Finite Group Databasebeta

← #162432

Subgroups — #162432

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35186C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7726C212yesmaximal, complemented
5#7716C212yesmaximal, complemented
6#7666C212yesmaximal, complemented
7#7656C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic, complemented
12#2861C2212yes—
13#2860C2212yes—
14#2651—22no—
15#2582C411yescharacteristic
16#2552C411yescharacteristic
17#2548C411yescharacteristic
18#2548C411yescharacteristic
19#2512—21no—
20#2468—22no—
21#2454C412yes—
22#2448—21no—
23#2447C412yes—
24#2326—21no—
25#2319—21no—
26C4×#127C2211yescharacteristic
27#2295C2211yescharacteristic
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#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C2×#127C2×C411yescharacteristic
54C2×#126—22no—
55C2×#125C2×C412yes—
56C2×#124—21no—
57C2×#124C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#399—22no—
74#398—22no—
75#398—22no—
76#397—22no—
77#397—22no—
78#396—22no—
79#394—22no—
80#394—22no—
81#391—21no—
82#391C2×C412yes—
83#391C2×C411yescharacteristic
84#391C2311yescharacteristic
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2312yes—
94#389—21no—
95#389—21no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C411yescharacteristic
99#389C2311yescharacteristic
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—21no—
108#387—21no—
109#385—22no—
110#385—22no—
111#384—21no—
112#384C2×C411yescharacteristic
113#381—22no—
114#380—22no—
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
123#376—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×C411yescharacteristic
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22×C412yes—
145C23×C4—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152#127—24no—
153#126—82no—
154#126—82no—
155#125—82no—
156#124—44no—
157#124—82no—
158C4×Q8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—81no—
166C4×D4—81no—
167C4×D4—82no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—42no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8—82no—
217C4×C8—42no—
218C4×C8—44no—
219C4×C8—42no—
220C4×C8—81no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C24—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—24no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—81no—
250C2×D4—81no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—44no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C42—24no—
322C42—22no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—21no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—42no—
333C42—42no—
334C42—41no—
335C42—42no—
336C42—42no—
337C42#13114yes—
338C42#13512yes—
339C42#13512yes—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23C22×C22⋊C411yescharacteristic
348Q8—44no—
349Q8—82no—
350D4—84no—
351D4—84no—
352D4—84no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—164no—
394C2×C4#40812yes—
395C2×C4#40812yes—
396C2×C4#41612yes—
397C2×C4#41612yes—
398C2×C4#42112yes—
399C2×C4C2×#13112yes—
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×#13511yescharacteristic
402C2×C4#57611yescharacteristic
403C2×C4#57711yescharacteristic
404C8—42no—
405C8—81no—
406C8—82no—
407C8—81no—
408C8—81no—
409C8—82no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22C2×#40811yescharacteristic
418C22#288411yescharacteristic
419C22#288811yescharacteristic
420C22C2×#41611yescharacteristic
421C22#289711yescharacteristic
422C22C2×#42111yescharacteristic
423C22#290111yescharacteristic
424C4—22no—
425C4—22no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—22no—
431C4—41no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—81no—
437C4—82no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4#148214yes—
442C2—162no—
443C2#1739011yescharacteristic
444C2C2×#148211yescharacteristic
445C2#1748311yescharacteristic
446C2#1760211yescharacteristic
447C2#1761611yescharacteristic
448C2#1768411yescharacteristic
449C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×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. #7726C2semidirect
5. #7716C2semidirect
6. #7666C2semidirect
7. #7656C2semidirect
9. #7577C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#351862
#228282
#220182
#77262
#77162
#76662
#76562
#75872
#75772
#75692
#75622