Finite Group Databasebeta

← #6484605

Subgroups — #6484605

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60864C212yesmaximal, complemented
2C2×#3184C211yesmaximal, characteristic
3#60192C211yesmaximal, characteristic, complemented
4#33125C214yesmaximal, complemented
5#29791C214yesmaximal, complemented
6#28763C212yesmaximal, complemented
7#28700C212yesmaximal, complemented
8#28670C214yesmaximal, complemented
9#28299C214yesmaximal, complemented
10#28220C214yesmaximal, complemented
11#21878C212yesmaximal, complemented
12#20827C211yesmaximal, characteristic, complemented
13C2×#533C2211yescharacteristic
14C2×#532C2212yes—
15#3235—24no—
16#3234—24no—
17#3228—24no—
18#3227—22no—
19#3226—22no—
20#3184—24no—
21#3184—24no—
22#3165C2214yes—
23C2×#498C2211yescharacteristic
24C2×#493C2211yescharacteristic
25#3138C2214yes—
26C2×#486C2212yes—
27#3094C2218yes—
28#3077C2218yes—
29C2×#469C2212yes—
30C2×#469C2212yes—
31Q8×M16—24no—
32#2962—24no—
33#2961C2212yes—
34#2961C2212yes—
35#2960C2212yes—
36#2960C2212yes—
37C2×#442C2212yes—
38C2×#442C2212yes—
39#2920C2212yes—
40#2905C2212yes—
41C2×#423C2211yescharacteristic
42#2873C2212yes—
43C2×C4×M16C2211yescharacteristic
44#2090C2218yes—
45#2063C22116yes—
46#2045C2218yes—
47#1997C2214yes—
48#1978C2218yes—
49#1965C2214yes—
50#1951C2218yes—
51#1931C22116yes—
52#1900C2218yes—
53#1849C2212yes—
54#1838C2212yes—
55#1811C2212yes—
56#1811C2212yes—
57#1800C2218yes—
58C4×#108C2212yes—
59C4×#108C2212yes—
60C22×#41—22no—
61C22×D8C2312yes—
62C22×D8C2312yes—
63C22×M16C2312yes—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66#533—22no—
67#533—22no—
68#533D412yes—
69#533D412yes—
70#533C2312yes—
71#533C2312yes—
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80C2×C4⋊Q8D412yes—
81C2×C4⋊Q8C2312yes—
82C2×#127C2312yes—
83C2×#125D412yes—
84C2×#124—22no—
85C2×C22⋊Q8—22no—
86C2×#121—22no—
87C2×#121C2314yes—
88C2×C4×Q8—22no—
89C2×C4×D4—22no—
90C2×C4×D4C2314yes—
91C2×#117C2312yes—
92C22×C42C2311yescharacteristic
93#501—28no—
94#500—24no—
95#498—24no—
96#498—24no—
97#493—24no—
98#490—28no—
99#486—24no—
100#486—24no—
101#486—24no—
102#486—24no—
103#482—216no—
104#481—28no—
105#480—28no—
106#478—216no—
107#476—28no—
108#474—28no—
109#469—24no—
110#469—24no—
111#469—24no—
112#469—24no—
113#446—28no—
114#446—28no—
115#446—24no—
116#446—24no—
117C8×Q8—24no—
118C8×Q8—24no—
119#442—28no—
120#442—28no—
121#432—22no—
122#432—22no—
123#431—22no—
124#431—22no—
125C2×#108C2314yes—
126C2×#108C2314yes—
127#428C23116yes—
128C2×#106C2312yes—
129C2×#106C2312yes—
130#424—24no—
131#424C2314yes—
132#424C2314yes—
133#423—24no—
134#423—22no—
135#423C2314yes—
136#423C2312yes—
137#423C2312yes—
138C2×C4⋊C8—22no—
139#418C23116yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#100C23116yes—
143#407C2314yes—
144#405—24no—
145#405C2314yes—
146#405C2314yes—
147C4×M16—24no—
148C4×M16D414yes—
149C4×M16D414yes—
150C4×M16C2314yes—
151C2×#97C2311yescharacteristic
152C2×#97C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#343C23116yes—
156#341C2314yes—
157#341C2314yes—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41—42no—
161C2×#41—42no—
162C2×#41—88no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4C2414yes—
169C23×C4—24no—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C2×D8—24no—
173C2×D8—24no—
174C2×D8—28no—
175C2×D8—44no—
176C2×M16—22no—
177C2×M16—24no—
178C2×M16—42no—
179C2×M16—44no—
180C2×M16C2×D412yes—
181C2×M16C2×D412yes—
182C2×M16C2×D412yes—
183C2×M16C2×D412yes—
184C2×M16C2414yes—
185C2×M16C2418yes—
186C22×C8—22no—
187C22×C8C2411yescharacteristic
188C22×C8C2411yescharacteristic
189C22×C8C2411yescharacteristic
190C22×C8C2411yescharacteristic
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8C2×D414yes—
198C4⋊Q8C2×D414yes—
199#127—22no—
200#127—22no—
201#127—22no—
202#127—22no—
203#125—24no—
204#125C2×D414yes—
205#125C2×D414yes—
206#124—42no—
