Finite Group Databasebeta

← #164842

Subgroups — #164842

Order 512.

735 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36300C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#22022C211yesmaximal, characteristic, complemented
4#7732C212yesmaximal, complemented
5#7726C212yesmaximal, complemented
6#7679C211yesmaximal, characteristic
7#7679C211yesmaximal, characteristic
8#7642C212yesmaximal, complemented
9#7636C211yesmaximal, characteristic, complemented
10#7594C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12#2860C2212yes—
13#2858C2211yescharacteristic
14#2567C411yescharacteristic
15#2548C411yescharacteristic
16#2521—22no—
17#2513C412yes—
18#2512—21no—
19#2474—22no—
20#2454C412yes—
21#2448—21no—
22#2369C411yescharacteristic
23#2367—21no—
24#2366C411yescharacteristic
25#2354—21no—
26C4×#125C2211yescharacteristic
27#2293C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1839C2212yes—
36#1839C2212yes—
37#1831C2212yes—
38#1831C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#126—22no—
56C2×#125C2×C411yescharacteristic
57C2×#124—21no—
58C2×#124—21no—
59C2×#124—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×#117—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×#97—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#399—22no—
75#397—22no—
76#397—22no—
77#396—22no—
78#395—22no—
79#394—22no—
80#391—21no—
81#391—22no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2312yes—
91#389—21no—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2311yescharacteristic
96#389C2311yescharacteristic
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#387—21no—
105#387—21no—
106#386—22no—
107#385—22no—
108#384—21no—
109#382C2×C411yescharacteristic
110#380—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C43C2311yescharacteristic
125C43C2312yes—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×Q8C22⋊C411yescharacteristic
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22×C412yes—
145C23×C4—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—42no—
153#127—42no—
154#126—82no—
155#126—82no—
156#125—22no—
157#125—41no—
158#124—44no—
159#124—81no—
160#124—81no—
161#124—81no—
162#124—81no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×D4—24no—
167C4×D4—42no—
168C4×D4—44no—
169#117—82no—
170#117—82no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42C22×C412yes—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2411yescharacteristic
217C4⋊C8—42no—
218C4⋊C8—42no—
219C4⋊C8—82no—
220C4⋊C8—82no—
221#97—81no—
222#97—81no—
223C4×C8—42no—
224C4×C8—42no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238C24—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—81no—
254C2×D4—81no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—82no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4#13112yes—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C42—22no—
329C42—21no—
330C42—22no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—41no—
347C42—42no—
348C42—42no—
349C42—82no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23C22×C22⋊C411yescharacteristic
358Q8—44no—
359Q8—162no—
360D4—82no—
361D4—82no—
362D4—84no—
363D4—162no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—88no—
407C2×C4#41312yes—
408C2×C4#42112yes—
409C2×C4#42112yes—
410C2×C4C2×#13111yescharacteristic
411C2×C4#56811yescharacteristic
412C2×C4#57611yescharacteristic
413C2×C4#57611yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57711yescharacteristic
416C8—81no—
417C8—81no—
418C8—82no—
419C8—82no—
420C8—81no—
421C8—81no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22#288011yescharacteristic
430C22#288411yescharacteristic
431C22C2×#41311yescharacteristic
432C22#289711yescharacteristic
433C22#289711yescharacteristic
434C22C2×#42111yescharacteristic
435C22C2×#42111yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—41no—
444C4—41no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4#153514yes—
455C2—162no—
456C2#1742811yescharacteristic
457C2#1752211yescharacteristic
458C2#1752611yescharacteristic
459C2#1760211yescharacteristic
460C2#1760211yescharacteristic
461C2C2×#153511yescharacteristic
462C2#1777611yescharacteristic
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. #22022C2semidirect
3. #22022C2semidirect
4. #7732C2semidirect
5. #7726C2semidirect
8. #7642C2semidirect
9. #7636C2semidirect
10. #7594C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#363002
#220222
#77322
#77262
#76792
#76422
#76362
#75942
#75682