Finite Group Databasebeta

← #630400

Subgroups — #630400

Order 512.

2634 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3#15631C218yesmaximal, complemented
4#15448C218yesmaximal, complemented
5#15224C212yesmaximal, complemented
6#15144C212yesmaximal, complemented
7#14700C212yesmaximal, complemented
8#14260C212yesmaximal, complemented
9#14160C212yesmaximal, complemented
10#14154C212yesmaximal
11#13759C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#127C2211yescharacteristic
14C22×#125C2211yescharacteristic
15#2641—28no—
16C2×#394C2214yes—
17C2×#394C2214yes—
18C2×#393C2218yes—
19#2331—28no—
20C2×#391C2214yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2218yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#388C2218yes—
30C2×#387C2214yes—
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×#376C2211yescharacteristic
35C2×#376C2211yescharacteristic
36C2×C43C2211yescharacteristic
37#1441C2218yes—
38#1440C2218yes—
39#1440C2218yes—
40#1440C2218yes—
41#1439C2218yes—
42#1439C2218yes—
43#1438C2218yes—
44#1437C2218yes—
45#1436C2214yes—
46#1436C2218yes—
47#1436C2214yes—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C4⋊#95C2212yes—
51C4⋊#95C2212yes—
52C4×#95C2212yes—
53C23×Q8C2312yes—
54C23×D4C2312yes—
55C23×D4C2314yes—
56C2×C4⋊Q8—28no—
57C2×C4⋊Q8—24no—
58C2×#127—24no—
59C2×#127—28no—
60C2×#125—24no—
61C2×#125—28no—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C22⋊C4C2318yes—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2318yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2318yes—
76C22×C22⋊C4C2318yes—
77C22×C42C2312yes—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2314yes—
82C22×C42C2314yes—
83C22×C42C2311yescharacteristic
84#394—216no—
85#394—216no—
86#393—416no—
87#391—216no—
88#390—28no—
89#390—28no—
90#390—28no—
91#390—28no—
92#390—44no—
93#390—44no—
94#390—416no—
95#389—28no—
96#388—416no—
97#387—216no—
98#382—24no—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—28no—
101#376—42no—
102#376D418yes—
103C2×#95C2314yes—
104C2×#95C2318yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2312yes—
109C2×#95C2312yes—
110C2×#95C2318yes—
111C2×#95C2314yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C2×#95C2314yes—
115C2×#95C2312yes—
116C2×#95C2312yes—
117C43—42no—
118C25C2414yes—
119C22×Q8—24no—
120C22×Q8—28no—
121C22×Q8—48no—
122C22×D4—24no—
123C22×D4—28no—
124C22×D4—28no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—28no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—48no—
133C22×D4—48no—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2411yescharacteristic
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2412yes—
143C23×C4C2411yescharacteristic
144C4⋊Q8—232no—
145#127—416no—
146#125—416no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4#4118yes—
166C2×C22⋊C4—216no—
167C2×C22⋊C4—216no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—216no—
173C2×C22⋊C4—216no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—48no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4—48no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4—48no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—28no—
196C2×C42—28no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—48no—
200C2×C42C2×D414yes—
201C2×C42C2×D414yes—
202C2×C42C2×D414yes—
203C2×C42#4118yes—
204#95—28no—
205#95—24no—
206#95—24no—
207#95—28no—
208#95—28no—
209#95—216no—
210#95—28no—
211#95—44no—
212#95—48no—
213#95—44no—
214#95—48no—
215#95—48no—
216#95C2×D418yes—
217#95C2×D418yes—
218C24—24no—
219C24—24no—
220C24—24no—
221C24—24no—
222C24—24no—
223C24—28no—
224C24—28no—
225C24—24no—
226C24—24no—
227C24—28no—
228C24—28no—
229C24C2511yescharacteristic
230C2×Q8—216no—
231C2×Q8—88no—
232C2×Q8—816no—
233C2×D4—432no—
234C2×D4—48no—
235C2×D4—416no—
236C2×D4—48no—
237C2×D4—48no—
238C2×D4—48no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—48no—
242C2×D4—816no—
243C2×D4—88no—
244C2×D4—88no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—24no—
281C22×C4—28no—
282C22×C4C22×D412yes—
283C22×C4C22×D414yes—
284C22×C4C22×D412yes—
285C22×C4C22×D414yes—
286C22×C4C22×D412yes—
287C22×C4C2×#4118yes—
288C22×C4C2×#4114yes—
289C22×C4C2×#4114yes—
290C22×C4C2×#4114yes—
291C22×C4C2×#4114yes—
292C22×C4H(5,𝔽2)18yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C42−1+412yes—
295C4⋊C4—216no—
296C4⋊C4—216no—
297C4⋊C4—416no—
298C4⋊C4—416no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—88no—
302C22⋊C4—432no—
303C22⋊C4—88no—
304C22⋊C4—88no—
305C22⋊C4—88no—
306C22⋊C4—88no—
307C22⋊C4—816no—
308C22⋊C4—816no—
309C22⋊C4—816no—
310C42—28no—
311C42—28no—
312C42—416no—
313C42—48no—
314C42—48no—
315C42—88no—
316C42#124116yes—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—48no—
321C23—48no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—48no—
327C23—48no—
328C23—44no—
329C23—44no—
330C23—44no—
331C23—44no—
332C23—48no—
333C23—48no—
334C23—48no—
335C23—48no—
336C23—48no—
337C23—48no—
338C23—48no—
339C23—48no—
340C23C23×D411yescharacteristic
341C23C22×#4112yes—
342C23C22×#4112yes—
343C23C2×H(5,𝔽2)11yescharacteristic
344C23C2×H(5,𝔽2)14yes—
345C23C2×2−1+411yescharacteristic
346C23#58514yes—
347Q8—816no—
348D4—832no—
349D4—816no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—216no—
365C2×C4—28no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4#52514yes—
395C2×C4C2×#12418yes—
396C2×C4C2×#12414yes—
397C2×C4#53714yes—
398C2×C4#55618yes—
399C2×C4#55918yes—
400C22—84no—
401C22—84no—
402C22—88no—
403C22—88no—
404C22—88no—
405C22—88no—
406C22—84no—
407C22—84no—
408C22—84no—
409C22—84no—
410C22—84no—
411C22C2×#52511yescharacteristic
412C22C22×#12411yescharacteristic
413C22C2×#53711yescharacteristic
414C22#346014yes—
415C22C2×#55612yes—
416C22C2×#55912yes—
417C22#349414yes—
418C22#349812yes—
419C22#350012yes—
420C22#351914yes—
421C22#352414yes—
422C22#354214yes—
423C22#355414yes—
424C4—28no—
425C4—42no—
426C4—48no—
427C4—44no—
428C4—48no—
429C4—42no—
430C4—84no—
431C4—84no—
432C4—84no—
433C4#259218yes—
434C2—164no—
435C2C2×#259211yescharacteristic
436C2#3526412yes—
437C2#3553711yescharacteristic
438C2#3567811yescharacteristic
439C2#3730112yes—
440C2#3823814yes—
441C2#3927514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2331C2semidirect
3. #15631C2semidirect
4. #15448C2semidirect
5. #15224C2semidirect
6. #15144C2semidirect
7. #14700C2semidirect
8. #14260C2semidirect
9. #14160C2semidirect
11. #13759C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
C2×#23312
#156312
#154482
#152242
#151442
#147002
#142602
#141602
#141542
#137592