Finite Group Databasebeta

← #165054

Subgroups — #165054

Order 512.

833 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36271C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3#22828C211yesmaximal, characteristic, complemented
4#7726C212yesmaximal, complemented
5#7692C212yesmaximal, complemented
6#7659C211yesmaximal, characteristic
7#7659C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic, complemented
9#7608C212yesmaximal, complemented
10#7580C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12C2×#556—21no—
13#2860C2212yes—
14#2858C2211yescharacteristic
15#2601—22no—
16#2592—21no—
17#2560—22no—
18#2548C411yescharacteristic
19#2546C411yescharacteristic
20#2545—21no—
21#2512C412yes—
22#2452C412yes—
23#2367—21no—
24#2366C411yescharacteristic
25#2365C411yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27#2293C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1917—22no—
32#1916—22no—
33#1839C2212yes—
34#1839C2212yes—
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
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—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×#97—22no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#398—22no—
82#397—22no—
83#396—22no—
84#396—22no—
85#394—22no—
86#393—22no—
87#391—21no—
88#391—21no—
89#391—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390C2×C412yes—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2×C412yes—
98#390C2312yes—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2311yescharacteristic
104#389C2311yescharacteristic
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#386—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—22no—
126C43C2311yescharacteristic
127C43C2312yes—
128#336—21no—
129#336—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C412yes—
148C23×C4—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—41no—
157C4⋊Q8—42no—
158C4⋊Q8—44no—
159#127—42no—
160#126—44no—
161#126—44no—
162#126—48no—
163#124—44no—
164#124—42no—
165#124—42no—
166#124—42no—
167#124—42no—
168C22⋊Q8—48no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—44no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×D4—24no—
178C4×D4—42no—
179C4×D4—44no—
180#117—44no—
181#117—44no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—22no—
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×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C22×C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2411yescharacteristic
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233#97—44no—
234C4×C8—44no—
235C4×C8—42no—
236C4×C8—42no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C24—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—81no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—81no—
267C2×D4—81no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—82no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—42no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—81no—
328C4⋊C4#13112yes—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—22no—
343C42—21no—
344C42—22no—
345C42—21no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—42no—
352C42—42no—
353C42—41no—
354C42—41no—
355C42—42no—
356C42—41no—
357C42—44no—
358C42—42no—
359C42—42no—
360C42#13512yes—
361C42#13512yes—
362C42AGL(1,ℤ/8)12yes—
363C42AGL(1,ℤ/8)12yes—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23C22×C22⋊C411yescharacteristic
372Q8—44no—
373Q8—84no—
374Q8—84no—
375D4—82no—
376D4—82no—
377D4—84no—
378D4—162no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—88no—
420C2×C4#41312yes—
421C2×C4#41612yes—
422C2×C4#41612yes—
423C2×C4#41812yes—
424C2×C4#41812yes—
425C2×C4C2×#13111yescharacteristic
426C2×C4#56811yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
430C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
431C8—42no—
432C8—42no—
433C8—81no—
434C8—82no—
435C8—81no—
436C8—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22#288011yescharacteristic
445C22#288411yescharacteristic
446C22C2×#41311yescharacteristic
447C22C2×#41611yescharacteristic
448C22C2×#41811yescharacteristic
449C22#290111yescharacteristic
450C22#290111yescharacteristic
451C4—21no—
452C4—21no—
453C4—22no—
454C4—22no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—42no—
460C4—41no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—82no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4#155314yes—
470C2—162no—
471C2#1742811yescharacteristic
472C2#1751011yescharacteristic
473C2#1751111yescharacteristic
474C2#1761611yescharacteristic
475C2#1763111yescharacteristic
476C2#1769111yescharacteristic
477C2C2×#155311yescharacteristic
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. #22832C2semidirect
3. #22828C2semidirect
4. #7726C2semidirect
5. #7692C2semidirect
8. #7652C2semidirect
9. #7608C2semidirect
10. #7580C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#362712
#228322
#228282
#77262
#76922
#76592
#76522
#76082
#75802
#75702