Finite Group Databasebeta

← #163433

Subgroups — #163433

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×#1839C211yesmaximal, characteristic, complemented
3#20752C211yesmaximal, characteristic, complemented
4#7723C212yesmaximal, complemented
5#7719C212yesmaximal, complemented
6#7678C214yesmaximal, complemented
7#7642C214yesmaximal, complemented
8C2×#529—21no—
9C2×#423—21no—
10#2861—21no—
11#2860—21no—
12#2860C412yes—
13#2859C2211yescharacteristic
14#2858C2211yescharacteristic
15#2857C411yescharacteristic
16#2856C411yescharacteristic
17#2854—21no—
18#2854C2212yes—
19#2850—21no—
20C4×C4⋊Q8C411yescharacteristic
21C4×#127C411yescharacteristic
22C4×#124—21no—
23C4×#121—21no—
24C4×#121C412yes—
25#2293—21no—
26#2293C2212yes—
27C2×C43C2211yescharacteristic
28#1839—21no—
29#1839—21no—
30#1839C2212yes—
31#1839C2212yes—
32#1831—21no—
33#1831—21no—
34#1831C2212yes—
35#1831C2212yes—
36#1728—21no—
37#1728C2212yes—
38C2×#336—21no—
39C2×#336—21no—
40C2×#336C2212yes—
41#1271C2218yes—
42#1270C2218yes—
43C22×M16—22no—
44C23×C8—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#127C2×C411yescharacteristic
47#529—44no—
48#529—44no—
49C2×#126—22no—
50C2×#125—21no—
51C2×#124—21no—
52C2×#124—21no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×#117—22no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71#423—44no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74#391—21no—
75#391—21no—
76#390—21no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—21no—
81#390—21no—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#390C2312yes—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2312yes—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#386—22no—
103#386—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#382—21no—
107#380—21no—
108#380C2×C412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C43C2×C411yescharacteristic
114C43C2×C411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336C2314yes—
130#336C2314yes—
131C22×Q8—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—44no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—44no—
150C22×C8—44no—
151C22×C8—44no—
152C4⋊Q8—42no—
153#127—42no—
154#126—48no—
155#125—42no—
156#125—42no—
157#124—44no—
158#124—42no—
159#124—42no—
160#123—44no—
161#123—44no—
162C22⋊Q8—44no—
163#121—44no—
164#121—44no—
165#121—44no—
166C4×Q8—44no—
167C4×D4—44no—
168C4×D4—44no—
169C4×D4—48no—
170C4×D4—44no—
171#117—44no—
172#117—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
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×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42D814yes—
216C2×C42SD1614yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—44no—
234C4⋊C8—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241C24—22no—
242C24C2×C22⋊C411yescharacteristic
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—24no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—42no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4#10214yes—
283C22×C4#10214yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×D812yes—
296C22×C4C2×SD1612yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300M16—88no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—88no—
308C4⋊C4—24no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—44no—
358C42#10214yes—
359C42#10214yes—
360C42C2×D812yes—
361C42C2×D812yes—
362C42C2×SD1612yes—
363C42C2×SD1612yes—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23#41012yes—
376C23C2×#10212yes—
377C23#42112yes—
378C23C22×C22⋊C411yescharacteristic
379Q8—84no—
380D4—84no—
381D4—84no—
382D4—84no—
383D4—88no—
384C2×C4—28no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—88no—
434C2×C4#41312yes—
435C2×C4C2×#10212yes—
436C2×C4C2×#10212yes—
437C2×C4C2×#10212yes—
438C2×C4C2×#10212yes—
439C2×C4C2×#10212yes—
440C2×C4#42112yes—
441C2×C4#56812yes—
442C2×C4C22×D811yescharacteristic
443C2×C4C22×SD1611yescharacteristic
444C2×C4#57512yes—
445C8—84no—
446C8—84no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22#151714yes—
461C22#287911yescharacteristic
462C22C2×#41011yescharacteristic
463C22C2×#41311yescharacteristic
464C22C22×#10211yescharacteristic
465C22#289611yescharacteristic
466C22C2×#42111yescharacteristic
467C22C2×#42111yescharacteristic
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—82no—
484C4—82no—
485C4—82no—
486C2—24no—
487C2—162no—
488C2#1739511yescharacteristic
489C2#1748511yescharacteristic
490C2#1749411yescharacteristic
491C2C2×#151711yescharacteristic
492C2#1759311yescharacteristic
493C2#1765911yescharacteristic
494C2#1770411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1839C2semidirect
3. #20752C2semidirect
4. #7723C2semidirect
5. #7719C2semidirect
6. #7678C2semidirect
7. #7642C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×#18392
#207522
#77232
#77192
#76782
#76422