Finite Group Databasebeta

← #99732

Subgroups — #99732

Order 512.

760 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊C8C211yesmaximal, characteristic
2#14699C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic
4C2×#1271C212yesmaximal
5#7025C212yesmaximal
6C22×C4⋊C8—21no—
7C22×C4×C8—21no—
8C2×#389C411yescharacteristic
9C2×#389C2211yescharacteristic
10C2×#387—21no—
11C2×C43C2211yescharacteristic
12C4×C4⋊C8—24no—
13C2×#336—21no—
14C2×#336—21no—
15C2×#336C2212yes—
16C2×#336C2212yes—
17#1436—21no—
18#1436—21no—
19C4⋊#95C411yescharacteristic
20C4⋊#95C411yescharacteristic
21C4⋊#95C411yescharacteristic
22C4⋊#95C2211yescharacteristic
23#1434—21no—
24#1271—24no—
25C23×C8—22no—
26C23×C8—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42C2311yescharacteristic
40C2×C4⋊C8—24no—
41C2×C4⋊C8—24no—
42C2×C4⋊C8—24no—
43C2×C4×C8—24no—
44C2×C4×C8—42no—
45C2×C4×C8—42no—
46#389—22no—
47#389—22no—
48#389—22no—
49#389C814yescomplemented
50#387—42no—
51#387—42no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C43—22no—
63C43—22no—
64#336—24no—
65#336—22no—
66#336—22no—
67#336—22no—
68#336—42no—
69#336D414yes—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C22×C8—24no—
82C22×C8—24no—
83C22×C8—24no—
84C22×C8—24no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—42no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2×C812yes—
117C2×C4⋊C4C2×C812yes—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—22no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42C2×C812yes—
139C2×C42M1612yes—
140C2×C42D812yes—
141C2×C42SD1612yes—
142C2×C42SD1612yes—
143C2×C42Q1612yes—
144C2×C42C2×D412yes—
145C2×C42#4112yes—
146C4⋊C8—216no—
147C4×C8—48no—
148#95—24no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—41no—
157#95—41no—
158#95—42no—
159C24C2×C22⋊C411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4C22⋊C812yes—
219C22×C4C22⋊C812yes—
220C22×C4#10112yes—
221C22×C4#10212yes—
222C22×C4#10212yes—
223C22×C4#10312yes—
224C22×C4#10312yes—
225C22×C4#10412yes—
226C22×C4#10412yes—
227C22×C4#10412yes—
228C22×C4C4×D412yes—
229C22×C4#12011yescharacteristic
230C22×C4#12112yes—
231C22×C4#12311yescharacteristic
232C22×C4C22×C811yescharacteristic
233C22×C4C2×M1611yescharacteristic
234C22×C4#13112yes—
235C22×C4C2×D811yescharacteristic
236C22×C4C2×SD1611yescharacteristic
237C22×C4C2×SD1611yescharacteristic
238C22×C4C2×Q1611yescharacteristic
239C22×C4#13512yes—
240C22×C4AGL(1,ℤ/8)11yescharacteristic
241C22×C4#13711yescharacteristic
242C2×C8—28no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—44no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—82no—
256C4⋊C4—24no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—48no—
260C4⋊C4—42no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—22no—
273C42—24no—
274C42—24no—
275C42—22no—
276C42—22no—
277C42—42no—
278C42—42no—
279C42—44no—
280C42—44no—
281C23#38611yescharacteristic
282C23C2×C22⋊C811yescharacteristic
283C23#40811yescharacteristic
284C23#41011yescharacteristic
285C23#41011yescharacteristic
286C23C2×#10111yescharacteristic
287C23#41311yescharacteristic
288C23C2×#10211yescharacteristic
289C23C2×#10311yescharacteristic
290C23#41611yescharacteristic
291C23#41711yescharacteristic
292C23C2×#10411yescharacteristic
293C23C2×#10411yescharacteristic
294C23C2×#10411yescharacteristic
295C23#42111yescharacteristic
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4D4⋊C814yes—
378C2×C4Q8⋊C814yes—
379C2×C4#33214yes—
380C2×C4C8×D412yes—
381C2×C4#43512yes—
382C2×C4#44711yescharacteristic
383C2×C4#44811yescharacteristic
384C2×C4#46512yes—
385C2×C4#46712yes—
386C2×C4#48011yescharacteristic
387C2×C4#48111yescharacteristic
388C8—48no—
389C8—84no—
390C22#145811yescharacteristic
391C22#146111yescharacteristic
392C22#147011yescharacteristic
393C22#147311yescharacteristic
394C22C2×D4⋊C811yescharacteristic
395C22C2×Q8⋊C811yescharacteristic
396C22#147611yescharacteristic
397C22#147711yescharacteristic
398C22#148011yescharacteristic
399C22#148311yescharacteristic
400C22#149311yescharacteristic
401C22#149411yescharacteristic
402C22#150011yescharacteristic
403C22#150211yescharacteristic
404C22#150211yescharacteristic
405C22#150311yescharacteristic
406C22#150911yescharacteristic
407C22#150911yescharacteristic
408C22#151811yescharacteristic
409C22#152111yescharacteristic
410C22#152411yescharacteristic
411C22#152611yescharacteristic
412C22#152811yescharacteristic
413C22#152911yescharacteristic
414C22C2×#33211yescharacteristic
415C22#154011yescharacteristic
416C22#155011yescharacteristic
417C22#155211yescharacteristic
418C22#186911yescharacteristic
419C22#187111yescharacteristic
420C22#187411yescharacteristic
421C22#189211yescharacteristic
422C22#189611yescharacteristic
423C22#189911yescharacteristic
424C22#190111yescharacteristic
425C4—24no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—24no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—24no—
436C4—42no—
437C4—82no—
438C4—82no—
439C4—161no—
440C4—161no—
441C2#756011yescharacteristic
442C2#757011yescharacteristic
443C2#757711yescharacteristic
444C2#758511yescharacteristic
445C2#766311yescharacteristic
446C2#767211yescharacteristic
447C2#767711yescharacteristic
448C2#768911yescharacteristic
449C2#893911yescharacteristic
450C2#907111yescharacteristic
451C2#908211yescharacteristic
452C2#911211yescharacteristic
453C2#912211yescharacteristic
454C2#913111yescharacteristic
455C2#914811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
49. #389C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊C82
#146992
#107952
C2×#12712
#70252