Finite Group Databasebeta

← #164781

Subgroups — #164781

Order 512.

750 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#36315C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20774C211yesmaximal, characteristic, complemented
4#7716C212yesmaximal, complemented
5#7687C211yesmaximal, characteristic, complemented
6#7667C211yesmaximal, characteristic, complemented
7#7656C212yesmaximal, complemented
8#7644C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7581C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12#2861C2212yes—
13#2857C2211yescharacteristic
14#2552C411yescharacteristic
15#2547C411yescharacteristic
16#2531—22no—
17#2529—22no—
18#2497—21no—
19#2489C412yes—
20#2485—21no—
21#2426C412yes—
22#2365C411yescharacteristic
23#2354—21no—
24#2354C411yescharacteristic
25#2346—21no—
26C4×#121C2211yescharacteristic
27#2295C2212yes—
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#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×#124C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×#121C2×C411yescharacteristic
60C2×C4×Q8C2×C412yes—
61C2×C4×D4—21no—
62C2×C4×D4C2×C411yescharacteristic
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C4×C8—22no—
72C2×C4×C8—21no—
73#400—22no—
74#398—22no—
75#397—22no—
76#397—22no—
77#396—22no—
78#394—22no—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2312yes—
82#390—22no—
83#390C2×C412yes—
84#390C2311yescharacteristic
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389C2×C412yes—
90#389C2311yescharacteristic
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—21no—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#386—21no—
103#384—21no—
104#384C2×C411yescharacteristic
105#382—22no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380C2×C411yescharacteristic
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4D412yes—
123#376—22no—
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×Q8C22×C412yes—
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—42no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—42no—
159C4⋊Q8—42no—
160#125—82no—
161#124—82no—
162C22⋊Q8—82no—
163#121—22no—
164#121—41no—
165C4×Q8—24no—
166C4×Q8—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170#117—22no—
171#117—22no—
172#117—42no—
173#117—44no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C412yes—
210C2×C42C22×C412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—42no—
219C4⋊C8—82no—
220C22⋊C8—42no—
221C22⋊C8—81no—
222C4×C8—42no—
223C4×C8—44no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239C24—21no—
240C24C2×C22⋊C411yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—81no—
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—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4#13512yes—
278C22×C4#13512yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—84no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—84no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—44no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—81no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4#13112yes—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42—41no—
346C42—41no—
347C42—42no—
348C42—42no—
349C42—44no—
350C42—44no—
351C42—42no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23#41612yes—
366C23C22×C22⋊C411yescharacteristic
367Q8—82no—
368Q8—82no—
369D4—81no—
370D4—81no—
371D4—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
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—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—84no—
418C2×C4#41012yes—
419C2×C4#41612yes—
420C2×C4#42112yes—
421C2×C4C2×#13111yescharacteristic
422C2×C4#56811yescharacteristic
423C2×C4C2×#13511yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57811yescharacteristic
427C8—42no—
428C8—81no—
429C8—82no—
430C8—81no—
431C8—81no—
432C8—82no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22#288011yescharacteristic
448C22C2×#41011yescharacteristic
449C22#288811yescharacteristic
450C22C2×#41611yescharacteristic
451C22#289811yescharacteristic
452C22C2×#42111yescharacteristic
453C22#290111yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—82no—
468C4—82no—
469C4—81no—
470C4—81no—
471C4#152114yes—
472C2—42no—
473C2—161no—
474C2#1741811yescharacteristic
475C2#1750811yescharacteristic
476C2#1752511yescharacteristic
477C2C2×#152111yescharacteristic
478C2#1760511yescharacteristic
479C2#1768411yescharacteristic
480C2#1772611yescharacteristic
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. #7716C2semidirect
5. #7687C2semidirect
6. #7667C2semidirect
7. #7656C2semidirect
8. #7644C2semidirect
10. #7581C2semidirect

Maximal subgroups

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

subgroupindex
#363152
#226322
#207742
#77162
#76872
#76672
#76562
#76442
#76162
#75812
#75652