Finite Group Databasebeta

← #6630317

Subgroups — #6630317

Order 512.

2380 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3184C211yesmaximal, characteristic, complemented
2C2×#2875C211yesmaximal, characteristic, complemented
3#34609C212yesmaximal, complemented
4#33213C212yesmaximal, complemented
5#33179C212yesmaximal, complemented
6#33140C214yesmaximal, complemented
7#33045C214yesmaximal, complemented
8#32939C214yesmaximal, complemented
9#21538C214yesmaximal, complemented
10#21526C214yesmaximal, complemented
11#21408C212yesmaximal, complemented
12#20825C211yesmaximal, characteristic, complemented
13C22×#131C2212yes—
14C2×#532C2212yes—
15C2×C4×#41C2211yescharacteristic
16#3184—28no—
17C2×#498C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×#486C2212yes—
20C2×#469C2214yes—
21C2×#469C2214yes—
22C2×#466C2214yes—
23C2×#442C2214yes—
24#2875—28no—
25C2×#405C2212yes—
26C2×#405C2212yes—
27C2×#405C2212yes—
28C2×C4×M16C2212yes—
29C2×C4×M16C2211yescharacteristic
30C22×#97C2212yes—
31C22×#97C2211yescharacteristic
32#2635—24no—
33#2632—22no—
34#2631—22no—
35#2628C2214yes—
36#2624C2214yes—
37#2619C2214yes—
38#2615C2214yes—
39#2398—22no—
40#2397—22no—
41#2388—24no—
42C4×#121C2218yes—
43#2283C2212yes—
44#1993C2214yes—
45#1972C2214yes—
46#1970C2214yes—
47#1961C2214yes—
48#1943C2214yes—
49#1941C2214yes—
50#1936C2218yes—
51#1934C2218yes—
52#1895C2218yes—
53#1893C2218yes—
54#1893C2218yes—
55#1846C2212yes—
56#1835C2212yes—
57#1809C2214yes—
58#1809C2214yes—
59#1807C2214yes—
60#1776C2214yes—
61#1762C2214yes—
62#1762C2214yes—
63C22×#41—22no—
64C22×#41C2311yescharacteristic
65C22×D8C2314yes—
66C2×#131—28no—
67C2×#131—28no—
68C2×#131—28no—
69C22×M16C2312yes—
70C22×M16C2312yes—
71C22×M16C2312yes—
72C23×C8C2312yes—
73C23×C8C2312yes—
74C23×C8C2312yes—
75#532—28no—
76#532—28no—
77C2×C4⋊Q8C2312yes—
78C2×#127C2312yes—
79C2×#124—22no—
80C2×#121—22no—
81C2×#121C2318yes—
82C2×#121C2314yes—
83C4×#41—24no—
84C4×#41—24no—
85C2×C4×Q8D412yes—
86C2×C4×D4—28no—
87C2×C4×D4—22no—
88C2×C4×D4D412yes—
89C2×C4×D4D414yes—
90C2×C4×D4C2314yes—
91C2×#117C2312yes—
92C2×#117C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#498—24no—
96#493—24no—
97#486—28no—
98#469—216no—
99#469—216no—
100#466—216no—
101#442—216no—
102C2×#107C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×#102C2318yes—
106C2×#102C2318yes—
107C2×#102C2318yes—
108#405—28no—
109#405—28no—
110#405—28no—
111C4×M16—28no—
112C4×M16D418yescomplemented
113C2×#97—24no—
114C2×#97—28no—
115C2×#97C2311yescharacteristic
116C2×#97C2311yescharacteristic
117C2×#97C2312yes—
118C2×#97C2312yes—
119C2×#97C2312yes—
120C2×#97C2312yes—
121C2×C4×C8C2312yes—
122C2×C4×C8C2312yes—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#400—24no—
126#399—24no—
127#399—24no—
128#397—24no—
129#395—24no—
130#394—24no—
131#394—24no—
132#394—28no—
133#393—24no—
134#393—24no—
135#392—24no—
136#392—28no—
137#392—24no—
138#392—24no—
139#390C2318yes—
140#389C2318yes—
141#388—28no—
142#386—28no—
143#383—22no—
144#382C2314yes—
145#381—22no—
146#380C2318yes—
147#380C2314yes—
148C4×C4⋊C4C2314yes—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—28no—
151#376—22no—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—24no—
158C2×#41—24no—
159C2×#41—48no—
160C2×#41—88no—
161C22×Q8—22no—
162C22×Q8C2×D411yescharacteristic
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C2×D411yescharacteristic
168C22×D4C2×D412yes—
169C22×D4C2414yes—
170C23×C4—24no—
171C23×C4C2×D414yes—
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C2×D8—28no—
176C2×D8—28no—
177C2×D8—28no—
178#131—264no—
179C2×M16—24no—
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—24no—
183C2×M16—24no—
184C2×M16—24no—
185C2×M16—24no—
186C2×M16C2×D418yes—
187C2×M16#4118yescomplemented
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8—24no—
196C22×C8#4118yescomplemented
197C22×C8C2412yes—
198C22×C8C2412yes—
199C22×C8C2412yes—
200C22×C8C2412yes—
201C4⋊Q8—24no—
202C4⋊Q8—24no—
203#127—24no—
204#127—24no—
205#124—48no—
206#121—28no—
207#121—28no—
208#121—216no—
