Finite Group Databasebeta

← #117716

Subgroups — #117716

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21323C211yesmaximal, characteristic
2#17787C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#10848C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7C2×#1286C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1993C412yescomplemented
16#1993C412yescomplemented
17#1991—22no—
18C2×#344—21no—
19C2×#344—21no—
20#1736—21no—
21#1736—21no—
22#1565—22no—
23#1565—22no—
24#1564—22no—
25#1520—22no—
26#1518—22no—
27#1503—22no—
28#1493—22no—
29#1487—22no—
30#1481—22no—
31#1294—22no—
32#1286—22no—
33C22×M16—21no—
34C22×M16D411yescharacteristic
35C22×M16D411yescharacteristic
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533D412yes—
43#533Q812yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
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#431D412yes—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423D412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C2×#97C2×C412yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8D412yes—
81#384—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#382C2×C412yes—
87#380—21no—
88#380D412yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C2×#95—21no—
92C2×#95—21no—
93#344—42no—
94#344—42no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#335—24no—
100#334—24no—
101#328—28no—
102Q8⋊C8—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—22no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16SD1612yes—
126C2×M16Q1612yes—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8C22⋊C412yes—
131C22×C8C22⋊C412yes—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8C4⋊C412yes—
139C4⋊Q8C4⋊C412yes—
140#125—22no—
141#125—22no—
142#125—41no—
143#125—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—41no—
160#117—22no—
161#117—22no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—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
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211#97—22no—
212#97—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218C4×C8—41no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95—41no—
224#95—41no—
225C24#12111yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—82no—
231C2×Q8—82no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12611yescharacteristic
259C22×C4#12711yescharacteristic
260C22×C4C2×SD1611yescharacteristic
261C22×C4C2×Q1611yescharacteristic
262C22×C4#13511yescharacteristic
263C22×C4#13511yescharacteristic
264C22×C4AGL(1,ℤ/8)11yescharacteristic
265C22×C4#13711yescharacteristic
266C22×C4#13711yescharacteristic
267C22×C4#13711yescharacteristic
268M16—22no—
269M16—22no—
270M16—22no—
271M16—22no—
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276M16—42no—
277M16—42no—
278M16—81no—
279M16—81no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—81no—
291C2×C8—81no—
292C2×C8#10314yes—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C4⋊C4—81no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C42—42no—
318C42—42no—
319C42—81no—
320C42—81no—
321C42C2×C4⋊C411yescharacteristic
322C42#11711yescharacteristic
323C42C4×D411yescharacteristic
324C42C4×D411yescharacteristic
325C42#12011yescharacteristic
326C42#12011yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42#12411yescharacteristic
336C42#12611yescharacteristic
337C23—41no—
338C23—41no—
339C23#39011yescharacteristic
340C23#39411yescharacteristic
341C23#39611yescharacteristic
342C23#46111yescharacteristic
343C23#46211yescharacteristic
344C23#46311yescharacteristic
345C23#46411yescharacteristic
346Q8—44no—
347Q8—82no—
348Q8—82no—
349Q8—82no—
350C2×C4—41no—
351C2×C4—41no—
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—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—162no—
373C2×C4—162no—
374C2×C4#38011yescharacteristic
375C2×C4#38511yescharacteristic
376C2×C4#38511yescharacteristic
377C2×C4#38611yescharacteristic
378C2×C4#38811yescharacteristic
379C2×C4#38811yescharacteristic
380C2×C4#39211yescharacteristic
381C2×C4#39311yescharacteristic
382C2×C4#39411yescharacteristic
383C2×C4#39611yescharacteristic
384C2×C4#39711yescharacteristic
385C2×C4#39911yescharacteristic
386C2×C4C2×#10311yescharacteristic
387C2×C4#41611yescharacteristic
388C2×C4#41711yescharacteristic
389C2×C4#41911yescharacteristic
390C2×C4#46011yescharacteristic
391C2×C4#46111yescharacteristic
392C2×C4#46211yescharacteristic
393C2×C4#46211yescharacteristic
394C2×C4#46311yescharacteristic
395C2×C4#46411yescharacteristic
396C2×C4#46411yescharacteristic
397C2×C4#46411yescharacteristic
398C2×C4#46511yescharacteristic
399C2×C4#46511yescharacteristic
400C2×C4#46711yescharacteristic
401C2×C4#46711yescharacteristic
402C2×C4#46811yescharacteristic
403C2×C4#46811yescharacteristic
404C2×C4#47011yescharacteristic
405C2×C4#47011yescharacteristic
406C2×C4#47511yescharacteristic
407C2×C4#47711yescharacteristic
408C2×C4#47911yescharacteristic
409C2×C4#47911yescharacteristic
410C8—24no—
411C8—42no—
412C8—82no—
413C8—82no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22#143911yescharacteristic
426C22#198111yescharacteristic
427C22#198311yescharacteristic
428C22#205511yescharacteristic
429C22#205611yescharacteristic
430C22#206611yescharacteristic
431C22#206811yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—162no—
443C4—162no—
444C4#180311yescharacteristic
445C4#180511yescharacteristic
446C4#189211yescharacteristic
447C4#189411yescharacteristic
448C4#205511yescharacteristic
449C4#205611yescharacteristic
450C4#206611yescharacteristic
451C4#206811yescharacteristic
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1107711yescharacteristic
459C2#1113111yescharacteristic
460C2#1113211yescharacteristic
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
15. #1993C4semidirect
16. #1993C4semidirect

Maximal subgroups

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

subgroupindex
#213232
#177872
#175752
#174862
#108482
#80262
C2×#12862