Finite Group Databasebeta

← #242690

Subgroups — #242690

Order 512.

687 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#21177C211yesmaximal, characteristic
3#17813C211yesmaximal, characteristic
4C2×#1565C211yesmaximal, characteristic
5#17708C211yesmaximal, characteristic
6C2×#1550C211yesmaximal, characteristic
7#10610C214yesmaximal, complemented
8#10609C214yesmaximal, complemented
9C2×#1292C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#430—21no—
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18#1998—22no—
19#1997C2212yes—
20#1997C2212yes—
21#1950—22no—
22#1950C2214yes—
23#1742—21no—
24#1742—21no—
25#1738—21no—
26#1570—22no—
27#1570C2214yes—
28#1566—22no—
29#1565—22no—
30#1565C2214yes—
31C2×#335—21no—
32C2×#334—21no—
33#1550—22no—
34#1550—22no—
35#1550—21no—
36#1550—21no—
37#1549C2212yes—
38#1548C2212yes—
39#1542—22no—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1292C2214yes—
43C22×M16—21no—
44C22×M16—21no—
45C22×M16—21no—
46C22×M16D411yescharacteristic
47C22×M16D411yescharacteristic
48C23×C8—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8Q812yes—
62C2×C4×Q8—21no—
63C2×C4×Q8Q812yes—
64C2×#117—21no—
65C22×C42C2311yescharacteristic
66#431—21no—
67#431—21no—
68#431D412yes—
69#431C2312yes—
70#430—44no—
71C2×#108—24no—
72#423—21no—
73#423—21no—
74#423—21no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423—21no—
80#423D412yes—
81#423D412yes—
82#423C2312yes—
83#423C2312yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16D412yes—
102C4×M16C2312yes—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106#343—22no—
107#343—22no—
108#343—22no—
109#343—22no—
110#341—24no—
111#335—24no—
112#334—24no—
113#334—28no—
114#333—24no—
115#333—22no—
116#333—22no—
117#332—24no—
118#332—22no—
119#332—22no—
120C22×Q8—21no—
121C22×Q8C2×Q812yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×Q811yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—21no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—42no—
135C2×M16—42no—
136C2×M16—41no—
137C2×M16—41no—
138C2×M16—42no—
139C2×M16—42no—
140C2×M16—42no—
141C2×M16—42no—
142C2×M16SD1614yes—
143C2×M16C2×D412yes—
144C2×M16C2×D412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—82no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164#125—41no—
165#125—41no—
166#125—41no—
167#125—41no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184#117—41no—
185#117—41no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Q812yes—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×Q811yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#108—48no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8C2×D412yes—
227C4⋊C8C2×D414yes—
228C4⋊C8C2×D412yes—
229C22⋊C8—22no—
230C22⋊C8—22no—
231#97—22no—
232#97—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—22no—
237C24C22⋊Q811yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—81no—
243C2×Q8—81no—
244C2×Q8—82no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—81no—
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4#12611yescharacteristic
265C22×C4C2×SD1612yes—
266C22×C4#13512yes—
267C22×C4#13712yes—
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272M16—22no—
273M16—24no—
274M16—22no—
275M16—22no—
276M16—44no—
277M16—44no—
278M16—42no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—82no—
284M16—82no—
285M16C2×SD1614yes—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—81no—
299C2×C8—81no—
300C2×C8—82no—
301C2×C8—82no—
302C2×C8—82no—
303C2×C8—82no—
304C2×C8—82no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C42—42no—
320C42—42no—
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42#12111yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42#12611yescharacteristic
332C42#12611yescharacteristic
333C42C22×D411yescharacteristic
334C42C22×D411yescharacteristic
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C23—41no—
338C23—41no—
339C23#39611yescharacteristic
340C23#39611yescharacteristic
341C23#47512yes—
342C23#47912yes—
343C23C2×C22⋊Q811yescharacteristic
344Q8—82no—
345Q8—82no—
346Q8—82no—
347Q8—81no—
348Q8—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
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—82no—
367C2×C4—82no—
368C2×C4#39611yescharacteristic
369C2×C4#39611yescharacteristic
370C2×C4#39611yescharacteristic
371C2×C4#39611yescharacteristic
372C2×C4#39611yescharacteristic
373C2×C4#39611yescharacteristic
374C2×C4#46112yes—
375C2×C4#46112yes—
376C2×C4#46412yes—
377C2×C4#46412yes—
378C2×C4#46512yes—
379C2×C4#46512yes—
380C2×C4#47012yes—
381C2×C4#47012yes—
382C2×C4#47512yes—
383C2×C4#47912yes—
384C2×C4C2×#12111yescharacteristic
385C2×C4C2×#12111yescharacteristic
386C2×C4C2×#12111yescharacteristic
387C2×C4C2×C22⋊Q811yescharacteristic
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4C2×#12611yescharacteristic
390C2×C4C22×SD1611yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4C2×#13711yescharacteristic
394C8—24no—
395C8—44no—
396C8—42no—
397C8—42no—
398C8—42no—
399C8—82no—
400C8—82no—
401C8—82no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—42no—
409C22—42no—
410C22#206612yes—
411C22#206612yes—
412C22C2×#39611yescharacteristic
413C22C2×#47511yescharacteristic
414C22C2×#47911yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—82no—
423C4—82no—
424C4#206612yes—
425C4#206612yes—
426C4C2×#46111yescharacteristic
427C4C2×#46411yescharacteristic
428C4C2×#46511yescharacteristic
429C4C2×#47011yescharacteristic
430C4#306811yescharacteristic
431C4#306911yescharacteristic
432C2—21no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2#2360411yescharacteristic
437C2C2×#206611yescharacteristic
438C2#2362211yescharacteristic
subgroupG/H
center, socleC22C2×#396
commutator, Frattini, cosocleC2×C42C24
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10610C2semidirect
8. #10609C2semidirect

Maximal subgroups

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

subgroupindex
#213372
#211772
#178132
C2×#15652
#177082
C2×#15502
#106102
#106092
C2×#12922