Finite Group Databasebeta

← #215243

Subgroups — #215243

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35317C211yesmaximal, characteristic
2#21924C212yesmaximal, complemented
3#9707C212yesmaximal, complemented
4#9702C212yesmaximal, complemented
5#9699C212yesmaximal, complemented
6#9691C212yesmaximal, complemented
7#8344C211yesmaximal, characteristic
8#8344C211yesmaximal, characteristic
9#8342C212yesmaximal, complemented
10C2×#536C412yes—
11C2×#536C412yes—
12C2×#532C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14#2868C2212yes—
15#2858C2212yes—
16#2658C414yes—
17#2542—22no—
18#2542—22no—
19#2501—22no—
20#2501—22no—
21C4×#117C2211yescharacteristic
22#1924—22no—
23#1923—22no—
24#1917—22no—
25#1916—22no—
26#1842C2212yes—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32C2×#333C2212yes—
33C2×#333C2212yes—
34C2×#333C2212yes—
35C2×#333C2212yes—
36C2×#333C2211yescharacteristic
37C2×#333C2212yes—
38C2×#333C2211yescharacteristic
39C2×#331C2211yescharacteristic
40C2×#331C2212yes—
41C2×#331C2211yescharacteristic
42C22×#41C2×C412yes—
43C23×Q8C2×C412yes—
44#536—48no—
45#536—48no—
46#532—41no—
47#532—42no—
48#532—41no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8—41no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2312yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#124C2×C414yes—
60C2×#124C2×C414yes—
61C2×#124C2×C412yes—
62C2×#123—22no—
63C2×#123—24no—
64C2×#123—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—22no—
68C2×#121—22no—
69C2×C4×D4—22no—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C22×C4⋊C4—22no—
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84#399—24no—
85#399—24no—
86#399—24no—
87#399—24no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390—22no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384—22no—
96C4×C4⋊C4D412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4D412yes—
99C4×C22⋊C4D412yes—
100#376C2×C412yes—
101C43C2312yes—
102#336—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#333—42no—
109#333—42no—
110#333—42no—
111#333—42no—
112#333—41no—
113#333—42no—
114#333—41no—
115#331—41no—
116#331—42no—
117#331—41no—
118C2×#41—42no—
119C2×#41—42no—
120C2×#41—44no—
121C2×#41—44no—
122C22×Q8—22no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8C22×C414yes—
128C22×Q8C22×C412yes—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C22×C412yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—21no—
145C4⋊Q8—22no—
146C4⋊Q8—21no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—44no—
150C4⋊Q8—44no—
151C4⋊Q8—48no—
152C4⋊Q8—48no—
153#127—21no—
154#127—22no—
155#127—21no—
156#124—42no—
157#124—42no—
158#124—48no—
159#124—48no—
160#123—48no—
161#123—416no—
162#123—48no—
163C22⋊Q8—48no—
164C22⋊Q8—416no—
165C22⋊Q8—48no—
166#121—44no—
167#121—44no—
168C4×D4—44no—
169C4×D4—44no—
170#117—24no—
171#117—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4C22⋊C411yescharacteristic
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C22⋊C411yescharacteristic
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42C22×C412yes—
197C2×C42C22×C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2411yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—82no—
214C4⋊C8—82no—
215C22⋊C8—82no—
216C22⋊C8—82no—
217#97—82no—
218C4×C8—82no—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223C24—22no—
224C24C2×C22⋊C411yescharacteristic
225#41—88no—
226#41—88no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—88no—
239C2×Q8—88no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D412yes—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4#10014yes—
305C22⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C22⋊C4#10014yes—
313C42—21no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—24no—
326C42—24no—
327C42H(5,𝔽2)11yescharacteristic
328C42H(5,𝔽2)12yes—
329C42H(5,𝔽2)11yescharacteristic
330C23—21no—
331C23—21no—
332C23—22no—
333C23—22no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—44no—
338C23—44no—
339C23C2×#10012yes—
340C23C22×C22⋊C411yescharacteristic
341Q8—84no—
342Q8—84no—
343Q8—82no—
344Q8—82no—
345Q8—88no—
346D4—82no—
347D4—82no—
348D4—88no—
349D4—82no—
350D4—82no—
351D4—82no—
352D4—82no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
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—22no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—88no—
383C2×C4—88no—
384C2×C4C2×#10014yes—
385C2×C4C2×#10014yes—
386C2×C4C2×#10012yes—
387C2×C4#51811yescharacteristic
388C2×C4#51811yescharacteristic
389C2×C4#53512yes—
390C2×C4#53512yes—
391C2×C4#57712yes—
392C2×C4#57712yes—
393C2×C4#57812yes—
394C2×C4#57812yes—
395C8—82no—
396C8—82no—
397C8—82no—
398C8—82no—
399C22—22no—
400C22—24no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—82no—
405C22—84no—
406C22—82no—
407C22—82no—
408C22—84no—
409C22#230911yescharacteristic
410C22C22×#10011yescharacteristic
411C22C22×#10011yescharacteristic
412C22#289812yes—
413C22#289812yes—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—24no—
418C4—24no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—82no—
423C4—84no—
424C4#322112yes—
425C4#322312yes—
426C4#322512yes—
427C4#322512yes—
428C2—42no—
429C2—162no—
430C2#1777311yescharacteristic
431C2#1777911yescharacteristic
432C2#1777912yes—
433C2#2179711yescharacteristic
434C2#2181412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21924C2semidirect
3. #9707C2semidirect
4. #9702C2semidirect
5. #9699C2semidirect
6. #9691C2semidirect
9. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#353172
#219242
#97072
#97022
#96992
#96912
#83442
#83422