Finite Group Databasebeta

← #114365

Subgroups — #114365

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21329C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#8001C211yesmaximal, characteristic
7#7924C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3230—22no—
11#3230—22no—
12#3230—22no—
13#3230—22no—
14C2×#500C412yescomplemented
15C2×#487C412yescomplemented
16C2×#484—22no—
17C2×#477—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1995—22no—
24#1994—22no—
25#1988—22no—
26#1987—22no—
27#1736—22no—
28#1733—22no—
29#1548—22no—
30#1548—22no—
31C2×#333—22no—
32#1493—22no—
33#1487—22no—
34#1481—22no—
35#1292—22no—
36#1282—22no—
37C23×Q8—21no—
38C22×M16—21no—
39C22×M16—21no—
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×#117D412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42C2311yescharacteristic
72#500—24no—
73#500—24no—
74#487—24no—
75#487—24no—
76#484—28no—
77#484—28no—
78#477—28no—
79#477—28no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431D412yes—
88C2×#107—24no—
89#423—21no—
90#423—21no—
91#423—21no—
92#423—21no—
93#423—21no—
94#423—21no—
95#423—21no—
96#423D412yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8D412yes—
102C2×C4⋊C8D412yes—
103C2×#103—24no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8D412yes—
106C2×C4×C8—21no—
107C2×C4×C8C2×C412yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112#376—21no—
113C43—21no—
114#340—28no—
115#339—28no—
116#333—28no—
117Q8⋊C8—28no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—41no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—84no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C2×M16—22no—
132C2×M16—41no—
133C2×M16—41no—
134C2×M16—41no—
135C2×M16—42no—
136C2×M16—42no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8C22⋊C414yes—
143C4⋊Q8—22no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—22no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8C22⋊C414yes—
165C4⋊Q8C22⋊C412yes—
166C4⋊Q8C22⋊C412yes—
167#125—22no—
168#125—22no—
169#125—41no—
170#125—41no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—41no—
193#117—41no—
194#117Q1614yes—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—41no—
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×C22⋊C4—22no—
215C2×C22⋊C4—41no—
216C2×C42—22no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232#107—216no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8—41no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245#103—216no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—41no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—41no—
254#95—41no—
255#95—41no—
256C24#12111yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—44no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—82no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8—84no—
272C2×Q8—84no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—84no—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12311yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12411yescharacteristic
297C22×C4#12411yescharacteristic
298C22×C4#12711yescharacteristic
299C22×C4#12711yescharacteristic
300C22×C4C2×Q1612yes—
301C22×C4#13512yes—
302C22×C4#13712yes—
303C22×C4#13712yes—
304M16—44no—
305M16—82no—
306M16—81no—
307M16—81no—
308M16—81no—
309M16—81no—
310C2×C8—24no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—44no—
316C2×C8—82no—
317C2×C8—82no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C42—24no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—82no—
349C42—82no—
350C42C2×C22⋊C411yescharacteristic
351C42C2×C22⋊C411yescharacteristic
352C42C4×D411yescharacteristic
353C42C4×D411yescharacteristic
354C42#12011yescharacteristic
355C42#12011yescharacteristic
356C42#12011yescharacteristic
357C42#12011yescharacteristic
358C42#12111yescharacteristic
359C42#12111yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42#12111yescharacteristic
364C42#12311yescharacteristic
365C42#12311yescharacteristic
366C23—41no—
367C23—41no—
368C23#39011yescharacteristic
369C23#39211yescharacteristic
370C23#39411yescharacteristic
371C23#46712yes—
372C23#47012yes—
373Q8—44no—
374Q8—44no—
375Q8—84no—
376Q8—82no—
377Q8—82no—
378Q8—84no—
379Q8—84no—
380Q8—84no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4#37911yescharacteristic
405C2×C4#38611yescharacteristic
406C2×C4#38611yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38611yescharacteristic
409C2×C4#39011yescharacteristic
410C2×C4#39211yescharacteristic
411C2×C4#39211yescharacteristic
412C2×C4#39211yescharacteristic
413C2×C4#39411yescharacteristic
414C2×C4#39411yescharacteristic
415C2×C4#39411yescharacteristic
416C2×C4C4×Q1612yes—
417C2×C4#44112yes—
418C2×C4#45112yes—
419C2×C4#45112yes—
420C2×C4#45212yes—
421C2×C4#45212yes—
422C2×C4#46212yes—
423C2×C4#46212yes—
424C2×C4#46412yes—
425C2×C4#46412yes—
426C2×C4#49411yescharacteristic
427C2×C4#49511yescharacteristic
428C2×C4#49711yescharacteristic
429C2×C4#49711yescharacteristic
430C8—44no—
431C8—82no—
432C8—82no—
433C8—82no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—42no—
441C22—42no—
442C22#143811yescharacteristic
443C22#196911yescharacteristic
444C22#197111yescharacteristic
445C22#200112yes—
446C22#205612yes—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—84no—
454C4—84no—
455C4—82no—
456C4—82no—
457C4#179612yes—
458C4#187812yes—
459C4#187812yes—
460C4#190312yes—
461C4#201511yescharacteristic
462C4#201511yescharacteristic
463C4#201611yescharacteristic
464C4#201611yescharacteristic
465C2—22no—
466C2—22no—
467C2—21no—
468C2—21no—
469C2#1100711yescharacteristic
470C2#1101411yescharacteristic
471C2#1102411yescharacteristic
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
14. C2×#500C4semidirect
15. C2×#487C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#213292
#213132
C2×#15482
#174862
#80012
#79242