Finite Group Databasebeta

← #164818

Subgroups — #164818

Order 512.

775 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36453C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#22012C211yesmaximal, characteristic, complemented
4#7721C212yesmaximal, complemented
5#7715C212yesmaximal, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7647C212yesmaximal, complemented
9#7635C211yesmaximal, characteristic, complemented
10#7599C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic, complemented
12#2859C2211yescharacteristic
13#2855C2212yes—
14#2576—22no—
15#2538C411yescharacteristic
16#2537C411yescharacteristic
17#2494—21no—
18#2494C412yes—
19#2460—22no—
20#2427C412yes—
21#2420—21no—
22#2361—21no—
23#2355C411yescharacteristic
24D4×C4⋊C4C411yescharacteristic
25#2345—21no—
26C4×#124C2211yescharacteristic
27C4×#121C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1924—22no—
34#1924—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C2×#125—21no—
54C2×#124C2×C411yescharacteristic
55C2×C22⋊Q8C2×C412yes—
56C2×#121—21no—
57C2×#121C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×D4—22no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C42C2312yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×#97—21no—
76C2×#97—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#401—22no—
80#397—22no—
81#397—22no—
82#396—22no—
83#394—22no—
84#394—22no—
85#391C2×C412yes—
86#391C2311yescharacteristic
87#390—21no—
88#390C2311yescharacteristic
89#389—21no—
90#389—21no—
91#389—21no—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2312yes—
95#388—22no—
96#388—21no—
97#388—21no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#386—22no—
104#386—22no—
105#385—22no—
106#385—22no—
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#382—21no—
110#382—21no—
111#382C2×C411yescharacteristic
112#381—22no—
113#380—21no—
114#380—21no—
115#380—22no—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C43C2311yescharacteristic
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×Q8C22×C411yescharacteristic
141C22×D4—22no—
142C22×D4C22⋊C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4—22no—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155#125—81no—
156#125—81no—
157#124—22no—
158#124—41no—
159C22⋊Q8—82no—
160#121—24no—
161#121—42no—
162#121—44no—
163C4×Q8—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—82no—
169C4×D4—82no—
170C4×D4—82no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—82no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8—82no—
217C4⋊C8—82no—
218C22⋊C8—82no—
219C22⋊C8—82no—
220C22⋊C8—82no—
221C22⋊C8—82no—
222#97—81no—
223#97—81no—
224C4×C8—42no—
225C4×C8—42no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95—21no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C24—21no—
242C24C2×C22⋊C412yes—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—82no—
256C2×D4#13112yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C4⋊C4—81no—
324C4⋊C4—82no—
325C4⋊C4—81no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—42no—
351C42—42no—
352C42—41no—
353C42—42no—
354C42—42no—
355C42—82no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C22×C22⋊C411yescharacteristic
371Q8—81no—
372Q8—81no—
373D4—42no—
374D4—42no—
375D4—44no—
376D4—88no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4—164no—
424C2×C4#41312yes—
425C2×C4#42112yes—
426C2×C4#42112yes—
427C2×C4C2×#13111yescharacteristic
428C2×C4#56811yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57811yescharacteristic
433C8—81no—
434C8—82no—
435C8—81no—
436C8—81no—
437C8—82no—
438C8—81no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22#288011yescharacteristic
454C22#288311yescharacteristic
455C22C2×#41311yescharacteristic
456C22#289711yescharacteristic
457C22#289811yescharacteristic
458C22C2×#42111yescharacteristic
459C22C2×#42111yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—82no—
475C4—82no—
476C4—81no—
477C4#153514yes—
478C2—82no—
479C2—82no—
480C2#1742411yescharacteristic
481C2#1752211yescharacteristic
482C2#1752311yescharacteristic
483C2#1759911yescharacteristic
484C2#1760011yescharacteristic
485C2C2×#153511yescharacteristic
486C2#1777211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22012C2semidirect
3. #22012C2semidirect
4. #7721C2semidirect
5. #7715C2semidirect
6. #7683C2semidirect
7. #7680C2semidirect
8. #7647C2semidirect
9. #7635C2semidirect
10. #7599C2semidirect
11. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364532
#220122
#77212
#77152
#76832
#76802
#76472
#76352
#75992
#75672