Finite Group Databasebeta

← #115474

Subgroups — #115474

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3230—22no—
11#3230—22no—
12#3230—22no—
13#3230C414yescomplemented
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477—22no—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#1997—22no—
24#1997—22no—
25#1992—22no—
26#1990—22no—
27#1990—22no—
28#1990—21no—
29#1990—21no—
30C2×#343—22no—
31#1736—22no—
32C2×#340—22no—
33#1493—22no—
34#1493—22no—
35C2×Q8⋊C8—22no—
36#1296—22no—
37#1278—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C22×M16D412yes—
41C23×C8—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
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⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D412yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42C2311yescharacteristic
76#500—24no—
77#500—24no—
78#487—24no—
79#487—24no—
80#484—28no—
81#484—28no—
82#477—28no—
83#477—28no—
84#431—21no—
85#431—21no—
86#431C2×C412yes—
87#431D412yes—
88C2×#107—24no—
89#423—21no—
90#423—21no—
91#423—21no—
92#423D412yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D412yes—
96C2×#103—24no—
97C2×C22⋊C8—21no—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8D412yes—
111#384—21no—
112#384—21no—
113#384—21no—
114#384D412yes—
115#343—28no—
116#340—28no—
117#340—24no—
118#340—24no—
119Q8⋊C8—28no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—82no—
129C22×Q8—82no—
130C23×C4—22no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—41no—
142C2×M16—42no—
143C2×M16—42no—
144C2×M16Q1614yes—
145C22×C8—24no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—42no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8C22⋊C412yes—
176C4⋊Q8C22⋊C412yes—
177#125—22no—
178#125—22no—
179#125—41no—
180#125—41no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197#117—22no—
198#117—22no—
199#117—41no—
200#117—41no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
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—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C22⋊C414yes—
229C2×C22⋊C4—41no—
230C2×C42—41no—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243#107—216no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8C22⋊C414yes—
250#103—216no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253#97—41no—
254#97—41no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—41no—
260C4×C8—41no—
261C4×C8—41no—
262C4×C8—41no—
263C24#12111yescharacteristic
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—82no—
279C2×Q8—82no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—82no—
283C2×Q8—82no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
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×Q1612yes—
318C22×C4#13512yes—
319C22×C4#13712yes—
320C22×C4#13712yes—
321M16—24no—
322M16—24no—
323M16—42no—
324M16—42no—
325M16—44no—
326M16—82no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—42no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—81no—
340C2×C8—81no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C42—42no—
371C42—42no—
372C42C2×C22⋊C411yescharacteristic
373C42C2×C22⋊C411yescharacteristic
374C42C4×D411yescharacteristic
375C42C4×D411yescharacteristic
376C42#12011yescharacteristic
377C42#12011yescharacteristic
378C42#12011yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C42#12111yescharacteristic
384C42#12111yescharacteristic
385C42#12311yescharacteristic
386C42#12411yescharacteristic
387C42#12711yescharacteristic
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#38611yescharacteristic
393C23#39211yescharacteristic
394C23#39411yescharacteristic
395C23#46212yes—
396C23#46412yes—
397Q8—44no—
398Q8—44no—
399Q8—84no—
400Q8—84no—
401Q8—82no—
402Q8—84no—
403Q8—84no—
404Q8—82no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—164no—
432C2×C4#37911yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#39011yescharacteristic
437C2×C4#39011yescharacteristic
438C2×C4#39211yescharacteristic
439C2×C4#39211yescharacteristic
440C2×C4#39211yescharacteristic
441C2×C4#39411yescharacteristic
442C2×C4#39411yescharacteristic
443C2×C4#39411yescharacteristic
444C2×C4C4×Q1612yes—
445C2×C4#44112yes—
446C2×C4#45112yes—
447C2×C4#45112yes—
448C2×C4#45212yes—
449C2×C4#45212yes—
450C2×C4#45412yes—
451C2×C4#45612yes—
452C2×C4#46212yes—
453C2×C4#46412yes—
454C2×C4#46712yes—
455C2×C4#47012yes—
456C2×C4#47112yes—
457C2×C4#47212yes—
458C2×C4#49411yescharacteristic
459C2×C4#49511yescharacteristic
460C2×C4#49711yescharacteristic
461C2×C4#49711yescharacteristic
462C8—44no—
463C8—44no—
464C8—44no—
465C8—82no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22#143811yescharacteristic
475C22#187812yes—
476C22#200311yescharacteristic
477C22#200511yescharacteristic
478C22#201511yescharacteristic
479C22#201611yescharacteristic
480C22#201811yescharacteristic
481C22#201911yescharacteristic
482C22#205612yes—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—84no—
488C4—82no—
489C4—82no—
490C4—162no—
491C4#179312yes—
492C4#187812yes—
493C4#190712yes—
494C4#196911yescharacteristic
495C4#197111yescharacteristic
496C4#200112yes—
497C4#201511yescharacteristic
498C4#201611yescharacteristic
499C4#203812yes—
500C4#205112yes—
501C2—22no—
502C2—22no—
503C2#1014911yescharacteristic
504C2#1015711yescharacteristic
505C2#1016511yescharacteristic
506C2#1017311yescharacteristic
507C2#1099811yescharacteristic
508C2#1102011yescharacteristic
509C2#1102411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#19972
#213222
C2×#19902
C2×#14932
#80432
C2×#12782