Finite Group Databasebeta

← #6621196

Subgroups — #6621196

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#33290C212yesmaximal
4#33289C211yesmaximal, characteristic
5#29220C212yesmaximal
6#29219C211yesmaximal, characteristic
7#28719C212yesmaximal
8#28717C211yesmaximal, characteristic
9#27732C218yesmaximal
10#27730C214yesmaximal
11#22521C214yesmaximal
12C4×#487C211yesmaximal, characteristic
13C4×#487C212yesmaximal
14C4×C8×Q8C211yesmaximal, characteristic
15C2×#557C2212yes—
16C2×#487C2212yes—
17C2×#487C2211yescharacteristic
18C2×C8×Q8C2212yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2862C2212yes—
22#2859C2212yes—
23#2582—21no—
24#2571C2212yes—
25#2570C2214yes—
26#2568—21no—
27#2568C2212yes—
28#2565C2214yes—
29#2546—21no—
30#2546—21no—
31#2511—22no—
32#2506—22no—
33#2506—22no—
34#2446—22no—
35#2365—21no—
36#2365—21no—
37#2365—21no—
38#2355—21no—
39#2355C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C42×Q8C2211yescharacteristic
45#2101C2218yes—
46#2099C2214yes—
47#2074C2218yes—
48#2072C2214yes—
49#1991C2218yes—
50#1990C2212yes—
51#1990C2212yes—
52#1990C2214yes—
53#1990C2211yescharacteristic
54#1990C2212yes—
55#1990C2212yes—
56#1989C2218yes—
57#1988C2214yes—
58#1956C2211yescharacteristic
59#1933C2218yes—
60#1933C2214yes—
61#1835C2218yes—
62#1834C2214yes—
63#1834C2212yes—
64#1834C2211yescharacteristic
65#1805C2218yes—
66#1803C2214yes—
67C8×C4⋊C4C2212yes—
68C8×C4⋊C4C2214yes—
69C4×C4⋊C8C2211yescharacteristic
70C4×C4⋊C8C2212yes—
71C4×#103C2214yes—
72C4×#103C2218yes—
73C42×C8C2211yescharacteristic
74C42×C8C2212yes—
75#557—216no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#125—24no—
86C2×#125C2314yes—
87C2×C4×Q8—22no—
88C2×C4×Q8—21no—
89C2×C4×Q8—22no—
90C2×C4×Q8D411yescharacteristic
91C2×C4×Q8D411yescharacteristic
92C2×C4×Q8C2314yes—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2311yescharacteristic
95#487—28no—
96#487—24no—
97C8×Q8—28no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2314yes—
100C2×#103C2318yes—
101C2×#103C2314yes—
102C2×#103C2314yes—
103C2×C4×C8C2312yes—
104C2×C4×C8C2312yes—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2314yes—
107C2×C4×C8C2312yes—
108C2×C4×C8C2314yes—
109C2×C4×C8C2311yescharacteristic
110#400—24no—
111#400—24no—
112#400—24no—
113#398—24no—
114#398—24no—
115#398—24no—
116#391—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#391—21no—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391C2312yes—
126#391C2314yes—
127#389—22no—
128#389—22no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389C2312yes—
139#389C2314yes—
140#387—24no—
141#387—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#384—21no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#384C2312yes—
155#382—22no—
156#382C2314yes—
157#382C2314yes—
158#382C2314yes—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2311yescharacteristic
174C43C2312yes—
175C43C2311yescharacteristic
176#340C2318yes—
177#340C2318yes—
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#340C2314yes—
182#340C2314yes—
183#336C2312yes—
184#336C2314yes—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—21no—
188C22×Q8C2×D411yescharacteristic
189C22×Q8C2412yes—
190C22×C8C2414yes—
191C22×C8C2412yes—
192C22×C8C2412yes—
193C4⋊Q8—28no—
194C4⋊Q8—22no—
195C4⋊Q8—24no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8—24no—
199C4⋊Q8—48no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203C4⋊Q8—82no—
204C4⋊Q8#4114yes—
205#125—216no—
206#125—416no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—216no—
210C4×Q8—28no—
211C4×Q8—48no—
212C4×Q8—44no—
213C4×Q8—84no—
214C4×Q8#4118yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2412yes—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2412yes—
262C2×C42C2411yescharacteristic
263C2×C42C2412yes—
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—28no—
267C4⋊C8—24no—
268#103—216no—
269#103—28no—
270#103—28no—
271C4×C8—24no—
272C4×C8—28no—
273C4×C8—28no—
274C4×C8—24no—
275C4×C8—24no—
276C4×C8—22no—
277C4×C8—22no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95C2×D412yes—
293#95C2×D412yes—
294#95C2×D412yes—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—44no—
303C2×Q8—42no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8#12414yes—
310C2×Q8C2×#4114yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2511yescharacteristic
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C82−1+414yes—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—44no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4#12414yes—
372C4⋊C4#12418yes—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—22no—
377C42—22no—
378C42—24no—
379C42—22no—
380C42—24no—
381C42—48no—
382C42—44no—
383C42—44no—
384C42—84no—
385C42#13514yes—
386C42#13514yes—
387C42C2×#4118yes—
388C42C2×#4114yes—
389C42C2×#4112yes—
390C422−1+412yes—
391C23C23×D411yescharacteristic
392Q8—28no—
393Q8—28no—
394Q8—44no—
395Q8—84no—
396Q8—88no—
397Q8—88no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4C2×#12414yes—
433C2×C4C2×#12412yes—
434C2×C4C2×#12414yes—
435C2×C4C2×#12412yes—
436C2×C4#54812yes—
437C2×C4#54812yes—
438C2×C4C2×#13512yes—
439C2×C4C2×#13512yes—
440C2×C4C2×#13512yes—
441C2×C4C2×#13512yes—
442C2×C4C2×#13512yes—
443C2×C4C2×#13512yes—
444C2×C4#57712yes—
445C2×C4#57812yes—
446C2×C4C22×#4111yescharacteristic
447C2×C4C22×#4112yes—
448C2×C4C2×2−1+412yes—
449C2×C4C2×2−1+411yescharacteristic
450C8—24no—
451C8—44no—
452C8—42no—
453C22C22×#12411yescharacteristic
454C22C2×#54811yescharacteristic
455C22#349911yescharacteristic
456C22C22×#13511yescharacteristic
457C22C22×#13511yescharacteristic
458C22C2×#57711yescharacteristic
459C22C2×#57811yescharacteristic
460C4—22no—
461C4—22no—
462C4—24no—
463C4—24no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—44no—
468C4—82no—
469C4—82no—
470C4—84no—
471C4—84no—
472C4—82no—
473C4#313114yes—
474C4#336414yes—
475C4#336514yes—
476C4#348212yes—
477C4#352312yes—
478C2#3644611yescharacteristic
479C2C2×#313111yescharacteristic
480C2#6062311yescharacteristic
481C2C2×#336411yescharacteristic
482C2C2×#336511yescharacteristic
483C2#6247011yescharacteristic
484C2#6247311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#365982
#365602
#332902
#332892
#292202
#292192
#287192
#287172
#277322
#277302
#225212
C4×#4872
C4×C8×Q82