Finite Group Databasebeta

← #493726

Subgroups — #493726

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34208C211yesmaximal, characteristic
2C2×#2132C212yesmaximal
3#20640C212yesmaximal, complemented
4#12804C214yesmaximal, complemented
5#12801C214yesmaximal, complemented
6#12587C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#520C411yescharacteristic
9C2×#520C2211yescharacteristic
10#2905—22no—
11C2×#413C2214yes—
12#2364—24no—
13#2352—22no—
14#2352—22no—
15#2328C412yes—
16#2327C2212yes—
17#2311C411yescharacteristic
18#2311C2211yescharacteristic
19C4×#124C412yes—
20C4×#124C2212yes—
21C2×#382C411yescharacteristic
22C4×#117C411yescharacteristic
23#2132—28no—
24C2×#356C2214yes—
25C2×#355C2214yes—
26#1801C2218yes—
27#1800—24no—
28C2×#343—22no—
29#1394C2218yes—
30C22×#41C2×C411yescharacteristic
31C22×#41C2311yescharacteristic
32C22×M16—24no—
33#532—24no—
34#532D414yes—
35#532Q814yes—
36C2×C4⋊Q8C2311yescharacteristic
37C2×#127C2×C411yescharacteristic
38C2×#124C2×C411yescharacteristic
39C2×#124C2311yescharacteristic
40C2×#121—22no—
41#520—44no—
42#520—44no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2312yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4C2×C412yes—
49C2×C4×D4D414yes—
50C2×#117—21no—
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117D412yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42C2×C411yescharacteristic
59#424—48no—
60#423—28no—
61#413—28no—
62#413—28no—
63C2×#101C2318yes—
64C2×#100—24no—
65#391C2×C412yes—
66#391C2312yes—
67#390C2×C412yes—
68#390C2×C412yes—
69#388—24no—
70#388—24no—
71#387—24no—
72#387—24no—
73#386—22no—
74#386—22no—
75#386—24no—
76#386—24no—
77#382—22no—
78#382C2×C412yes—
79#382C2×C412yes—
80#380—22no—
81#380—22no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—24no—
87#376C2×C412yes—
88C2×#95—22no—
89C43C2×C412yes—
90#356—28no—
91#355—28no—
92#343—28no—
93C2×#41—22no—
94C2×#41—42no—
95C2×#41—42no—
96C2×#41—42no—
97C2×#41—42no—
98C2×#41C4⋊C414yes—
99C2×#41C2×D412yes—
100C2×#41C2×Q812yes—
101C2×#41#4114yes—
102C22×Q8C22×C411yescharacteristic
103C22×Q8C2411yescharacteristic
104C22×D4—21no—
105C22×D4—22no—
106C22×D4C22×C411yescharacteristic
107C22×D4C22×C411yescharacteristic
108C22×D4C2×D412yes—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C414yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D412yes—
117C2×M16—28no—
118C2×M16—44no—
119C2×M16—48no—
120C2×M16—48no—
121C22×C8—44no—
122C4⋊Q8—22no—
123C4⋊Q8C2×D412yes—
124C4⋊Q8C2×Q812yes—
125#127—22no—
126#127C4⋊C414yes—
127#124—22no—
128#124—22no—
129#124C4⋊C414yes—
130#124C2×D412yes—
131#124C2×Q812yes—
132#121—28no—
133#121—28no—
134C4×Q8—44no—
135C4×Q8—44no—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4—44no—
139C4×D4—48no—
140C4×D4—44no—
141C4×D4—48no—
142#117—24no—
143#117—28no—
144#117—44no—
145#117—42no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C2×D414yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C22⋊C414yes—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C4⋊C414yes—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C412yes—
178C2×C42C2×D414yes—
179C4⋊C8—416no—
180#101—28no—
181#101—28no—
182#100—48no—
183#95—24no—
184#95—24no—
185#95—22no—
186#95—22no—
187#95—24no—
188#95—24no—
189C24—21no—
190C24C2×C22⋊C412yes—
191C24C2×C22⋊C411yescharacteristic
192#41—44no—
193#41—44no—
194#41—48no—
195#41—48no—
196#41—48no—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×C4⋊C412yes—
203C2×Q8C22×D411yescharacteristic
204C2×Q8C22×Q811yescharacteristic
205C2×Q8C2×#4112yes—
206C2×D4—24no—
207C2×D4—28no—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—82no—
218C2×D4—82no—
219C2×D4C2×C4⋊C412yes—
220C2×D4C2×C4⋊C412yes—
221C2×D4#11712yes—
222C2×D4#12114yes—
223C2×D4C22⋊Q814yes—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—84no—
244C22×C4—82no—
245C22×C4—84no—
246C22×C4—82no—
247C22×C4C2×C22⋊C414yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C4⋊C412yes—
252C22×C4#11712yes—
253C22×C4#12114yes—
254C22×C4C22⋊Q814yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D412yes—
257M16—48no—
258M16—48no—
259M16—88no—
260C2×C8—48no—
261C2×C8—88no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4#9914yes—
273C22⋊C4—28no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—28no—
277C22⋊C4—24no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—84no—
281C22⋊C4—84no—
282C42—21no—
283C42—21no—
284C42—24no—
285C42—28no—
286C42—24no—
287C42—42no—
288C42—44no—
289C42—82no—
290C42—82no—
291C42C2×C4⋊C412yes—
292C42C2×C4⋊C412yes—
293C23—21no—
294C23—22no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—41no—
305C23—41no—
306C23#38014yes—
307C23#38012yes—
308C23C22×C22⋊C411yescharacteristic
309Q8—41no—
310Q8—41no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—42no—
315D4—42no—
316D4—44no—
317D4—44no—
318D4—42no—
319D4—42no—
320D4—48no—
321D4—88no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—84no—
351C2×C4—164no—
352C2×C4—164no—
353C2×C4—162no—
354C2×C4—162no—
355C2×C4#38012yes—
356C2×C4C2×#9914yes—
357C2×C4C2×#9912yes—
358C2×C4#41012yes—
359C2×C4C22×C4⋊C411yescharacteristic
360C2×C4C2×#11711yescharacteristic
361C2×C4C2×#12112yes—
362C2×C4C2×C22⋊Q812yes—
363C8—88no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—24no—
368C22—42no—
369C22—44no—
370C22—42no—
371C22—44no—
372C22—41no—
373C22—42no—
374C22—41no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—81no—
380C22—81no—
381C22C2×#38011yescharacteristic
382C22C22×#9911yescharacteristic
383C22C2×#41011yescharacteristic
384C4—24no—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—44no—
393C4—42no—
394C4—82no—
395C4—82no—
396C4#179914yes—
397C2—21no—
398C2—21no—
399C2—42no—
400C2—44no—
401C2—82no—
402C2C2×#179911yescharacteristic
403C2#3337011yescharacteristic
404C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20640C2semidirect
4. #12804C2semidirect
5. #12801C2semidirect
6. #12587C2semidirect

Maximal subgroups

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

subgroupindex
#342082
C2×#21322
#206402
#128042
#128012
#125872