Finite Group Databasebeta

← #109653

Subgroups — #109653

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1855C211yesmaximal, characteristic
2#20389C211yesmaximal, characteristic
3#10732C211yesmaximal, characteristic
4#8051C211yesmaximal, characteristic
5#7982C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C23×M16C2211yescharacteristic
9C22×#108C412yescomplemented
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C2×#405—21no—
14#2280C2211yescharacteristic
15#2280C2211yescharacteristic
16C2×C4×C22⋊C4C2211yescharacteristic
17#1855—21no—
18#1855—22no—
19#1855—21no—
20#1753—22no—
21#1752—21no—
22C8×C22⋊C4—21no—
23C2×#341C412yescomplemented
24C2×#339—22no—
25#1729—21no—
26#1297—22no—
27#1289C414yescomplemented
28#1282—21no—
29#1282—21no—
30#1282C414yescomplemented
31C24×C4C2311yescharacteristic
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—44no—
36C22×M16C2×C412yes—
37C22×M16C2×C411yescharacteristic
38C22×M16C2×C411yescharacteristic
39C22×M16D411yescharacteristic
40C22×M16D412yes—
41C22×M16D411yescharacteristic
42C23×C8—21no—
43C23×C8—21no—
44C23×C8C2×C412yes—
45C2×#117—21no—
46C2×#117—21no—
47C22×C22⋊C4—21no—
48C22×C42—21no—
49C2×#108—22no—
50C2×#108—22no—
51C2×#108—44no—
52#407—21no—
53#407—21no—
54#407—21no—
55#407—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8C2×C412yes—
66C2×C22⋊C8C2×C412yes—
67C2×C22⋊C8C2×C412yes—
68C2×C22⋊C8D411yescharacteristic
69C2×C22⋊C8D411yescharacteristic
70C2×C22⋊C8Q812yes—
71#405—44no—
72C2×#97—22no—
73C2×C4×C8—22no—
74#381—21no—
75#381Q812yes—
76#380—21no—
77#380—21no—
78#380C2×C412yes—
79#380C2×C412yes—
80#379—21no—
81#379—21no—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4C2×C412yes—
89C4×C22⋊C4D412yes—
90C2×#95—21no—
91#341—22no—
92#341—22no—
93#339—44no—
94#336—22no—
95#336—22no—
96C25#4111yescharacteristic
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—41no—
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×Q811yescharacteristic
108C23×C4C2×Q811yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C2×M16—21no—
115C2×M16—24no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—21no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—44no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—44no—
130C2×M16—44no—
131C2×M16—42no—
132C2×M16—42no—
133C2×M16C22⋊C412yes—
134C2×M16C22⋊C412yes—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—44no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8C4214yes—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—41no—
174C2×C42—41no—
175#108—44no—
176#108—44no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—21no—
184C22⋊C8—21no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8C4⋊C412yes—
190C22⋊C8C4⋊C412yes—
191#97—44no—
192C4×C8—44no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—41no—
197#95—41no—
198#95C4⋊C414yes—
199C24—22no—
200C24—22no—
201C24—42no—
202C24—81no—
203C24#11711yescharacteristic
204C24C4×D411yescharacteristic
205C24C4×Q811yescharacteristic
206C24C22⋊Q811yescharacteristic
207C24#12311yescharacteristic
208C24#12411yescharacteristic
209C24#12611yescharacteristic
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C2×C4211yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×Q811yescharacteristic
251C22×C4C4×Q811yescharacteristic
252C22×C4C4×Q811yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12511yescharacteristic
262C22×C4#12511yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4C4⋊Q811yescharacteristic
265M16—24no—
266M16—22no—
267M16—22no—
268M16—22no—
269M16—44no—
270M16—44no—
271M16—42no—
272M16—42no—
273M16—42no—
274M16—81no—
275M16—81no—
276M16#9914yes—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—44no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—44no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—81no—
299C2×C8—81no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C22⋊C4—24no—
303C22⋊C4—44no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—42no—
314C42—42no—
315C42—44no—
316C42—81no—
317C42—81no—
318C42—81no—
319C42—81no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—84no—
343C23—84no—
344C23C4×C4⋊C411yescharacteristic
345C23#38511yescharacteristic
346C23#38711yescharacteristic
347C23#38711yescharacteristic
348C23#38811yescharacteristic
349C23#38811yescharacteristic
350C23#39111yescharacteristic
351C23#44312yes—
352C23#44412yes—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—88no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4C4×C22⋊C411yescharacteristic
397C2×C4C4×C4⋊C411yescharacteristic
398C2×C4#38511yescharacteristic
399C2×C4#38611yescharacteristic
400C2×C4#38711yescharacteristic
401C2×C4#38711yescharacteristic
402C2×C4#38911yescharacteristic
403C2×C4#38911yescharacteristic
404C2×C4C2×#9912yes—
405C2×C4#41012yes—
406C8—24no—
407C8—42no—
408C8—42no—
409C8—42no—
410C8—44no—
411C8—81no—
412C8—81no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—44no—
433C22—44no—
434C22—42no—
435C22—42no—
436C22—82no—
437C22—82no—
438C22—81no—
439C22—81no—
440C22#143611yescharacteristic
441C22#177211yescharacteristic
442C22#177311yescharacteristic
443C22#186612yes—
444C22#192912yes—
445C22#199612yes—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—84no—
453C4—84no—
454C4—162no—
455C4C4×#9912yes—
456C4#189811yescharacteristic
457C4#189911yescharacteristic
458C2—22no—
459C2—22no—
460C2—22no—
461C2—21no—
462C2—21no—
463C2—41no—
464C2—41no—
465C2—41no—
466C2#1090511yescharacteristic
467C2#1091011yescharacteristic
468C2#1091211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1436
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×#108C4semidirect
23. C2×#341C4semidirect
27. #1289C4semidirect
30. #1282C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18552
#203892
#107322
#80512
#79822
#79252
C2×#12822