465C4#201611yescharacteristic
466C4#204312yes—
467C2—21no—
468C2—22no—
469C2—21no—
470C2—22no—
471C2#1101011yescharacteristic
472C2#1101211yescharacteristic
473C2#1101511yescharacteristic
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
15. C2×#500C4semidirect
16. C2×#498C4semidirect
321. C2×C8#103semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#218042
C2×#15492
#175752
#174862
#108482
#78602