Finite Group Databasebeta

← #501761

Subgroups — #501761

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×#2151C211yesmaximal, characteristic, complemented
3#33216C211yesmaximal, characteristic, complemented
4#21524C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13514C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13476C211yesmaximal, characteristic, complemented
9#13277C211yesmaximal, characteristic, complemented
10#13177C211yesmaximal, characteristic, complemented
11#13061C211yesmaximal, characteristic
12#13059C211yesmaximal, characteristic
13#12836C211yesmaximal, characteristic, complemented
14#12835C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2629—21no—
23#2324—21no—
24#2228—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29#2151—24no—
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38#1992—21no—
39#1970C2211yescharacteristic
40#1943—21no—
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1936—21no—
45#1895—21no—
46#1845C2211yescharacteristic
47#1809—21no—
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1762—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—21no—
65C2×#121—21no—
66C2×C4×D4—21no—
67C2×#117D411yescharacteristic
68C2×M32—22no—
69C2×M32—22no—
70C2×M32—41no—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16—41no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#499—22no—
79#499—22no—
80#466—22no—
81#466Q814yes—
82C4×D8—22no—
83C4×D8—22no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96#405—22no—
97#405—22no—
98C2×#97D411yescharacteristic
99C2×#97D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#398—41no—
103#398—41no—
104#396—41no—
105#396—41no—
106#390—21no—
107#390—41no—
108#389—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#386—41no—
113#385—41no—
114#384—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119#367—22no—
120#367—41no—
121#367—41no—
122#366—22no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126C4⋊C16D414yescomplemented
127#357—22no—
128#357—22no—
129#357—41no—
130#357—41no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138C22×D4—21no—
139C22×D4—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C2×Q812yes—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—41no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C22×C8C2411yescharacteristic
161#125—42no—
162#125—42no—
163#121—24no—
164#121—42no—
165C4×D4—42no—
166C4×D4—42no—
167#117—22no—
168#117—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4D811yescharacteristic
184C2×C4⋊C4D811yescharacteristic
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4D811yescharacteristic
193C2×C22⋊C4D811yescharacteristic
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197M32—44no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—44no—
209C2×C16C2×D412yes—
210C2×C16C2×D412yes—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
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—42no—
225#107C2×Q812yes—
226#106—42no—
227#102—24no—
228#102—42no—
229#102—42no—
230#97—22no—
231#97—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8C2×D412yes—
235C4×C8#4112yes—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243C24—41no—
244C24C2×D811yescharacteristic
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—161no—
253C2×D4—161no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—161no—
267C22×C4—161no—
268C22×C4C22⋊Q812yes—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4#13712yes—
275C22×C4C22×D411yescharacteristic
276D8—22no—
277D8—22no—
278D8—41no—
279D8—41no—
280M16—44no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8#12112yes—
293C2×C8C22⋊Q812yes—
294C2×C8C22×D411yescharacteristic
295C2×C8C22×Q811yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8H(5,𝔽2)11yescharacteristic
299C2×C82−1+411yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—161no—
319C4⋊C4—161no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C4⋊C4—161no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—161no—
328C22⋊C4—161no—
329C22⋊C4—161no—
330C22⋊C4—161no—
331C22⋊C4—161no—
332C42—21no—
333C42—21no—
334C42—161no—
335C42#12112yes—
336C42#13512yes—
337C16—22no—
338C16—22no—
339C16—41no—
340C16—41no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23#47412yes—
355C23C22×D811yescharacteristic
356D4—42no—
357D4—42no—
358D4—81no—
359D4—81no—
360D4—164no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—81no—
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—81no—
397C2×C4—81no—
398C2×C4—164no—
399C2×C4#46612yes—
400C2×C4#46612yes—
401C2×C4#47412yes—
402C2×C4C2×#12111yescharacteristic
403C2×C4C2×C22⋊Q811yescharacteristic
404C2×C4#53711yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4C2×#13711yescharacteristic
407C2×C4#57511yescharacteristic
408C8—22no—
409C8—22no—
410C8—41no—
411C8#54611yescharacteristic
412C8D4×Q811yescharacteristic
413C8#55211yescharacteristic
414C8#55411yescharacteristic
415C22—22no—
416C22—22no—
417C22—22no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22C2×#46611yescharacteristic
430C22C2×#47411yescharacteristic
431C22#324511yescharacteristic
432C22#341511yescharacteristic
433C22#341611yescharacteristic
434C22#341611yescharacteristic
435C22#341611yescharacteristic
436C4—22no—
437C4—22no—
438C4—22no—
439C4—41no—
440C4—41no—
441C4—82no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4#241711yescharacteristic
449C4#305611yescharacteristic
450C4#307611yescharacteristic
451C4#324811yescharacteristic
452C2—42no—
453C2—162no—
454C2#2483511yescharacteristic
455C2#3361111yescharacteristic
456C2#3361211yescharacteristic
457C2#3363111yescharacteristic
458C2#3363411yescharacteristic
459C2#3382311yescharacteristic
460C2#3382611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2228C2semidirect
2. C2×#2151C2semidirect
3. #33216C2semidirect
5. #13516C2semidirect
6. #13514C2semidirect
7. #13476C2semidirect
8. #13476C2semidirect
9. #13277C2semidirect
10. #13177C2semidirect
13. #12836C2semidirect
14. #12835C2semidirect
15. #12811C2semidirect
126. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#21512
#332162
#215242
#135162
#135142
#134762
#132772
#131772
#130612
#130592
#128362
#128352
#128112