Finite Group Databasebeta

← #164673

Subgroups — #164673

Order 512.

761 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36301C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20774C211yesmaximal, characteristic, complemented
4#7717C212yesmaximal, complemented
5#7688C211yesmaximal, characteristic, complemented
6#7665C211yesmaximal, characteristic, complemented
7#7657C212yesmaximal, complemented
8#7642C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7580C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12#2864C2212yes—
13#2857C2211yescharacteristic
14#2571C411yescharacteristic
15#2563C411yescharacteristic
16#2527—22no—
17#2494—21no—
18#2494C412yes—
19#2485—21no—
20#2477—22no—
21#2420C412yes—
22#2369C411yescharacteristic
23#2352—21no—
24D4×C4⋊C4C411yescharacteristic
25#2344—21no—
26C4×#121C2211yescharacteristic
27#2293C2212yes—
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1923—22no—
31#1840—22no—
32#1839C2212yes—
33#1839C2212yes—
34#1831C2212yes—
35#1797C2211yescharacteristic
36C8×C4⋊C4—21no—
37C4×C22⋊C8C2211yescharacteristic
38#1744C2212yes—
39#1729—21no—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×#121C2×C411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×#117—22no—
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C4×C8—22no—
71C2×C4×C8—21no—
72#401—22no—
73#400—22no—
74#397—22no—
75#397—22no—
76#394—22no—
77#393—22no—
78#391—21no—
79#391—22no—
80#390C2×C412yes—
81#390C2311yescharacteristic
82#389—21no—
83#389—21no—
84#389—21no—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2312yes—
89#389C2311yescharacteristic
90#388—22no—
91#388—22no—
92#388—22no—
93#388—21no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#386—21no—
100#386—22no—
101#385—22no—
102#384C2×C411yescharacteristic
103#382—21no—
104#382C2×C412yes—
105#382C2×C412yes—
106#381—22no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380C2×C412yes—
112#380C2×C411yescharacteristic
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4D412yes—
124#376D412yes—
125C43C2312yes—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—21no—
133#336—22no—
134#336C2312yes—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—42no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—42no—
160C4⋊Q8—42no—
161C22⋊Q8—81no—
162C22⋊Q8—81no—
163#121—22no—
164#121—41no—
165C4×D4—24no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170#117—22no—
171#117—22no—
172#117—42no—
173#117—82no—
174#117—82no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—42no—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2411yescharacteristic
217C4⋊C8—42no—
218C4⋊C8—82no—
219C22⋊C8—42no—
220C22⋊C8—81no—
221C4×C8—42no—
222C4×C8—44no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24—21no—
241C24C2×C22⋊C411yescharacteristic
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—81no—
251C2×D4#13112yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—82no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4#13512yes—
281C22×C4#13512yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—84no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—84no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—81no—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—42no—
348C42—41no—
349C42—41no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—82no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23#41612yes—
368C23C22×C22⋊C411yescharacteristic
369Q8—162no—
370D4—42no—
371D4—42no—
372D4—84no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—88no—
419C2×C4—84no—
420C2×C4#41012yes—
421C2×C4#41612yes—
422C2×C4#42112yes—
423C2×C4C2×#13111yescharacteristic
424C2×C4#56811yescharacteristic
425C2×C4C2×#13511yescharacteristic
426C2×C4C2×#13511yescharacteristic
427C2×C4#57611yescharacteristic
428C2×C4#57711yescharacteristic
429C8—42no—
430C8—81no—
431C8—82no—
432C8—81no—
433C8—81no—
434C8—82no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22#288011yescharacteristic
450C22C2×#41011yescharacteristic
451C22#288911yescharacteristic
452C22C2×#41611yescharacteristic
453C22#289711yescharacteristic
454C22C2×#42111yescharacteristic
455C22#290111yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—41no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—41no—
468C4—81no—
469C4—81no—
470C4—82no—
471C4—81no—
472C4—82no—
473C4#152114yes—
474C2—42no—
475C2—82no—
476C2#1741311yescharacteristic
477C2#1750811yescharacteristic
478C2#1752211yescharacteristic
479C2C2×#152111yescharacteristic
480C2#1761011yescharacteristic
481C2#1768311yescharacteristic
482C2#1772711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22632C2semidirect
3. #20774C2semidirect
4. #7717C2semidirect
5. #7688C2semidirect
6. #7665C2semidirect
7. #7657C2semidirect
8. #7642C2semidirect
10. #7580C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#226322
#207742
#77172
#76882
#76652
#76572
#76422
#76162
#75802
#75652