209#121—216no—
210#121—48no—
211C4×Q8—28no—
212C4×D4—216no—
213C4×D4—28no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—48no—
217C4×D4—816no—
218#117—22no—
219#117—22no—
220#117—24no—
221#117—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C2418yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4C2×D414yes—
236C2×C22⋊C4C2×D412yes—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42C2×D414yes—
241C2×C42C2×D412yes—
242C2×C42C2×D414yes—
243C2×C42#4114yes—
244C2×C42C2412yes—
245C2×C42C2411yescharacteristic
246C2×C42C2411yescharacteristic
247#107—28no—
248#106—28no—
249#106—28no—
250#102—216no—
251#102—216no—
252#102—216no—
253#97—24no—
254#97—24no—
255#97—22no—
256#97—22no—
257#97—24no—
258#97C2×D418yes—
259C4×C8—24no—
260C4×C8—24no—
261C4×C8C2×D414yes—
262C4×C8C2×D414yes—
263#95—24no—
264#95—24no—
265#95—28no—
266#95—24no—
267#95—28no—
268#95—24no—
269C24—24no—
270C24C22×D411yescharacteristic
271C24C22×D412yes—
272#41—216no—
273#41—28no—
274#41—28no—
275#41—832no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—44no—
279C2×Q8—84no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—24no—
289C2×D4—48no—
290C2×D4—416no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—84no—
294C2×D4—84no—
295C2×D4—88no—
296C2×D4—88no—
297C2×D4—88no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—44no—
308C22×C4—48no—
309C22×C4—416no—
310C22×C4—48no—
311C22×C4#12118yes—
312C22×C4C22×D414yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D412yes—
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4114yes—
320C22×C4C2×#4114yes—
321C22×C4C2511yescharacteristic
322D8—416no—
323M16—216no—
324M16—216no—
325M16—216no—
326C2×C8—216no—
327C2×C8—216no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—216no—
335C2×C8C22×D414yes—
336C2×C8C22×D414yes—
337C2×C8C2×#4114yes—
338C2×C8C2×#4114yes—
339C2×C8C2×#4114yes—
340C2×C8C2×#4114yes—
341C4⋊C4—28no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—44no—
348C4⋊C4—88no—
349C22⋊C4—24no—
350C22⋊C4—28no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—416no—
354C22⋊C4—88no—
355C22⋊C4—88no—
356C22⋊C4—84no—
357C22⋊C4—88no—
358C42—28no—
359C42—22no—
360C42—22no—
361C42—48no—
362C42#12118yes—
363C42C22×D412yes—
364C42C22×D412yes—
365C42C2×#4112yes—
366C42C2×#4112yes—
367C23—28no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—44no—
373C23—48no—
374C23—44no—
375C23—44no—
376C23—48no—
377C23C2×#12112yes—
378C23#53214yes—
379C23C23×D411yescharacteristic
380Q8—28no—
381Q8—88no—
382D4—28no—
383D4—216no—
384D4—48no—
385D4—44no—
386D4—44no—
387D4—88no—
388D4—88no—
389D4—832no—
390D4—816no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—24no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—88no—
412C2×C4—816no—
413C2×C4C2×#12114yes—
414C2×C4C2×#12112yes—
415C2×C4C2×#12114yes—
416C2×C4#53214yes—
417C2×C4C23×D411yescharacteristic
418C2×C4C22×#4112yes—
419C2×C4C22×#4112yes—
420C2×C4C22×#4111yescharacteristic
421C8—24no—
422C8—24no—
423C8—24no—
424C8#52518yes—
425C22—22no—
426C22—24no—
427C22—28no—
428C22—24no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22—84no—
433C22—88no—
434C22—88no—
435C22—84no—
436C22—84no—
437C22C22×#12111yescharacteristic
438C22C2×#53212yes—
439C22#358514yes—
440C4—22no—
441C4—24no—
442C4—24no—
443C4—24no—
444C4—44no—
445C4—88no—
446C4—84no—
447C4C2×#52512yes—
448C4#344411yescharacteristic
449C4#344411yescharacteristic
450C2—42no—
451C2—44no—
452C2—164no—
453C2#3662411yescharacteristic
454C2#6045412yes—
455C2#6069814yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3184C2semidirect
2. C2×#2875C2semidirect
3. #34609C2semidirect
4. #33213C2semidirect
5. #33179C2semidirect
6. #33140C2semidirect
7. #33045C2semidirect
8. #32939C2semidirect
9. #21538C2semidirect
10. #21526C2semidirect
11. #21408C2semidirect
12. #20825C2semidirect
112. C4×M16D4semidirect
187. C2×M16#41semidirect
196. C22×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31842
C2×#28752
#346092
#332132
#331792
#331402
#330452
#329392
#215382
#215262
#214082
#208252