Finite Group Databasebeta

← #7234169

Subgroups — #7234169

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46346C212yesmaximal
2#33267C212yesmaximal
3#32863C212yesmaximal
4#32630C212yesmaximal
5#32549C212yesmaximal
6#31975C214yesmaximal
7#31813C214yesmaximal
8#28737C211yesmaximal, characteristic
9#28675C211yesmaximal, characteristic
10#28268C212yesmaximal
11#28012C212yesmaximal
12#27993C212yesmaximal
13#26485C212yesmaximal
14#26255C212yesmaximal
15C8×C4⋊Q8C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2212yes—
21#2863C2214yes—
22#2833—22no—
23#2794—22no—
24#2766—24no—
25#2758—24no—
26#2734—22no—
27#2581C2214yes—
28#2568C2212yes—
29#2546C2212yes—
30#2514C2214yes—
31#2509C2214yes—
32#2451C2214yes—
33#2446C2214yes—
34#2381—22no—
35C4×C4⋊Q8C2211yescharacteristic
36#2101C2214yes—
37#2088C2214yes—
38#2086C2212yes—
39#2086C2212yes—
40#2074C2214yes—
41#2068C2214yes—
42#2066C2214yes—
43#2066C2214yes—
44#2066C2214yes—
45#2056C2212yes—
46#2056C2212yes—
47#2056C2214yes—
48#2043C2214yes—
49#2024C2214yes—
50#1988C2212yes—
51#1984C2211yescharacteristic
52#1984C2211yescharacteristic
53#1981C2212yes—
54#1969C2212yes—
55#1969C2212yes—
56#1969C2212yes—
57#1969C2212yes—
58#1969C2212yes—
59#1942C2212yes—
60#1942C2212yes—
61#1942C2212yes—
62#1942C2212yes—
63#1942C2212yes—
64#1939C2212yes—
65#1939C2212yes—
66#1933C2214yes—
67#1933C2214yes—
68#1933C2214yes—
69#1927C2214yes—
70#1919C2214yes—
71#1878C2214yes—
72#1863C2214yes—
73#1846C2212yes—
74#1842C2211yescharacteristic
75#1805C2214yes—
76C4×#107C2211yescharacteristic
77C4×#107C2211yescharacteristic
78C8×C4⋊C4C2212yes—
79C8×C4⋊C4C2212yes—
80C42×C8C2211yescharacteristic
81C22×Q16C2312yes—
82C22×Q16C2312yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8D411yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4×Q8C2314yes—
88C2×C4×Q8C2311yescharacteristic
89C2×C4×Q8C2311yescharacteristic
90#500—42no—
91#494—42no—
92C8×Q8—42no—
93C8×Q8—42no—
94C4×Q16—44no—
95C2×#107C2312yes—
96C2×#107C2312yes—
97C2×#107C2312yes—
98C2×#107C2312yes—
99C2×#106C2314yes—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2312yes—
118C2×C4×C8C2312yes—
119C2×C4×C8C2311yescharacteristic
120#401—24no—
121#400—22no—
122#400—22no—
123#400—24no—
124#400—24no—
125#400—24no—
126#400—24no—
127#400—24no—
128#398—22no—
129#398—24no—
130#398—22no—
131#398—24no—
132#395—24no—
133#395—24no—
134#391—22no—
135#391—22no—
136#391—24no—
137#391D411yescharacteristic
138#391D411yescharacteristic
139#391C2314yes—
140#391C2314yes—
141#391C2314yes—
142#391C2314yes—
143#391C2314yes—
144#389—24no—
145#389D412yes—
146#389D412yes—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#387—24no—
153#387—24no—
154#387—24no—
155#387—24no—
156#387—24no—
157#387—24no—
158#387—24no—
159#387—24no—
160#387—24no—
161#387—24no—
162#384D411yescharacteristic
163#383—24no—
164#383—24no—
165#382—22no—
166#382C2314yes—
167C4×C4⋊C4C2312yes—
168C4×C4⋊C4C2312yes—
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2312yes—
171C43C2311yescharacteristic
172#340C2312yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2312yes—
177#340C2314yes—
178#340C2314yes—
179#336C2312yes—
180#336C2312yes—
181#336C2312yes—
182#336C2312yes—
183C22×Q8—22no—
184C22×Q8C2×D411yescharacteristic
185C22×Q8C2×D411yescharacteristic
186C22×Q8C2414yes—
187C2×Q16—24no—
188C2×Q16—24no—
189C2×Q16—24no—
190C2×Q16—44no—
191C2×Q16—42no—
192C22×C8C2411yescharacteristic
193C22×C8C2412yes—
194C22×C8C2412yes—
195C22×C8C2411yescharacteristic
196C22×C8C2411yescharacteristic
197C22×C8C2411yescharacteristic
198C4⋊Q8—24no—
199C4⋊Q8—42no—
200C4⋊Q8—42no—
201C4⋊Q8—42no—
202C4⋊Q8—42no—
203C4⋊Q8—48no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—44no—
209C4×Q8—44no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2414yes—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2412yes—
238C2×C42C2411yescharacteristic
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241C2×C42C2412yes—
242#107—24no—
243#107—24no—
244#107—24no—
245#107—42no—
246#106—44no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—42no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254#103—44no—
255#103—44no—
256#103—44no—
257#103—44no—
258#103—44no—
259#103—44no—
260C4×C8—24no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—24no—
265C4×C8—42no—
266C4×C8—41no—
267#95—22no—
268#95—22no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95C2×D412yes—
287#95C2×D412yes—
288#95C2×D412yes—
289#95C2×D412yes—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—44no—
302C2×Q8—82no—
303C2×Q8—82no—
304C2×Q8#13514yes—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D412yes—
311C22×C4C22×D412yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D412yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4C2511yescharacteristic
320Q16—48no—
321Q16—84no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8H(5,𝔽2)12yes—
335C2×C82−1+414yes—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C42—22no—
361C42—24no—
362C42—21no—
363C42—22no—
364C42—21no—
365C42—21no—
366C42—22no—
367C42—22no—
368C42—21no—
369C42—22no—
370C42—21no—
371C42—21no—
372C42—41no—
373C42—44no—
374C42—82no—
375C42—84no—
376C23C23×D411yescharacteristic
377Q8—44no—
378Q8—44no—
379Q8—44no—
380Q8—44no—
381Q8—44no—
382Q8—44no—
383Q8—164no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4C2×#13512yes—
420C2×C4C2×#13512yes—
421C2×C4C2×#13512yes—
422C2×C4#57814yes—
423C2×C4C2×H(5,𝔽2)11yescharacteristic
424C2×C4C2×2−1+412yes—
425C2×C4#58512yes—
426C2×C4#58511yescharacteristic
427C8—42no—
428C8—41no—
429C8—41no—
430C8—41no—
431C8—42no—
432C8#56314yes—
433C22#349611yescharacteristic
434C22#349712yes—
435C22C22×#13511yescharacteristic
436C22C2×#57812yes—
437C22#358411yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—24no—
442C4—42no—
443C4—42no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—42no—
448C4—84no—
449C4—84no—
450C4—84no—
451C4—84no—
452C4C2×#56311yescharacteristic
453C4#355111yescharacteristic
454C4#355712yes—
455C2#4632911yescharacteristic
456C2#6224711yescharacteristic
457C2#6232911yescharacteristic
458C2#6236412yes—
459C2#6238012yes—
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊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
#463462
#332672
#328632
#326302
#325492
#319752
#318132
#287372
#286752
#282682
#280122
#279932
#264852
#262552
C8×C4⋊Q82