Finite Group Databasebeta

← #479853

Subgroups — #479853

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11394C212yesmaximal
2#11390C211yesmaximal, characteristic
3#11381C211yesmaximal, characteristic
4#11381C212yesmaximal
5#11287C211yesmaximal, characteristic
6#11287C212yesmaximal
7#11279C212yesmaximal
8#11279C212yesmaximal
9#11279C212yesmaximal
10C2×#543C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12#2315C2211yescharacteristic
13#2315C2212yes—
14C2×#342C2212yes—
15C2×#342C2212yes—
16C2×#342C2212yes—
17C2×#342C2212yes—
18C2×#342C2212yes—
19C2×#342C2212yes—
20C2×#342C2212yes—
21C2×#342C2211yescharacteristic
22C2×#342C2212yes—
23C2×#342C2212yes—
24C2×#342C2212yes—
25C2×#342C2212yes—
26C2×#342C2211yescharacteristic
27C22⋊#95C2211yescharacteristic
28C22⋊#95C2212yes—
29C22⋊#95C2212yes—
30C22⋊#95C2211yescharacteristic
31C23×Q8Q811yescharacteristic, complemented
32C24×C4C2311yescharacteristic
33#543—24no—
34#543—22no—
35#543—41no—
36C2×C22⋊Q8—42no—
37C2×C22⋊Q8—41no—
38C2×C22⋊Q8—42no—
39C2×C22⋊Q8—41no—
40C2×C22⋊Q8—41no—
41C2×C22⋊Q8—42no—
42C22×C4⋊C4Q811yescharacteristic, complemented
43C22×C4⋊C4Q812yescomplemented
44C22×C22⋊C4C2312yes—
45C22×C22⋊C4C2312yes—
46C22×C22⋊C4C2312yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2311yescharacteristic
51C22×C22⋊C4C2312yes—
52#385—42no—
53#385—42no—
54#385—42no—
55#380—41no—
56#380—42no—
57C2×#95—41no—
58C2×#95—42no—
59C2×#95—42no—
60C2×#95—41no—
61#342—24no—
62#342—24no—
63#342—24no—
64#342—42no—
65#342—42no—
66#342—42no—
67#342—42no—
68#342—42no—
69#342—42no—
70#342—42no—
71#342—41no—
72#342—42no—
73#342Q814yescomplemented
74C25C2411yescharacteristic
75C22×Q8—42no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8—82no—
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—82no—
91C23×C4C2×Q811yescharacteristic
92C23×C4C2×Q812yes—
93C23×C4C2×Q812yes—
94C23×C4C2×Q811yescharacteristic
95C22⋊Q8—44no—
96C22⋊Q8—48no—
97C22⋊Q8—48no—
98C22⋊Q8—44no—
99C22⋊Q8—48no—
100C22⋊Q8—44no—
101C22⋊Q8—44no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—43no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—41no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4C2×Q814yes—
153C2×C22⋊C4C2×Q812yes—
154C2×C22⋊C4#4114yes—
155C2×C22⋊C4#4112yes—
156#95—44no—
157#95—42no—
158#95—82no—
159#95—84no—
160#95—84no—
161#95—82no—
162C24—81no—
163C24—81no—
164C24C22×Q811yescharacteristic
165C24C22×Q811yescharacteristic
166C24C2×#4112yes—
167C24C2×#4111yescharacteristic
168C24H(5,𝔽2)12yes—
169C24H(5,𝔽2)11yescharacteristic
170C24H(5,𝔽2)12yes—
171C24H(5,𝔽2)11yescharacteristic
172C24H(5,𝔽2)11yescharacteristic
173C242−1+411yescharacteristic
174C242−1+412yes—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—82no—
178C2×Q8—84no—
179C2×Q8—82no—
180C2×Q8—84no—
181C2×Q8—161no—
182C2×Q8—161no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4—81no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—81no—
235C22×C4—82no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C4⋊C4—44no—
244C4⋊C4—48no—
245C4⋊C4—412no—
246C4⋊C4—162no—
247C4⋊C4—161no—
248C4⋊C4—162no—
249C4⋊C4—161no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23—81no—
289C23—81no—
290C23—82no—
291C23—81no—
292C23—82no—
293C23—81no—
294C23#53812yes—
295C23#53811yescharacteristic
296C23#54212yes—
297C23#54212yes—
298C23#54212yes—
299C23#54312yes—
300C23#54311yescharacteristic
301C23#54412yes—
302C23#54411yescharacteristic
303C23#55111yescharacteristic
304C23#55112yes—
305C23#55411yescharacteristic
306C23#55412yes—
307C23#55711yescharacteristic
308C23#55712yes—
309C23Q8211yescharacteristic
310C23H(3,𝔽4)12yes—
311C23H(3,𝔽4)11yescharacteristic
312C23H(3,𝔽4)11yescharacteristic
313C23#56212yes—
314C23#56211yescharacteristic
315C23#56311yescharacteristic
316C23#56312yes—
317Q8—88no—
318Q8—164no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—162no—
369C2×C4—161no—
370C2×C4—162no—
371C2×C4—161no—
372C2×C4—162no—
373C2×C4—162no—
374C2×C4—162no—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—42no—
397C22—42no—
398C22—41no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22#273611yescharacteristic
404C22#276311yescharacteristic
405C22#276312yes—
406C22#279912yes—
407C22#279911yescharacteristic
408C22#280612yes—
409C22#280611yescharacteristic
410C22#283212yes—
411C22#283211yescharacteristic
412C22#284011yescharacteristic
413C22#284411yescharacteristic
414C4—42no—
415C4—84no—
416C4—82no—
417C4—84no—
418C4—82no—
419C4—81no—
420C4—82no—
421C4—82no—
422C4—162no—
423C4—162no—
424C4—162no—
425C2—21no—
426C2—22no—
427C2—21no—
428C2—21no—
429C2—22no—
430C2—21no—
431C2—21no—
432C2—22no—
433C2—21no—
434C2—22no—
435C2#1728111yescharacteristic
436C2#3287311yescharacteristic
437C2#3287411yescharacteristic
subgroupG/H
center, socleC22#2840
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. C23×Q8Q8semidirect
42. C22×C4⋊C4Q8semidirect
43. C22×C4⋊C4Q8semidirect
73. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#113942
#113902
#113812
#112872
#112792