Finite Group Databasebeta

← #198612

Subgroups — #198612

Order 512.

1072 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#9270C218yesmaximal, complemented
7#8079C211yesmaximal, characteristic
8#7932C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3230—22no—
12#3230—22no—
13#3230C414yes—
14#3230C2214yes—
15C2×#500—21no—
16C2×#487—21no—
17C2×#484C414yes—
18C2×#477—22no—
19Q8×M16—22no—
20Q8×M16—22no—
21#2962—22no—
22#2962C2214yes—
23#2961—22no—
24#2960—22no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2905—21no—
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30#2873—21no—
31C2×C4×M16C2211yescharacteristic
32#1997—22no—
33#1997C2214yes—
34C2×#343—21no—
35C2×#343—21no—
36#1550C2214yes—
37#1549—22no—
38#1549—22no—
39#1549—21no—
40#1548—21no—
41#1542—22no—
42C2×#332C2214yes—
43#1301C2214yes—
44#1283C2214yes—
45C23×Q8—21no—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16D411yescharacteristic
49C22×M16D411yescharacteristic
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533C2×C412yes—
55#533D412yes—
56#533D412yes—
57#533C2312yes—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8C2×C412yes—
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8C2312yes—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8C2×C412yes—
75C2×C4×Q8—21no—
76C2×C4×Q8D412yes—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2×C412yes—
80C22×C42C2311yescharacteristic
81#500—24no—
82#500—24no—
83#487—24no—
84#487—24no—
85#484—28no—
86#484C2×C4116yes—
87#477—28no—
88#477—28no—
89#446—24no—
90#446—28no—
91#446—28no—
92#446—24no—
93C8×Q8—24no—
94C8×Q8—24no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431C2×C412yes—
102#431C2312yes—
103C2×#107—24no—
104#424—24no—
105#424—24no—
106#423—24no—
107#423—21no—
108#423—21no—
109#423—21no—
110#423—21no—
111#423—21no—
112#423D412yes—
113#423D412yes—
114#423C2312yes—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#103—24no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8C2×C412yes—
124#405—24no—
125#405—24no—
126C4×M16—24no—
127C4×M16—21no—
128C4×M16—21no—
129C4×M16D412yes—
130C4×M16C2312yes—
131C2×#97—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135#343—24no—
136#343—24no—
137#333—24no—
138#332—24no—
139#332—28no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—41no—
144C22×Q8—41no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—84no—
148C22×Q8C22⋊C412yes—
149C23×C4—22no—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C2×M16—21no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—41no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16Q1614yes—
167C2×M16C2×D412yes—
168C2×M16C2×D412yes—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—41no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4⋊Q8—41no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8C22⋊C412yes—
196C4⋊Q8C22⋊C412yes—
197C4⋊Q8C2×D412yes—
198C4⋊Q8C2×D412yes—
199#125—22no—
200#125—22no—
201#125—22no—
202#125—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8C22⋊C414yes—
210C22⋊Q8C22×C414yes—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8C22⋊C414yes—
218C4×Q8C2×D414yes—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22×C414yes—
241C2×C22⋊C4—22no—
242C2×C42—22no—
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42C2411yescharacteristic
255#107—216no—
256C4⋊C8—28no—
257C4⋊C8—28no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8C2×D414yes—
270#103—216no—
271C22⋊C8—22no—
272C22⋊C8—41no—
273C22⋊C8—41no—
274C22⋊C8C22×C414yes—
275#97—28no—
276#97—22no—
277#97—22no—
278C4×C8—28no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—41no—
284C4×C8—41no—
285C24C4×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—44no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—82no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8C2×C22⋊C414yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—84no—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4C2×Q1612yes—
326C22×C4#13512yes—
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332M16—22no—
333M16—24no—
334M16—22no—
335M16—22no—
336M16—42no—
337M16—44no—
338M16—44no—
339M16—42no—
340M16—42no—
341M16—82no—
342M16C2×Q1614yes—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—44no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—82no—
355C2×C8—82no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4C2×C22⋊C414yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C42—24no—
379C42—24no—
380C42C2×C22⋊C411yescharacteristic
381C42C2×C22⋊C411yescharacteristic
382C42C4×D411yescharacteristic
383C42C4×D411yescharacteristic
384C42#12011yescharacteristic
385C42#12011yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42#12111yescharacteristic
389C42#12111yescharacteristic
390C42#12311yescharacteristic
391C42#12311yescharacteristic
392C42C22×D411yescharacteristic
393C42C22×D411yescharacteristic
394C42C22×D411yescharacteristic
395C42C2×#4111yescharacteristic
396C23—41no—
397C23—41no—
398C23#38611yescharacteristic
399C23#38611yescharacteristic
400C23C4×Q1612yes—
401C23#44112yes—
402C23C2×C4×D411yescharacteristic
403Q8—44no—
404Q8—44no—
405Q8—44no—
406Q8—82no—
407Q8—84no—
408Q8—84no—
409Q8—84no—
410Q8—84no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4C4×Q1612yes—
438C2×C4#44112yes—
439C2×C4#45112yes—
440C2×C4#45112yes—
441C2×C4#45212yes—
442C2×C4#45212yes—
443C2×C4#46212yes—
444C2×C4#46212yes—
445C2×C4#46412yes—
446C2×C4#46412yes—
447C2×C4C22×C22⋊C411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×#12111yescharacteristic
452C2×C4C2×#12311yescharacteristic
453C2×C4C22×Q1611yescharacteristic
454C2×C4C2×#13511yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4C2×#13711yescharacteristic
457C8—24no—
458C8—44no—
459C8—42no—
460C8—44no—
461C8—82no—
462C8—82no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—42no—
470C22—42no—
471C22#187812yes—
472C22#187812yes—
473C22C2×#38611yescharacteristic
474C22C2×C4×Q1611yescharacteristic
475C22C2×#44111yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—44no—
483C4—44no—
484C4—84no—
485C4#187812yes—
486C4#187812yes—
487C4#191512yes—
488C4C2×#45111yescharacteristic
489C4C2×#45211yescharacteristic
490C4C2×#46211yescharacteristic
491C4C2×#46411yescharacteristic
492C4#306812yes—
493C2—22no—
494C2—22no—
495C2—21no—
496C2—21no—
497C2C2×#187811yescharacteristic
498C2#2100511yescharacteristic
499C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9270C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#601922
C2×#19972
#177082
C2×#15492
#92702
#80792
#79322