Finite Group Databasebeta

← #215597

Subgroups — #215597

Order 512.

998 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
1C2×#2295C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3#20777C211yesmaximal, characteristic
4#9880C214yesmaximal
5#9879C212yesmaximal
6#9878C212yesmaximal
7#8699C214yesmaximal
8C22×C4×Q8C411yescharacteristic
9#2295—21no—
10#2295—21no—
11#2295—21no—
12#2295—21no—
13#2295—21no—
14#2295—21no—
15#2295—21no—
16#2295—21no—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389C411yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387C412yes—
22C2×#387C2212yes—
23C2×C4×C4⋊C4C412yes—
24C2×C4×C4⋊C4C411yescharacteristic
25C2×#376C2212yes—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1831—21no—
30#1831—21no—
31#1817C2212yes—
32#1817C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35C4×C4⋊C8—21no—
36C4×C4⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744—21no—
40#1744—21no—
41#1280C2218yes—
42#1280C2218yes—
43C23×Q8C2×C411yescharacteristic
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—24no—
49C2×C4×Q8C818yescomplemented
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2×C412yes—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8C2314yes—
68C2×C22⋊C8C2314yes—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103#376—22no—
104#376—22no—
105#376—22no—
106#376—22no—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2312yes—
113C2×#95C2312yes—
114C43D411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8C2×C814yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—24no—
148C22×C8—24no—
149C4×Q8—28no—
150C4×Q8—28no—
151C4×Q8—28no—
152C4×Q8—28no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C2×C814yes—
172C2×C4⋊C4C2×C818yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—44no—
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C2×C818yes—
202C2×C42C2×C814yes—
203C2×C42M1614yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C4⋊C8—44no—
217C4⋊C8—44no—
218C22⋊C8—28no—
219C22⋊C8—28no—
220C4×C8—44no—
221C4×C8—44no—
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—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236C24C23×C411yescharacteristic
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—48no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—48no—
243C2×Q8—42no—
244C2×Q8—42no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
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×C4C22×C812yes—
291C22×C4C22×C812yes—
292C22×C4C22×C814yes—
293C22×C4C22×C814yes—
294C22×C4C22×C812yes—
295C22×C4C2×M1612yes—
296C22×C4C2×M1612yes—
297C22×C4C2×M1612yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C42−1+414yes—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—28no—
324C42—28no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—44no—
346C42—44no—
347C42C22⋊C814yes—
348C42C22⋊C814yes—
349C42C22⋊C814yes—
350C42C22⋊C814yes—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23C22×C22⋊C411yescharacteristic
356C23#51912yes—
357C23C23×C811yescharacteristic
358C23C22×M1611yescharacteristic
359C23#56812yes—
360Q8—48no—
361Q8—48no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
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—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4C2×C22⋊C812yes—
421C2×C4C2×C22⋊C812yes—
422C2×C4C2×C22⋊C812yes—
423C2×C4C2×C22⋊C812yes—
424C2×C4C2×C22⋊C812yes—
425C2×C4C2×C22⋊C812yes—
426C2×C4#41012yes—
427C2×C4#41012yes—
428C2×C4#41312yes—
429C2×C4#41312yes—
430C2×C4#53612yes—
431C2×C4#53612yes—
432C8—84no—
433C8—84no—
434C22—22no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22#231011yescharacteristic
443C22C22×C22⋊C811yescharacteristic
444C22#287911yescharacteristic
445C22C2×#41011yescharacteristic
446C22C2×#41011yescharacteristic
447C22C2×#41311yescharacteristic
448C22C2×#41311yescharacteristic
449C22#298212yes—
450C22#298312yes—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—44no—
463C4—44no—
464C4—41no—
465C4—41no—
466C4—82no—
467C4—82no—
468C4#145814yes—
469C2—22no—
470C2—22no—
471C2C2×#145811yescharacteristic
472C2#1734911yescharacteristic
473C2#1739511yescharacteristic
474C2#1739911yescharacteristic
475C2#2181811yescharacteristic
476C2#2182611yescharacteristic
477C2#2183111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
49. C2×C4×Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22952
#207772
#98802
#98792
#98782
#86992