Finite Group Databasebeta

← #117536

Subgroups — #117536

Order 512.

640 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21332C211yesmaximal, characteristic
2#21172C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7#7854C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#423C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1996—22no—
17#1996—22no—
18#1949—22no—
19#1949—22no—
20#1740—21no—
21#1740—21no—
22#1736—21no—
23#1550—22no—
24#1549C412yescomplemented
25#1548C412yescomplemented
26#1542—22no—
27#1518—22no—
28#1518—22no—
29C2×#328—22no—
30#1303—22no—
31#1294—22no—
32#1274—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8D412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8Q812yes—
49C2×#117—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#431C2×C412yes—
62#431D412yes—
63#428—42no—
64#428—42no—
65C2×#107—22no—
66C2×#106—22no—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70#423D412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#384—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83#382—21no—
84#382D412yes—
85#380—21no—
86#380D412yes—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
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#342—24no—
98#342—24no—
99#340—24no—
100#333—24no—
101#332—24no—
102#328—28no—
103C22×Q8—21no—
104C22×Q8C4⋊C412yes—
105C23×C4—41no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—41no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—41no—
127#125—41no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—21no—
147#117—21no—
148#117—21no—
149#117—21no—
150#117—21no—
151#117—21no—
152#117—41no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157#117—41no—
158#117—41no—
159#117—41no—
160#117SD1612yes—
161#117Q1612yes—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—41no—
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⋊C4C4⋊C412yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×Q811yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201#107—44no—
202#106—44no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8C22⋊C414yes—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—41no—
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—
225#95—41no—
226#95—41no—
227C24#12011yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—82no—
233C2×Q8—82no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12411yescharacteristic
263C22×C4#12711yescharacteristic
264C22×C4C2×SD1611yescharacteristic
265C22×C4C2×Q1611yescharacteristic
266C22×C4#13511yescharacteristic
267C22×C4#13511yescharacteristic
268C22×C4AGL(1,ℤ/8)11yescharacteristic
269C22×C4#13711yescharacteristic
270C22×C4#13711yescharacteristic
271C22×C4#13711yescharacteristic
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—82no—
282C2×C8—82no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C42—22no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—41no—
320C42—41no—
321C42—82no—
322C42—82no—
323C42—81no—
324C42—81no—
325C42C2×C22⋊C411yescharacteristic
326C42#11711yescharacteristic
327C42C4×D411yescharacteristic
328C42C4×D411yescharacteristic
329C42#12111yescharacteristic
330C42#12111yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42C22⋊Q811yescharacteristic
335C42#12311yescharacteristic
336C42#12311yescharacteristic
337C42#12411yescharacteristic
338C42#12411yescharacteristic
339C42#12611yescharacteristic
340C42#12611yescharacteristic
341C23—41no—
342C23—41no—
343C23#38511yescharacteristic
344C23#39211yescharacteristic
345C23#39411yescharacteristic
346C23#44811yescharacteristic
347C23#44911yescharacteristic
348C23#45111yescharacteristic
349C23#45211yescharacteristic
350Q8—82no—
351Q8—82no—
352Q8—82no—
353Q8—82no—
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—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—162no—
383C2×C4—162no—
384C2×C4#38011yescharacteristic
385C2×C4#38511yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#38811yescharacteristic
389C2×C4#39011yescharacteristic
390C2×C4#39311yescharacteristic
391C2×C4#39411yescharacteristic
392C2×C4#39611yescharacteristic
393C2×C4#39611yescharacteristic
394C2×C4#39711yescharacteristic
395C2×C4#39911yescharacteristic
396C2×C4C4×SD1611yescharacteristic
397C2×C4C4×Q1611yescharacteristic
398C2×C4#44011yescharacteristic
399C2×C4#44111yescharacteristic
400C2×C4#45011yescharacteristic
401C2×C4#45111yescharacteristic
402C2×C4#45211yescharacteristic
403C2×C4#45211yescharacteristic
404C2×C4#46111yescharacteristic
405C2×C4#46211yescharacteristic
406C2×C4#46311yescharacteristic
407C2×C4#46411yescharacteristic
408C2×C4#46511yescharacteristic
409C2×C4#46711yescharacteristic
410C2×C4#46811yescharacteristic
411C2×C4#47011yescharacteristic
412C2×C4#47511yescharacteristic
413C2×C4#47711yescharacteristic
414C2×C4#47911yescharacteristic
415C2×C4#47911yescharacteristic
416C2×C4#48711yescharacteristic
417C2×C4#48811yescharacteristic
418C2×C4#48911yescharacteristic
419C2×C4#49111yescharacteristic
420C8—44no—
421C8—82no—
422C8—82no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22#143911yescharacteristic
434C22#186311yescharacteristic
435C22#186511yescharacteristic
436C22#200011yescharacteristic
437C22#200111yescharacteristic
438C22#204211yescharacteristic
439C22#204311yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—82no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4—162no—
453C4—162no—
454C4#187711yescharacteristic
455C4#187811yescharacteristic
456C4#193311yescharacteristic
457C4#193511yescharacteristic
458C4#202411yescharacteristic
459C4#202611yescharacteristic
460C4#206611yescharacteristic
461C4#206811yescharacteristic
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2—21no—
467C2—21no—
468C2#1106711yescharacteristic
469C2#1113211yescharacteristic
470C2#1113811yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
#213322
#211722
#177082
C2×#15182
#80952
#80262
#78542