Finite Group Databasebeta

← #164073

Subgroups — #164073

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35267C211yesmaximal, characteristic
2#22643C211yesmaximal, characteristic, complemented
3#22639C211yesmaximal, characteristic, complemented
4#7726C212yesmaximal, complemented
5#7713C212yesmaximal, complemented
6#7686C212yesmaximal, complemented
7#7673C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic
9#7576C211yesmaximal, characteristic, complemented
10#7573C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic
12#2863C2212yes—
13#2860C2212yes—
14#2645C412yes—
15#2644C412yes—
16#2510—22no—
17#2497—21no—
18#2496—21no—
19#2495—21no—
20#2494—21no—
21#2481—22no—
22#2317C411yescharacteristic
23#2316C411yescharacteristic
24#2311C411yescharacteristic
25#2311C411yescharacteristic
26#2297C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1923—22no—
33#1839C2212yes—
34#1839C2212yes—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39#1767C2212yes—
40C4×C4⋊C8C2212yes—
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C2×#126—22no—
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×#97—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74#401—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#397—22no—
79#397—22no—
80#397—22no—
81#396—22no—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2×C411yescharacteristic
85#391C2311yescharacteristic
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2×C411yescharacteristic
89#390C2311yescharacteristic
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2312yes—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#386—21no—
107#386—21no—
108#386—22no—
109#386—22no—
110#385—22no—
111#384—21no—
112#383—22no—
113#382—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—22no—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D412yes—
127#376C2×C412yes—
128C2×#95—21no—
129C43C2312yes—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336C2312yes—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C2×#41—24no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8C22×C411yescharacteristic
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158#126—82no—
159#126—82no—
160#117—24no—
161#117—42no—
162#117—42no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4—81no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C42—21no—
195C2×C42—42no—
196C2×C42C22×C412yes—
197C2×C42C22×C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2411yescharacteristic
206C4⋊C8—42no—
207C4⋊C8—42no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8—81no—
213C22⋊C8—81no—
214#97—82no—
215C4×C8—42no—
216C4×C8—42no—
217C4×C8—82no—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236C24—21no—
237C24C2×C22⋊C411yescharacteristic
238#41—88no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4#13114yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C22⋊C4—24no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C42—24no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—42no—
339C42—42no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—42no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23#40812yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—82no—
360D4—82no—
361D4—84no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—28no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—162no—
409C2×C4#40812yes—
410C2×C4#41012yes—
411C2×C4#41312yes—
412C2×C4C2×#13112yes—
413C2×C4#57611yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57711yescharacteristic
416C2×C4#57811yescharacteristic
417C8—42no—
418C8—42no—
419C8—82no—
420C8—81no—
421C8—82no—
422C8—81no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22C2×#40811yescharacteristic
438C22C2×#41011yescharacteristic
439C22C2×#41311yescharacteristic
440C22#289711yescharacteristic
441C22#289811yescharacteristic
442C22#290111yescharacteristic
443C22#290111yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—22no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—41no—
454C4—41no—
455C4—81no—
456C4—82no—
457C4—81no—
458C4—82no—
459C4—81no—
460C4#147314yes—
461C2—42no—
462C2—161no—
463C2C2×#147311yescharacteristic
464C2#1746911yescharacteristic
465C2#1748311yescharacteristic
466C2#1760711yescharacteristic
467C2#1761211yescharacteristic
468C2#1768711yescharacteristic
469C2#1778111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×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. #22643C2semidirect
3. #22639C2semidirect
4. #7726C2semidirect
5. #7713C2semidirect
6. #7686C2semidirect
7. #7673C2semidirect
9. #7576C2semidirect
10. #7573C2semidirect

Maximal subgroups

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

subgroupindex
#352672
#226432
#226392
#77262
#77132
#76862
#76732
#75972
#75762
#75732
#75632