Finite Group Databasebeta

← #6806416

Subgroups — #6806416

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#33225C211yesmaximal, characteristic
4#32084C218yesmaximal
5#30220C212yesmaximal
6#29193C212yesmaximal
7#28748C211yesmaximal, characteristic
8#28748C212yesmaximal
9#28511C211yesmaximal, characteristic
10#25651C214yesmaximal
11#25278C214yesmaximal
12#22221C212yesmaximal
13C4×#487C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×#557C2212yes—
16C2×#487C2211yescharacteristic
17C2×#487C2212yes—
18C2×#487C2214yes—
19C2×#477C2214yes—
20#2868C2211yescharacteristic
21#2868C2212yes—
22#2868C2211yescharacteristic
23#2859C2212yes—
24#2582—21no—
25#2567C2212yes—
26#2552—21no—
27#2552—21no—
28#2552—22no—
29#2552—22no—
30#2552—21no—
31#2550—22no—
32#2546C2214yes—
33#2546C2212yes—
34#2507C2214yes—
35#2451—22no—
36#2369C2212yes—
37#2355—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C42×Q8C2211yescharacteristic
46#2095C2214yes—
47#2068C2218yes—
48#2043C2218yes—
49#2043C2218yes—
50#2043C2218yes—
51#2043C2214yes—
52#1990C2211yescharacteristic
53#1984C2212yes—
54#1981C2214yes—
55#1981C2218yes—
56#1956C2212yes—
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1956C2212yes—
60#1956C2212yes—
61#1956C2214yes—
62#1939C2212yes—
63#1933C2214yes—
64#1928C2218yes—
65#1916C2214yes—
66#1863C2218yes—
67#1863C2218yes—
68#1845C2212yes—
69#1834C2211yescharacteristic
70C4×C4⋊C8C2212yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#103C2214yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75#557—48no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8C2312yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2314yes—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2314yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×#125—22no—
87C2×#125—24no—
88C2×#125—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—21no—
91C2×C4×Q8—22no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2311yescharacteristic
97#487—24no—
98#487—28no—
99#487—48no—
100#477—48no—
101C2×#107C2318yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2314yes—
104C2×#103C2318yes—
105C2×#103C2314yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×#103C2314yes—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2312yes—
116#398—24no—
117#398—24no—
118#398—24no—
119#398—24no—
120#398—24no—
121#395—24no—
122#391—22no—
123#391—22no—
124#391—21no—
125#391—24no—
126#391C2314yes—
127#391C2312yes—
128#389—22no—
129#389—22no—
130#389—24no—
131#389—22no—
132#389—24no—
133#389—21no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—21no—
138#389—21no—
139#389—22no—
140#389—22no—
141#389—24no—
142#389C2314yes—
143#389C2312yes—
144#389C2314yes—
145#389C2314yes—
146#387—24no—
147#384—22no—
148#384—21no—
149#384—22no—
150#384—22no—
151#384—21no—
152#384—21no—
153#384—21no—
154#384—22no—
155#384—22no—
156#384—21no—
157#384—21no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#384C2312yes—
161#384C2312yes—
162#382—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C4⋊C4C2312yes—
178C43C2312yes—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#336C2312yes—
183#336C2314yes—
184#336C2314yes—
185#336C2314yes—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×Q8C2412yes—
190C22×Q8C2414yes—
191C22×C8C2412yes—
192C22×C8C2414yes—
193C22×C8C2412yes—
194C4⋊Q8—24no—
195C4⋊Q8—22no—
196C4⋊Q8—28no—
197C4⋊Q8—28no—
198C4⋊Q8—22no—
199C4⋊Q8—24no—
200C4⋊Q8—22no—
201C4⋊Q8—48no—
202C4⋊Q8—44no—
203C4⋊Q8—48no—
204C4⋊Q8—82no—
205C4⋊Q8—82no—
206C4⋊Q8#4114yes—
207#125—416no—
208#125—48no—
209#125—84no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—28no—
213C4×Q8—48no—
214C4×Q8—48no—
215C4×Q8—44no—
216C4×Q8—84no—
217C4×Q8Q1618yes—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2412yes—
269C2×C42C2412yes—
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273#107—48no—
274C4⋊C8—24no—
275C4⋊C8—44no—
276#103—216no—
277#103—28no—
278#103—44no—
279#103—48no—
280#103—48no—
281#103—48no—
282#103—48no—
283C4×C8—22no—
284C4×C8—24no—
285C4×C8—24no—
286C4×C8—22no—
287C4×C8—42no—
288#95—24no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D412yes—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—28no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—28no—
312C2×Q8—44no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8C2×Q1614yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D412yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2511yescharacteristic
335C2×C8—24no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—82no—
375C4⋊C4—84no—
376C4⋊C4—82no—
377C4⋊C4—84no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4C2×Q1618yes—
381C4⋊C4C2×Q1614yes—
382C4⋊C4C2×#4114yes—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—22no—
390C42—24no—
391C42—48no—
392C42—44no—
393C42—44no—
394C42—84no—
395C42C2×Q1618yes—
396C42C2×Q1614yes—
397C42#13714yes—
398C42C2×#4112yes—
399C42H(5,𝔽2)14yes—
400C422−1+412yes—
401C23C23×D411yescharacteristic
402Q8—28no—
403Q8—48no—
404Q8—44no—
405Q8—48no—
406Q8—44no—
407Q8—48no—
408Q8—164no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4#53512yes—
444C2×C4#53512yes—
445C2×C4#54812yes—
446C2×C4#54812yes—
447C2×C4C22×Q1614yes—
448C2×C4C22×Q1612yes—
449C2×C4C22×Q1614yes—
450C2×C4C22×Q1612yes—
451C2×C4C22×Q1612yes—
452C2×C4C2×#13712yes—
453C2×C4C2×#13712yes—
454C2×C4C2×#13712yes—
455C2×C4#57712yes—
456C2×C4#57712yes—
457C2×C4C22×#4111yescharacteristic
458C2×C4C2×H(5,𝔽2)12yes—
459C2×C4C2×2−1+411yescharacteristic
460C2×C4#58512yes—
461C8—42no—
462C8—44no—
463C8—42no—
464C22C2×#53511yescharacteristic
465C22C2×#54811yescharacteristic
466C22#350011yescharacteristic
467C22C23×Q1611yescharacteristic
468C22C22×#13711yescharacteristic
469C22C2×#57711yescharacteristic
470C22C2×#57711yescharacteristic
471C4—24no—
472C4—22no—
473C4—22no—
474C4—24no—
475C4—22no—
476C4—42no—
477C4—44no—
478C4—42no—
479C4—84no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—84no—
484C4#320914yes—
485C4#336314yes—
486C4#337214yes—
487C4#351712yes—
488C4#355412yes—
489C2#3917411yescharacteristic
490C2C2×#320911yescharacteristic
491C2#6082011yescharacteristic
492C2C2×#336311yescharacteristic
493C2C2×#337211yescharacteristic
494C2#6253011yescharacteristic
495C2#6254011yescharacteristic
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
#365932
#365632
#332252
#320842
#302202
#291932
#287482
#285112
#256512
#252782
#222212
C4×#4872
#213392