Finite Group Databasebeta

← #506455

Subgroups — #506455

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×#2228C211yesmaximal, characteristic
3#21723C211yesmaximal, characteristic, complemented
4#21696C211yesmaximal, characteristic
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic
7#13475C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic
9#13254C211yesmaximal, characteristic
10#13254C211yesmaximal, characteristic
11#13177C211yesmaximal, characteristic, complemented
12#13176C211yesmaximal, characteristic, complemented
13#12899C211yesmaximal, characteristic, complemented
14#12812C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#440—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2330—21no—
22#2326—21no—
23#2229—24no—
24#2228—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×C4⋊C16C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43#1992—21no—
44#1973—21no—
45#1970C2211yescharacteristic
46#1969C2211yescharacteristic
47#1959—21no—
48#1944—21no—
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1845C2211yescharacteristic
52#1845C2211yescharacteristic
53#1776—21no—
54C4×#107C2211yescharacteristic
55C4×#107C2211yescharacteristic
56#1726C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C2×#125—21no—
65C2×#124—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#499—22no—
73#499D414yes—
74#495—22no—
75#495Q814yes—
76#440—44no—
77C4×Q16—22no—
78C4×Q16—22no—
79C4×D8—22no—
80C4×D8—22no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#97D411yescharacteristic
95C2×#97D411yescharacteristic
96C2×#97D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—41no—
105#389—41no—
106#388—41no—
107#388—41no—
108#387—41no—
109#387—41no—
110#385—41no—
111#385—41no—
112#384—21no—
113#384—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116#376D411yescharacteristic
117#367—22no—
118#367—22no—
119#366—22no—
120#366—22no—
121C4⋊C16—22no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125C4⋊C16—41no—
126C4⋊C16—41no—
127#358—22no—
128#358—22no—
129#358—41no—
130#358—41no—
131#357—22no—
132#357—22no—
133#357—41no—
134#357—41no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139C22×Q8—21no—
140C22×D4—21no—
141C23×C4—41no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16C2×Q812yes—
148C2×SD16—24no—
149C2×SD16—42no—
150C2×SD16—42no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8C2×Q812yes—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2411yescharacteristic
161#125—24no—
162#125—42no—
163#124—24no—
164#124—42no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×D4—42no—
168C4×D4—42no—
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—41no—
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⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
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—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—42no—
218#107—42no—
219#107C2×D412yes—
220#107C2×D412yes—
221#106—24no—
222#103—42no—
223#103—42no—
224#102—42no—
225#102—42no—
226#97—22no—
227#97—22no—
228#97—41no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8C2×D412yes—
232C4×C8C2×Q812yes—
233C4×C8#4112yes—
234C4×C8#4112yes—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95D811yescharacteristic
242#95D811yescharacteristic
243#95D811yescharacteristic
244#95D811yescharacteristic
245C24—41no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
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×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C22×D411yescharacteristic
275Q16—22no—
276Q16—22no—
277Q16—41no—
278Q16—41no—
279SD16—48no—
280D8—22no—
281D8—22no—
282D8—41no—
283D8—41no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8#12112yes—
292C2×C8C22⋊Q812yes—
293C2×C8C22×D411yescharacteristic
294C2×C8C22×Q811yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C2×C82−1+411yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—44no—
304C4⋊C4—44no—
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—82no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C4⋊C4—161no—
318C4⋊C4—161no—
319C4⋊C4—161no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C42—21no—
326C42—21no—
327C42—82no—
328C42—161no—
329C42—161no—
330C42#12112yes—
331C42C22⋊Q812yes—
332C42AGL(1,ℤ/8)12yes—
333C42#13712yes—
334C16—41no—
335C16—41no—
336C16—41no—
337C16—41no—
338C23—42no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23C22×D811yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—81no—
349Q8—81no—
350D4—42no—
351D4—42no—
352D4—81no—
353D4—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
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—164no—
390C2×C4#45912yes—
391C2×C4#45912yes—
392C2×C4#47412yes—
393C2×C4#47412yes—
394C2×C4C2×#12111yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4#53711yescharacteristic
397C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57511yescharacteristic
400C8—22no—
401C8—22no—
402C8—41no—
403C8D4×Q811yescharacteristic
404C8#55011yescharacteristic
405C8#55411yescharacteristic
406C8#55611yescharacteristic
407C22—82no—
408C22—82no—
409C22—82no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22C2×#45911yescharacteristic
415C22C2×#47411yescharacteristic
416C22#324511yescharacteristic
417C22#341511yescharacteristic
418C22#341611yescharacteristic
419C22#341611yescharacteristic
420C22#341711yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4#253811yescharacteristic
435C4#303711yescharacteristic
436C4#307711yescharacteristic
437C4#325411yescharacteristic
438C2—162no—
439C2#2827911yescharacteristic
440C2#3358911yescharacteristic
441C2#3359011yescharacteristic
442C2#3363311yescharacteristic
443C2#3363411yescharacteristic
444C2#3382311yescharacteristic
445C2#3382411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2229C2semidirect
3. #21723C2semidirect
5. #13477C2semidirect
7. #13475C2semidirect
11. #13177C2semidirect
12. #13176C2semidirect
13. #12899C2semidirect
14. #12812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×#22282
#217232
#216962
#134772
#134762
#134752
#134742
#132542
#131772
#131762
#128992
#128122
#128112