Finite Group Databasebeta

← #207599

Subgroups — #207599

Order 512.

851 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20472C211yesmaximal, characteristic, complemented
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9648C211yesmaximal, characteristic, complemented
7#9644C211yesmaximal, characteristic, complemented
8#9078C212yesmaximal, complemented
9#9076C212yesmaximal, complemented
10#8179C212yesmaximal, complemented
11#8177C212yesmaximal, complemented
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2651—22no—
15#2648—22no—
16#2647C411yescharacteristic
17#2644C412yes—
18#2641C411yescharacteristic
19#2629C411yescharacteristic
20#2624C412yes—
21#2619C411yescharacteristic
22#2395—22no—
23#2392—22no—
24C4×#127C2211yescharacteristic
25C4×#125C2211yescharacteristic
26C4×#121C2212yes—
27#2293C2212yes—
28C4×#117C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1771—22no—
35#1767C2212yes—
36#1753—22no—
37#1744C2211yescharacteristic
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50C2×#127C2×C411yescharacteristic
51C2×#125C2×C411yescharacteristic
52C2×#121C2×C412yes—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2312yes—
56C2×#117—22no—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60#431—22no—
61#431—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—22no—
68C4×M16—22no—
69C4×M16—22no—
70C2×#97—21no—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8—22no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#400—22no—
79#399—22no—
80#398—22no—
81#398—22no—
82#397—22no—
83#397—22no—
84#397—22no—
85#397—22no—
86#396—22no—
87#396—22no—
88#394—22no—
89#394—22no—
90#394—22no—
91#394—22no—
92#394—22no—
93#392—22no—
94#390—22no—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2×C412yes—
98#390C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—22no—
107#386—22no—
108#384C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110#380C2×C412yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D412yes—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128D4⋊C8—24no—
129D4⋊C8—24no—
130D4⋊C8—24no—
131D4⋊C8—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22×C412yes—
135C23×C4—22no—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—44no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147#127—42no—
148#125—42no—
149#121—44no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—82no—
155C4×D4—82no—
156C4×D4—82no—
157C4×D4—82no—
158#117—42no—
159#117—42no—
160#117—82no—
161#117—82no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—24no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—82no—
209C22⋊C8—24no—
210C22⋊C8—82no—
211#97—24no—
212#97—44no—
213#97—82no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—44no—
219C4×C8#4112yes—
220C4×C8#4112yes—
221C4×C8#4112yes—
222C4×C8#4112yes—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24—22no—
235C24C2×C22⋊C411yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—82no—
243C2×D4—82no—
244C2×D4—82no—
245C2×D4—82no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—48no—
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
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—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C42C2×#4111yescharacteristic
339C23—21no—
340C23—21no—
341C23—22no—
342C23—22no—
343C23—21no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23C22×C22⋊C411yescharacteristic
352D4—44no—
353D4—44no—
354D4—88no—
355D4—88no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
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—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4C4×#4111yescharacteristic
397C2×C4C4×#4111yescharacteristic
398C2×C4#52511yescharacteristic
399C2×C4#52511yescharacteristic
400C2×C4#53211yescharacteristic
401C2×C4#53211yescharacteristic
402C2×C4#56812yes—
403C2×C4#56812yes—
404C2×C4#57612yes—
405C2×C4#57712yes—
406C8—24no—
407C8—42no—
408C8—42no—
409C8—42no—
410C8—82no—
411C8—82no—
412C22—42no—
413C22—42no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22#229711yescharacteristic
425C22#287911yescharacteristic
426C22#287911yescharacteristic
427C22#289712yes—
428C22#290112yes—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4#296712yes—
442C4#296712yes—
443C4#317411yescharacteristic
444C4#317611yescharacteristic
445C4#317912yes—
446C2—81no—
447C2—162no—
448C2#1750512yes—
449C2#1750512yes—
450C2#2137611yescharacteristic
451C2#2141411yescharacteristic
452C2#2141911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2619C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #20472C2semidirect
4. #9876C2semidirect
5. #9875C2semidirect
6. #9648C2semidirect
7. #9644C2semidirect
8. #9078C2semidirect
9. #9076C2semidirect
10. #8179C2semidirect
11. #8177C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#219262
#204722
#98762
#98752
#96482
#96442
#90782
#90762
#81792
#81772