Finite Group Databasebeta

← #240598

Subgroups — #240598

Order 512.

897 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21799C211yesmaximal, characteristic
3#17806C211yesmaximal, characteristic
4#17751C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#17464C211yesmaximal, characteristic
7#10493C218yesmaximal, complemented
8#8065C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—22no—
12#3209—22no—
13#3209—22no—
14#3209C2214yes—
15C2×#494—21no—
16C2×#487—21no—
17C2×#462—22no—
18C2×#451—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24#1966—22no—
25#1966C2214yes—
26#1784—22no—
27#1574C2214yes—
28#1573—22no—
29#1564—22no—
30#1561C2214yes—
31#1552—22no—
32#1549—21no—
33#1548—22no—
34#1548C2214yes—
35#1546—21no—
36C2×#333—22no—
37#1485C2214yes—
38#1481—22no—
39#1479—22no—
40#1299C2214yes—
41C23×Q8D412yes—
42C22×Q16—24no—
43C22×M16—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533D412yes—
51#533C2312yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#494—24no—
76#494—24no—
77#487—24no—
78#487—24no—
79#462—28no—
80#462—28no—
81#451—28no—
82#451—28no—
83#432—21no—
84#432—21no—
85#432—21no—
86#432—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431D412yes—
96#431D412yes—
97#431C2312yes—
98#431C2312yes—
99#423—21no—
100#423—21no—
101#423—21no—
102#423—21no—
103#423C2312yes—
104#423C2312yes—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D412yes—
108C2×C4⋊C8D412yes—
109C2×#103—24no—
110C2×#101—28no—
111#407—21no—
112#407—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C4×M16—21no—
116C4×M16C2312yes—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119#335—24no—
120#334—24no—
121#333—28no—
122#332—24no—
123#332—24no—
124Q8⋊C8—28no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—84no—
133C22×Q8C2×D414yes—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C2×Q16—216no—
139C2×Q16—88no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—41no—
144C2×M16—82no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—41no—
148C4⋊Q8—21no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8C2×D412yes—
171C4⋊Q8C2×D412yes—
172#125—22no—
173#125—22no—
174#125—41no—
175#125—41no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192#117—22no—
193#117—22no—
194#117—41no—
195#117—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42C2411yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8—41no—
238C4⋊C8C2×D414yes—
239C4⋊C8C2×D414yes—
240#103—216no—
241#101—88no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247C22⋊C8—41no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250#97—22no—
251#97—41no—
252C4×C8—22no—
253C4×C8—22no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—41no—
258C24#12311yescharacteristic
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—48no—
264C2×Q8—44no—
265C2×Q8—48no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—84no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—84no—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12311yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4#12411yescharacteristic
300C22×C4#12411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304Q16—816no—
305M16—44no—
306M16—44no—
307M16—82no—
308M16—82no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—82no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C42—42no—
340C42—42no—
341C42#12011yescharacteristic
342C42#12011yescharacteristic
343C42#12111yescharacteristic
344C42#12111yescharacteristic
345C42#12111yescharacteristic
346C42#12111yescharacteristic
347C42#12111yescharacteristic
348C42#12111yescharacteristic
349C42#12311yescharacteristic
350C42#12311yescharacteristic
351C42#12411yescharacteristic
352C42#12411yescharacteristic
353C42C22×D411yescharacteristic
354C42C22×D411yescharacteristic
355C42C22×D411yescharacteristic
356C42C2×#4111yescharacteristic
357C23—22no—
358C23—22no—
359C23—41no—
360C23#39411yescharacteristic
361C23#39411yescharacteristic
362C23C2×#12311yescharacteristic
363Q8—44no—
364Q8—48no—
365Q8—42no—
366Q8—42no—
367Q8—84no—
368Q8—82no—
369Q8—84no—
370Q8—84no—
371Q8—82no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—48no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#39411yescharacteristic
399C2×C4#39411yescharacteristic
400C2×C4#39411yescharacteristic
401C2×C4#39411yescharacteristic
402C2×C4#39411yescharacteristic
403C2×C4#39411yescharacteristic
404C2×C4#47112yes—
405C2×C4#47312yes—
406C2×C4#47314yes—
407C2×C4C2×#12011yescharacteristic
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4C2×#12111yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C2×C4C2×#12411yescharacteristic
413C8—44no—
414C8—44no—
415C8—82no—
416C8—82no—
417C8—82no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—44no—
426C22#204712yes—
427C22#205212yes—
428C22C2×#39411yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—84no—
437C4—82no—
438C4—82no—
439C4#204712yes—
440C4#205212yes—
441C4#301912yes—
442C4C2×#47111yescharacteristic
443C4C2×#47312yes—
444C4C2×#47311yescharacteristic
445C2—22no—
446C2—22no—
447C2—41no—
448C2#2349311yescharacteristic
449C2C2×#204711yescharacteristic
450C2C2×#205211yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#123
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. #10493C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#217992
#178062
#177512
C2×#15482
#174642
#104932
#80652