Finite Group Databasebeta

← #164897

Subgroups — #164897

Order 512.

873 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36271C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic, complemented
3#22734C211yesmaximal, characteristic, complemented
4#7672C212yesmaximal, complemented
5#7666C212yesmaximal, complemented
6#7659C211yesmaximal, characteristic
7#7659C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic, complemented
9#7642C212yesmaximal, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7560C212yesmaximal, complemented
12C2×#556—21no—
13C2×C8×Q8—21no—
14#2860C2212yes—
15#2858C2211yescharacteristic
16#2601—22no—
17#2592—21no—
18#2560—22no—
19#2548C411yescharacteristic
20#2546C411yescharacteristic
21#2545—21no—
22#2512C412yes—
23#2452C412yes—
24#2367—21no—
25#2366C411yescharacteristic
26#2365C411yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28#2293C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1939—22no—
33#1916—22no—
34#1839C2212yes—
35#1831C2211yescharacteristic
36#1831C2212yes—
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2211yescharacteristic
42#1744C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53#556—44no—
54#556—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#126—22no—
60C2×#126—22no—
61C2×#124—21no—
62C2×#124—21no—
63C2×#124—21no—
64C2×C22⋊Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×#117—22no—
73C8×Q8—44no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#398—22no—
85#397—22no—
86#396—22no—
87#396—22no—
88#394—22no—
89#393—22no—
90#391—21no—
91#391—21no—
92#391—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390C2×C412yes—
98#390C2×C412yes—
99#390C2×C412yes—
100#390C2×C412yes—
101#390C2312yes—
102#389—21no—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2311yescharacteristic
107#389C2311yescharacteristic
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#387—22no—
115#386—22no—
116#385—22no—
117#384C2×C411yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—22no—
129C43C2311yescharacteristic
130C43C2312yes—
131#336—21no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8C22⋊C411yescharacteristic
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—41no—
158C4⋊Q8—42no—
159C4⋊Q8—44no—
160#127—42no—
161#126—48no—
162#126—44no—
163#126—44no—
164#124—44no—
165#124—42no—
166#124—42no—
167#124—42no—
168#124—42no—
169C22⋊Q8—48no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—42no—
173C4×Q8—44no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8D814yes—
177C4×Q8SD1614yes—
178C4×D4—24no—
179C4×D4—42no—
180C4×D4—44no—
181#117—44no—
182#117—44no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2411yescharacteristic
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—44no—
233C4⋊C8—44no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4×C8—44no—
237C4×C8—44no—
238C4×C8—42no—
239C4×C8—42no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—81no—
260C2×Q8#10214yes—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—82no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—81no—
329C4⋊C4#10214yes—
330C4⋊C4#10218yes—
331C4⋊C4#13112yes—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—22no—
346C42—21no—
347C42—22no—
348C42—21no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—42no—
353C42—41no—
354C42—41no—
355C42—42no—
356C42—42no—
357C42—41no—
358C42—44no—
359C42—42no—
360C42—42no—
361C42C2×D814yes—
362C42C2×D812yes—
363C42C2×SD1614yes—
364C42C2×SD1612yes—
365C42#13512yes—
366C42#13512yes—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23C22×C22⋊C411yescharacteristic
375Q8—28no—
376Q8—84no—
377Q8—84no—
378D4—82no—
379D4—82no—
380D4—84no—
381D4—162no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—88no—
421C2×C4#41312yes—
422C2×C4C2×#10214yes—
423C2×C4C2×#10214yes—
424C2×C4C2×#10212yes—
425C2×C4C2×#10212yes—
426C2×C4#41612yes—
427C2×C4#41612yes—
428C2×C4C2×#13111yescharacteristic
429C2×C4#56811yescharacteristic
430C2×C4C22×D811yescharacteristic
431C2×C4C22×SD1611yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C8—42no—
435C8—42no—
436C8—81no—
437C8—82no—
438C8—82no—
439C8—81no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22#288011yescharacteristic
448C22#288411yescharacteristic
449C22C2×#41311yescharacteristic
450C22C22×#10211yescharacteristic
451C22C2×#41611yescharacteristic
452C22#290111yescharacteristic
453C22#290111yescharacteristic
454C4—21no—
455C4—21no—
456C4—24no—
457C4—24no—
458C4—22no—
459C4—22no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—42no—
468C4—82no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4#154014yes—
473C2—162no—
474C2#1742811yescharacteristic
475C2#1750711yescharacteristic
476C2#1751011yescharacteristic
477C2#1761611yescharacteristic
478C2#1761811yescharacteristic
479C2#1769111yescharacteristic
480C2C2×#154011yescharacteristic
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. #22734C2semidirect
4. #7672C2semidirect
5. #7666C2semidirect
8. #7652C2semidirect
9. #7642C2semidirect
10. #7570C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#362712
#228282
#227342
#76722
#76662
#76592
#76522
#76422
#75702
#75602