Finite Group Databasebeta

← #497616

Subgroups — #497616

Order 512.

780 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×#3380C211yesmaximal, characteristic
2C2×#2183C211yesmaximal, characteristic, complemented
3#22727C211yesmaximal, characteristic, complemented
4#13284C211yesmaximal, characteristic, complemented
5#13283C211yesmaximal, characteristic, complemented
6#13281C211yesmaximal, characteristic, complemented
7#13277C211yesmaximal, characteristic, complemented
8#13277C211yesmaximal, characteristic, complemented
9#13275C211yesmaximal, characteristic, complemented
10#12813C211yesmaximal, characteristic
11#12813C211yesmaximal, characteristic
12#12811C211yesmaximal, characteristic
13#12637C211yesmaximal, characteristic
14#12637C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic
16C2×#554—21no—
17#3380—24no—
18#3380C418yescomplemented
19C2×#500C2211yescharacteristic
20C2×#499C2211yescharacteristic
21C2×#499C2211yescharacteristic
22C2×#478—21no—
23C2×#478—21no—
24C2×#476—21no—
25C2×C8×Q8C2211yescharacteristic
26C2×C4×D8C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28C2×C4×D8C2211yescharacteristic
29#2355—21no—
30#2183—24no—
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#1949—21no—
42#1949—21no—
43#1948—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×D8C2311yescharacteristic
63#554—28no—
64#554—44no—
65C2×C4⋊Q8—21no—
66C2×#125—21no—
67C2×#125—21no—
68C2×C22⋊Q8—41no—
69C2×C22⋊Q8—41no—
70C2×C22⋊Q8—41no—
71C2×C4×Q8D411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C22×C4⋊C4—41no—
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#500—22no—
81#500C2×C414yes—
82#499—22no—
83#499—22no—
84#499C2×C414yes—
85#499C2×C414yes—
86#478—28no—
87#478—28no—
88#476—28no—
89C8×Q8C2×C414yes—
90C8×Q8D414yes—
91C4×D8—22no—
92C4×D8—22no—
93C4×D8—22no—
94C4×D8C2×C414yes—
95C4×D8C2×C414yes—
96C4×D8C2×C414yes—
97C2×#107—21no—
98C2×#107—21no—
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#106—21no—
107C2×#106—21no—
108C2×#106—21no—
109C2×#106—21no—
110C2×C4⋊C8D411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D411yescharacteristic
113C2×#102—21no—
114C2×#102—21no—
115C2×#102—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391—41no—
120#387—41no—
121#387—41no—
122#384—21no—
123#382—21no—
124#382—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—41no—
129C4⋊C16—22no—
130C4⋊C16—22no—
131C4⋊C16—22no—
132#357—22no—
133#357—22no—
134#357—22no—
135#357Q814yescomplemented
136#346—22no—
137#346—22no—
138C4×C16—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142C22×Q8—41no—
143C22×Q8D811yescharacteristic
144C22×D4—21no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8C22×C412yes—
152C2×D8C2×Q812yes—
153C2×D8#4112yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158C4⋊Q8—24no—
159C4⋊Q8—42no—
160#125—24no—
161#125—24no—
162#125—42no—
163#125—42no—
164C22⋊Q8—48no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C4×Q8C22⋊C414yes—
168C4×Q8SD1614yes—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4D811yescharacteristic
191C2×C4⋊C4D811yescharacteristic
192C2×C4⋊C4D811yescharacteristic
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16C2×Q812yes—
214#107—24no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—81no—
226#107C22×C412yes—
227#107C22×C412yes—
228#107C22×C412yes—
229#107C2×Q812yes—
230#106—24no—
231#106—24no—
232#106—81no—
233#106—81no—
234C4⋊C8C22⋊C414yes—
235C4⋊C8C22⋊C414yes—
236C4⋊C8C22⋊C414yes—
237#102—24no—
238#102—24no—
239#102—24no—
240C4×C8C22×C412yes—
241C4×C8C22×C412yes—
242C4×C8C22×C412yes—
243C4×C8C2×D412yes—
244C4×C8C2×D412yes—
245C4×C8C2×D412yes—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—44no—
257C2×Q8—161no—
258C2×Q8—161no—
259C2×Q8#10214yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—161no—
279C22×C4—161no—
280C22×C4—161no—
281C22×C4—161no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289D8—22no—
290D8—22no—
291D8—22no—
292D8C4×Q814yes—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8C2×C22⋊C412yes—
300C2×C8C2×C22⋊C412yes—
301C2×C8C2×C22⋊C412yes—
302C2×C8C23×C411yescharacteristic
303C2×C8C22×D411yescharacteristic
304C2×C8C22×Q811yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C82−1+411yescharacteristic
307C2×C82−1+411yescharacteristic
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—82no—
328C4⋊C4—161no—
329C4⋊C4#10214yes—
330C4⋊C4#10214yes—
331C4⋊C4#10214yes—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C42—82no—
336C42—161no—
337C42—161no—
338C42C2×C22⋊C412yes—
339C42C2×C22⋊C412yes—
340C42C2×C22⋊C412yes—
341C42C2×SD1612yes—
342C42C2×SD1612yes—
343C42C2×SD1612yes—
344C16—22no—
345C16—41no—
346C16—41no—
347C16—41no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23C22×D811yescharacteristic
356Q8—44no—
357Q8—164no—
358D4—24no—
359D4—42no—
360D4—42no—
361D4—42no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—88no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—164no—
397C2×C4—164no—
398C2×C4C2×#10212yes—
399C2×C4C2×#10212yes—
400C2×C4C2×#10212yes—
401C2×C4C2×#10212yes—
402C2×C4C2×#10212yes—
403C2×C4C2×#10212yes—
404C2×C4C22×C22⋊C411yescharacteristic
405C2×C4#54811yescharacteristic
406C2×C4D4×Q811yescharacteristic
407C2×C4C22×SD1611yescharacteristic
408C2×C4#57711yescharacteristic
409C2×C4#57811yescharacteristic
410C8—22no—
411C8—22no—
412C8—22no—
413C8C2×C4×Q811yescharacteristic
414C8#51911yescharacteristic
415C8#54811yescharacteristic
416C8D4×Q811yescharacteristic
417C22—44no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22C22×#10211yescharacteristic
425C22#335811yescharacteristic
426C22Q8×D811yescharacteristic
427C22#341411yescharacteristic
428C22#341411yescharacteristic
429C22#341511yescharacteristic
430C22#341611yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4Q8×C22⋊C411yescharacteristic
445C4#289811yescharacteristic
446C4#335911yescharacteristic
447C4Q8×SD1611yescharacteristic
448C2—84no—
449C2Q8×#10211yescharacteristic
450C2#3338011yescharacteristic
451C2#3338111yescharacteristic
452C2#3393711yescharacteristic
453C2#3394311yescharacteristic
454C2#3395511yescharacteristic
455C2#3395611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2183C2semidirect
3. #22727C2semidirect
4. #13284C2semidirect
5. #13283C2semidirect
6. #13281C2semidirect
7. #13277C2semidirect
8. #13277C2semidirect
9. #13275C2semidirect
18. #3380C4semidirect
135. #357Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
C2×#21832
#227272
#132842
#132832
#132812
#132772
#132752
#128132
#128112
#126372
C4×#3572