Finite Group Databasebeta

← #162426

Subgroups — #162426

Order 512.

693 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#35184C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7729C212yesmaximal, complemented
5#7716C212yesmaximal
6#7669C212yesmaximal, complemented
7#7656C212yesmaximal
8#7590C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic
10#7571C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12#2863C2212yes—
13#2861C2212yes—
14#2654—22no—
15#2568C411yescharacteristic
16#2552C411yescharacteristic
17#2547C411yescharacteristic
18#2538C411yescharacteristic
19#2525—22no—
20#2495—21no—
21#2453C412yes—
22#2448C412yes—
23#2423—21no—
24#2323—21no—
25#2314—21no—
26C4×C4⋊Q8C2211yescharacteristic
27#2293C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1916—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×#124C2×C412yes—
56C2×C22⋊Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×#117—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71#400—22no—
72#400—22no—
73#399—22no—
74#398—22no—
75#398—22no—
76#397—22no—
77#397—22no—
78#396—22no—
79#391—21no—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2312yes—
84#390C2×C412yes—
85#390C2×C411yescharacteristic
86#390C2311yescharacteristic
87#389—22no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C411yescharacteristic
94#389C2311yescharacteristic
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—21no—
101#388—21no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#384—21no—
108#384C2×C411yescharacteristic
109#383—22no—
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—21no—
124C43C2311yescharacteristic
125C43C2312yes—
126#336—21no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—22no—
134#336C2311yescharacteristic
135#336C2312yes—
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8C22⋊C411yescharacteristic
141C22×Q8C22×C412yes—
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C4⋊Q8—24no—
151#125—82no—
152#124—82no—
153C22⋊Q8—44no—
154C4×Q8—24no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—42no—
158C4×Q8—82no—
159C4×D4—24no—
160#117—42no—
161#117—42no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—42no—
210C4⋊C8—82no—
211C4×C8—42no—
212C4×C8—44no—
213C4×C8—42no—
214C4×C8—81no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231C24—21no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×D4—22no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—44no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—44no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—81no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C42—24no—
312C42—22no—
313C42—22no—
314C42—21no—
315C42—22no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—41no—
321C42—41no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—42no—
326C42—82no—
327C42#13114yes—
328C42#13512yes—
329C42#13512yes—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23C22×C22⋊C411yescharacteristic
338Q8—44no—
339Q8—84no—
340D4—82no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—84no—
382C2×C4#40812yes—
383C2×C4#40812yes—
384C2×C4#41612yes—
385C2×C4#41612yes—
386C2×C4#42112yes—
387C2×C4C2×#13112yes—
388C2×C4C2×#13511yescharacteristic
389C2×C4C2×#13511yescharacteristic
390C2×C4#57711yescharacteristic
391C2×C4#57811yescharacteristic
392C8—42no—
393C8—81no—
394C8—82no—
395C8—81no—
396C8—81no—
397C8—82no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C2×#40811yescharacteristic
406C22#288411yescharacteristic
407C22#288811yescharacteristic
408C22C2×#41611yescharacteristic
409C22#289811yescharacteristic
410C22C2×#42111yescharacteristic
411C22#290111yescharacteristic
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—82no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—82no—
429C4#148214yes—
430C2—161no—
431C2#1739011yescharacteristic
432C2C2×#148211yescharacteristic
433C2#1748411yescharacteristic
434C2#1760111yescharacteristic
435C2#1761611yescharacteristic
436C2#1768411yescharacteristic
437C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22828C2semidirect
3. #22018C2semidirect
4. #7729C2semidirect
6. #7669C2semidirect
8. #7590C2semidirect
10. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#351842
#228282
#220182
#77292
#77162
#76692
#76562
#75902
#75742
#75712
#75602