Finite Group Databasebeta

← #193403

Subgroups — #193403

Order 512.

814 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#21329C211yesmaximal, characteristic
3#17820C211yesmaximal, characteristic
4C2×#1574C211yesmaximal, characteristic
5#17715C211yesmaximal, characteristic
6#8936C214yesmaximal, complemented
7#8936C214yesmaximal, complemented
8#7870C211yesmaximal, characteristic
9#7869C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2986C2212yes—
12#2986C2212yes—
13#2985—22no—
14#2984—22no—
15#2983—21no—
16#2982—21no—
17#2962—21no—
18#2962—21no—
19#2960—21no—
20#2960—21no—
21#2924—22no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1995—22no—
29#1994C2214yes—
30#1733—22no—
31#1574—21no—
32#1574—21no—
33#1574C2212yes—
34#1574C2212yes—
35#1574C2214yes—
36#1571—21no—
37#1570—21no—
38#1569—22no—
39C2×#335—21no—
40C2×#334—21no—
41#1552C414yescomplemented
42#1549C412yescomplemented
43#1546C412yescomplemented
44#1544C2214yes—
45#1276C2212yes—
46#1276C2212yes—
47#1276C2214yes—
48C22×M16—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yes—
54#533D412yes—
55#533Q812yes—
56#533C2312yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125Q812yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×#117—21no—
65C2×#117D412yes—
66C22×C42C2311yescharacteristic
67#446—22no—
68#446—22no—
69#446—24no—
70#446—24no—
71#446—22no—
72#446—22no—
73#446—22no—
74#446—22no—
75C8×Q8—22no—
76C8×Q8—22no—
77C8×Q8—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81#433—24no—
82#433—24no—
83#432—24no—
84#432—21no—
85#432—21no—
86#432C2×C412yes—
87#432D412yes—
88#431—24no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431C2×C412yes—
96#431D411yescharacteristic
97#431D411yescharacteristic
98#431C2312yes—
99#431C2312yes—
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2312yes—
103#424—22no—
104#424—22no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423C2×C412yes—
109#423D411yescharacteristic
110#423D411yescharacteristic
111#423C2312yes—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2×C412yes—
115C2×C4⋊C8D411yescharacteristic
116C2×C4⋊C8D411yescharacteristic
117#407—21no—
118#407—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121#405—22no—
122#405—22no—
123C4×M16—21no—
124C4×M16—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43C2312yes—
133#339—28no—
134#335—22no—
135#335—22no—
136#335—24no—
137#334—22no—
138#334—22no—
139#334—24no—
140#332C2×C418yes—
141#332C2×C418yes—
142C22×Q8—21no—
143C22×Q8—21no—
144C23×C4C2×D411yescharacteristic
145C23×C4#4111yescharacteristic
146C23×C4#4111yescharacteristic
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—41no—
152C2×M16—44no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—41no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C4⋊C412yes—
162C4⋊Q8C4⋊C412yes—
163C4⋊Q8C22×C412yes—
164C4⋊Q8C22×C412yes—
165#125—22no—
166#125—22no—
167#125C4⋊C414yes—
168#125C2×Q814yes—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185#117—22no—
186#117—22no—
187#117—24no—
188#117—42no—
189#117C4⋊C414yes—
190#117C2×D414yes—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C4⋊C412yes—
197C2×C4⋊C4C4⋊C412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—41no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q811yescharacteristic
212C2×C42C2×Q811yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—41no—
229C4⋊C8—41no—
230C4⋊C8C22×C414yes—
231C4⋊C8C22×C414yes—
232C4⋊C8C2×D411yescharacteristic
233C4⋊C8C2×D411yescharacteristic
234C4⋊C8C2×D412yes—
235C4⋊C8C2×D411yescharacteristic
236C4⋊C8C2×D411yescharacteristic
237C4⋊C8C2×D412yes—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—22no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248#97—24no—
249#97—24no—
250#97—22no—
251#97—22no—
252C4×C8—24no—
253C4×C8—24no—
254C4×C8—22no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—22no—
262#95—41no—
263#95—41no—
264C24#12311yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—44no—
270C2×Q8—82no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4C2×C4⋊C411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4C22⋊Q811yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4C22⋊Q811yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12311yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296M16—42no—
297M16—42no—
298M16—44no—
299M16—82no—
300M16—82no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—82no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—44no—
325C4⋊C4C2×C4⋊C414yes—
326C4⋊C4C2×C4⋊C414yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—42no—
334C42—44no—
335C42—44no—
336C42C2×C4⋊C411yescharacteristic
337C42C2×C4⋊C411yescharacteristic
338C42C4×D411yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42C4×D411yescharacteristic
342C42#12011yescharacteristic
343C42#12011yescharacteristic
344C42C22⋊Q811yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42C22⋊Q811yescharacteristic
348C42C23×C411yescharacteristic
349C42C22×D411yescharacteristic
350C42C22×D411yescharacteristic
351C42C22×Q811yescharacteristic
352C23—22no—
353C23—41no—
354C23—41no—
355C23#38511yescharacteristic
356C23#38511yescharacteristic
357C23C2×#12311yescharacteristic
358Q8—44no—
359Q8—44no—
360Q8—44no—
361Q8—82no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#38511yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38511yescharacteristic
391C2×C4#38511yescharacteristic
392C2×C4#38511yescharacteristic
393C2×C4#38511yescharacteristic
394C2×C4#45412yes—
395C2×C4#45612yes—
396C2×C4C22×C4⋊C411yescharacteristic
397C2×C4C2×C4×D411yescharacteristic
398C2×C4C2×C4×D411yescharacteristic
399C2×C4C2×#12011yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×C22⋊Q811yescharacteristic
402C8—44no—
403C8—44no—
404C8—41no—
405C8—41no—
406C8—42no—
407C8—41no—
408C8—41no—
409C8—42no—
410C8—82no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—82no—
420C22#185712yes—
421C22C2×#38511yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—44no—
431C4—44no—
432C4—84no—
433C4#185712yes—
434C4#295512yes—
435C4#295712yes—
436C4C2×#45411yescharacteristic
437C4C2×#45611yescharacteristic
438C2—22no—
439C2—41no—
440C2—41no—
441C2C2×#185711yescharacteristic
442C2#2088411yescharacteristic
443C2#2088611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8936C2semidirect
7. #8936C2semidirect
41. #1552C4semidirect
42. #1549C4semidirect
43. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#213292
#178202
C2×#15742
#177152
#89362
#78702
#78692