Finite Group Databasebeta

← #117572

Subgroups — #117572

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21329C211yesmaximal, characteristic
2#21172C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#8079C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7924C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1995—22no—
17#1994—22no—
18#1949—22no—
19#1949—22no—
20#1736—21no—
21#1733—21no—
22#1733—21no—
23#1550—22no—
24#1549C412yescomplemented
25#1548C412yescomplemented
26#1542—22no—
27#1518—22no—
28#1518—22no—
29C2×#328—22no—
30#1301—22no—
31#1286—22no—
32#1282—22no—
33C22×M16—21no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533D412yes—
42#533Q812yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C2×#117D411yescharacteristic
52C2×#117D411yescharacteristic
53C22×C4⋊C4—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#428—42no—
64#428—42no—
65C2×#107—22no—
66C2×#106—22no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423C2×C412yes—
73#423D412yes—
74#423D412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×C22⋊C8—21no—
82C2×C22⋊C8D412yes—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#382—21no—
86#382—21no—
87#382—21no—
88#382D412yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93#376—21no—
94C2×#95—21no—
95C2×#95—21no—
96C43—21no—
97#340—24no—
98#339—24no—
99#339—24no—
100#333—24no—
101#332—24no—
102#328—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—41no—
124C22×C8—41no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C4⋊C412yes—
132C4⋊Q8C4⋊C412yes—
133#125—22no—
134#125—22no—
135#125—41no—
136#125—41no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—21no—
154#117—21no—
155#117—21no—
156#117—21no—
157#117—22no—
158#117—22no—
159#117—21no—
160#117—21no—
161#117—41no—
162#117—41no—
163#117SD1612yes—
164#117Q1612yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—41no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×Q811yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199#107—44no—
200#106—44no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8C22⋊C414yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—41no—
217C4×C8—22no—
218C4×C8—41no—
219C4×C8—41no—
220C4×C8—41no—
221#95—22no—
222#95—22no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227C24#12111yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—82no—
233C2×Q8—82no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4#11711yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4#12611yescharacteristic
262C22×C4#12711yescharacteristic
263C22×C4C2×SD1611yescharacteristic
264C22×C4C2×Q1611yescharacteristic
265C22×C4#13511yescharacteristic
266C22×C4#13511yescharacteristic
267C22×C4AGL(1,ℤ/8)11yescharacteristic
268C22×C4#13711yescharacteristic
269C22×C4#13711yescharacteristic
270C22×C4#13711yescharacteristic
271M16—42no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—42no—
277M16—81no—
278M16—81no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—82no—
286C2×C8—81no—
287C2×C8—81no—
288C2×C8—82no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—22no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—41no—
320C42—41no—
321C42—82no—
322C42—82no—
323C42C2×C22⋊C411yescharacteristic
324C42C2×C4⋊C411yescharacteristic
325C42C4×D411yescharacteristic
326C42C4×D411yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42#12111yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42#12311yescharacteristic
336C42#12311yescharacteristic
337C42#12411yescharacteristic
338C42#12411yescharacteristic
339C23—41no—
340C23—41no—
341C23#38811yescharacteristic
342C23#39211yescharacteristic
343C23#39611yescharacteristic
344C23#46511yescharacteristic
345C23#46711yescharacteristic
346C23#46811yescharacteristic
347C23#47011yescharacteristic
348Q8—82no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—162no—
378C2×C4—162no—
379C2×C4#38011yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38611yescharacteristic
383C2×C4#38811yescharacteristic
384C2×C4#39011yescharacteristic
385C2×C4#39311yescharacteristic
386C2×C4#39411yescharacteristic
387C2×C4#39411yescharacteristic
388C2×C4#39611yescharacteristic
389C2×C4#39711yescharacteristic
390C2×C4#39911yescharacteristic
391C2×C4C4×SD1611yescharacteristic
392C2×C4C4×Q1611yescharacteristic
393C2×C4#44011yescharacteristic
394C2×C4#44111yescharacteristic
395C2×C4#44811yescharacteristic
396C2×C4#44911yescharacteristic
397C2×C4#45011yescharacteristic
398C2×C4#45111yescharacteristic
399C2×C4#45111yescharacteristic
400C2×C4#45211yescharacteristic
401C2×C4#45211yescharacteristic
402C2×C4#45211yescharacteristic
403C2×C4#46111yescharacteristic
404C2×C4#46211yescharacteristic
405C2×C4#46311yescharacteristic
406C2×C4#46411yescharacteristic
407C2×C4#47511yescharacteristic
408C2×C4#47711yescharacteristic
409C2×C4#47911yescharacteristic
410C2×C4#47911yescharacteristic
411C2×C4#48711yescharacteristic
412C2×C4#48811yescharacteristic
413C2×C4#48911yescharacteristic
414C2×C4#49111yescharacteristic
415C8—44no—
416C8—82no—
417C8—82no—
418C8—82no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22#143911yescharacteristic
430C22#193311yescharacteristic
431C22#193511yescharacteristic
432C22#200011yescharacteristic
433C22#200111yescharacteristic
434C22#206611yescharacteristic
435C22#206811yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—162no—
449C4#186311yescharacteristic
450C4#186511yescharacteristic
451C4#187711yescharacteristic
452C4#187811yescharacteristic
453C4#202411yescharacteristic
454C4#202611yescharacteristic
455C4#204211yescharacteristic
456C4#204311yescharacteristic
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2#1106711yescharacteristic
464C2#1113211yescharacteristic
465C2#1114111yescharacteristic
subgroupG/H
center, socleC22#1439
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
24. #1549C4semidirect
25. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#213292
#211722
#177082
C2×#15182
#80792
C2×#12862
#79242