Finite Group Databasebeta

← #205967

Subgroups — #205967

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21337C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#17813C211yesmaximal, characteristic
5#17708C211yesmaximal, characteristic
6C2×#1549C211yesmaximal, characteristic
7#17319C211yesmaximal, characteristic
8#9810C214yesmaximal, complemented
9#9803C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—22no—
13#3230—22no—
14#3230—22no—
15#3230—22no—
16C2×#500C2212yes—
17C2×#487C2212yes—
18C2×#484C414yes—
19C2×#477C414yes—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#1998—22no—
26#1997—22no—
27#1997C2212yes—
28#1997C2214yes—
29#1997C2212yes—
30#1742—22no—
31C2×#343—22no—
32#1570—22no—
33#1570—22no—
34#1566C2214yes—
35#1550—22no—
36#1549—22no—
37#1549C2214yes—
38#1549C2212yes—
39#1548C2212yes—
40#1542—22no—
41C2×#332—22no—
42#1452—22no—
43#1451—22no—
44#1451C2214yes—
45C23×Q8—21no—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16D412yes—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8—21no—
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⋊Q8C2×C412yes—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8C2×C412yes—
74C2×C4×Q8—21no—
75C2×C4×Q8C2×C412yes—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2×C412yes—
79C22×C42C2311yescharacteristic
80#500—24no—
81#500—24no—
82#487—24no—
83#487—24no—
84#484—28no—
85#484—28no—
86#477—28no—
87#477—28no—
88#431—21no—
89#431—21no—
90#431D412yes—
91#431D412yes—
92#431D412yes—
93#431D412yes—
94#431D412yes—
95#431C2312yes—
96C2×#107C2×C418yes—
97#423—21no—
98#423—21no—
99#423—21no—
100#423—21no—
101#423—21no—
102#423—21no—
103#423C2312yes—
104#423C2312yes—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2×C412yes—
110C2×C4⋊C8D412yes—
111C2×#103C2×C418yes—
112C2×C22⋊C8—21no—
113C2×C22⋊C8C2×C412yes—
114C4×M16—21no—
115C4×M16—21no—
116C4×M16D412yes—
117C4×M16C2312yes—
118C2×#97—21no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2312yes—
122#343—28no—
123#343—24no—
124#343—24no—
125#334—28no—
126#333—24no—
127#332—28no—
128#332—24no—
129#322—28no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—84no—
138C22×Q8C22×C412yes—
139C23×C4—22no—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C2×M16—22no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—41no—
149C2×M16—41no—
150C2×M16—44no—
151C2×M16—44no—
152C2×M16C2×D414yes—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8C22×C414yes—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—22no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182#125—41no—
183#125—41no—
184#125—41no—
185#125—41no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
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—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—22no—
225C2×C42—22no—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42C2411yescharacteristic
238#107—216no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8C22⋊C414yes—
249C4⋊C8C22⋊C414yes—
250C4⋊C8C2×D414yes—
251#103—216no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8C22⋊C414yes—
255C22⋊C8C22⋊C414yes—
256#97—22no—
257#97—22no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8C2×D414yes—
263C4×C8C2×D414yes—
264C24C4×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—44no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—82no—
275C2×Q8—84no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—84no—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12411yescharacteristic
301C22×C4#12411yescharacteristic
302C22×C4#12711yescharacteristic
303C22×C4#12711yescharacteristic
304C22×C4#13714yes—
305C22×C4#13714yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4111yescharacteristic
309M16—24no—
310M16—24no—
311M16—44no—
312M16—44no—
313M16—44no—
314M16—82no—
315M16—82no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—82no—
323C2×C8—82no—
324C2×C8—82no—
325C2×C8C2×C22⋊C418yes—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C42—42no—
349C42—42no—
350C42C2×C22⋊C411yescharacteristic
351C42C2×C22⋊C411yescharacteristic
352C42C4×D411yescharacteristic
353C42C4×D411yescharacteristic
354C42#12111yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42#12111yescharacteristic
358C42#12411yescharacteristic
359C42#12411yescharacteristic
360C42#12711yescharacteristic
361C42#12711yescharacteristic
362C42C22×D411yescharacteristic
363C42C22×D411yescharacteristic
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C23—41no—
367C23—41no—
368C23#39011yescharacteristic
369C23#39011yescharacteristic
370C23#44114yes—
371C23C2×C4×D411yescharacteristic
372Q8—44no—
373Q8—82no—
374Q8—82no—
375Q8—84no—
376Q8—84no—
377Q8—84no—
378Q8—82no—
379Q8—84no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#39011yescharacteristic
401C2×C4#39011yescharacteristic
402C2×C4#39011yescharacteristic
403C2×C4#39011yescharacteristic
404C2×C4#39011yescharacteristic
405C2×C4#39011yescharacteristic
406C2×C4#44114yes—
407C2×C4#47014yes—
408C2×C4#47014yes—
409C2×C4#49712yes—
410C2×C4#49712yes—
411C2×C4#49712yes—
412C2×C4#49712yes—
413C2×C4C22×C22⋊C411yescharacteristic
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4C2×#12711yescharacteristic
419C2×C4C2×#13712yes—
420C2×C4C2×#13712yes—
421C8—28no—
422C8—44no—
423C8—44no—
424C8—82no—
425C8—82no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—44no—
433C22#197112yes—
434C22#197112yes—
435C22C2×#39011yescharacteristic
436C22C2×#44112yes—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4#197112yes—
447C4#197112yes—
448C4#288912yes—
449C4C2×#47012yes—
450C4#306812yes—
451C4C2×#49711yescharacteristic
452C4C2×#49711yescharacteristic
453C2—22no—
454C2—22no—
455C2—22no—
456C2#2122011yescharacteristic
457C2C2×#197111yescharacteristic
458C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
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
8. #9810C2semidirect
9. #9803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#213372
C2×#19972
#178132
#177082
C2×#15492
#173192
#98102
#98032