207#124—42no—
208#124—42no—
209#124—42no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214#121—24no—
215#121—24no—
216#121—24no—
217#121—24no—
218#121—42no—
219#121—42no—
220#121—42no—
221#121—42no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—42no—
231C4×D4—42no—
232C4×D4—42no—
233C4×D4—42no—
234#117—22no—
235#117—22no—
236#117C2×D414yes—
237#117C2414yes—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2414yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C42—22no—
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4112yes—
254C2×C42C2411yescharacteristic
255C2×C42C2411yescharacteristic
256#108—28no—
257#108—44no—
258#107—216no—
259#106—24no—
260#106—24no—
261#106—216no—
262C4⋊C8—28no—
263C4⋊C8—28no—
264C4⋊C8—24no—
265C4⋊C8—24no—
266C4⋊C8—24no—
267C4⋊C8—24no—
268#102—28no—
269#102—28no—
270#102—216no—
271#102—216no—
272#100—416no—
273C22⋊C8—24no—
274C22⋊C8—24no—
275#97—24no—
276#97—24no—
277#97—24no—
278#97C2×D414yes—
279#97C2×D414yes—
280C4×C8—24no—
281C4×C8—24no—
282C4×C8—24no—
283C4×C8C2×D414yes—
284C4×C8C2×D414yes—
285C24—24no—
286C24C2×#4111yescharacteristic
287#41—88no—
288#41—88no—
289#41—88no—
290#41—88no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—88no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—88no—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4112yes—
329C22×C4C2511yescharacteristic
330D8—48no—
331D8—48no—
332M16—24no—
333M16—24no—
334M16—28no—
335M16—28no—
336M16—28no—
337M16—42no—
338M16—42no—
339M16—44no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—28no—
345C2×C8—44no—
346C2×C8C22×D412yes—
347C2×C8C22×D412yes—
348C2×C8C22×D412yes—
349C2×C8C22×D412yes—
350C2×C82−1+414yes—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4C22×D418yes—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C42—24no—
368C42—24no—
369C42#12112yes—
370C42#12112yes—
371C42#12112yes—
372C42#12112yes—
373C42C22×D411yescharacteristic
374C42C22×D412yes—
375C42C22×D411yescharacteristic
376C42C22×D412yes—
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C23—22no—
380C23—42no—
381C23—44no—
382C23—48no—
383C23—84no—
384C23—84no—
385C23C2×#12111yescharacteristic
386C23C2×#12111yescharacteristic
387C23C22×#4111yescharacteristic
388Q8—48no—
389Q8—84no—
390Q8—84no—
391Q8—84no—
392D4—44no—
393D4—44no—
394D4—48no—
395D4—84no—
396D4—84no—
397D4—88no—
398D4—88no—
399D4—84no—
400D4—84no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—88no—
417C2×C4—88no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4C2×#12111yescharacteristic
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12112yes—
423C2×C4C2×#12112yes—
424C2×C4C2×#12112yes—
425C2×C4C2×#12112yes—
426C2×C4C23×D411yescharacteristic
427C2×C4C23×D411yescharacteristic
428C2×C4C2×2−1+412yes—
429C2×C4#58512yes—
430C8—22no—
431C8—22no—
432C8—22no—
433C8—44no—
434C8—48no—
435C8#53614yes—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—82no—
442C22—84no—
443C22—84no—
444C22—84no—
445C22—168no—
446C22C22×#12111yescharacteristic
447C22#350712yes—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—21no—
452C4—21no—
453C4—48no—
454C4—84no—
455C4#306814yes—
456C4C2×#53611yescharacteristic
457C4#344911yescharacteristic
458C4#349712yes—
459C2—22no—
460C2—41no—
461C2—41no—
462C2—164no—
463C2#3517811yescharacteristic
464C2C2×#306812yes—
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3184C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60864C2semidirect
3. #60192C2semidirect
4. #33125C2semidirect
5. #29791C2semidirect
6. #28763C2semidirect
7. #28700C2semidirect
8. #28670C2semidirect
9. #28299C2semidirect
10. #28220C2semidirect
11. #21878C2semidirect
12. #20827C2semidirect

Maximal subgroups

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

subgroupindex
#608642
C2×#31842
#601922
#331252
#297912
#287632
#287002
#286702
#282992
#282202
#218782
#208272