Finite Group Databasebeta

← #501712

Subgroups — #501712

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×C4×SD32C211yesmaximal, characteristic
3#28534C211yesmaximal, characteristic
4#22685C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13506C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13474C211yesmaximal, characteristic
9#13281C211yesmaximal, characteristic, complemented
10#13159C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic
12#13013C211yesmaximal, characteristic
13#12814C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic
15#12812C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#478—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2566—21no—
24#2351—21no—
25#2229—24no—
26C4×SD32—24no—
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#2094—21no—
41#2092—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1865—21no—
47#1865—21no—
48#1846C2211yescharacteristic
49#1846C2211yescharacteristic
50#1845C2211yescharacteristic
51#1805—21no—
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×SD32—22no—
69C2×SD32—22no—
70C2×SD32Q814yes—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—22no—
77#478—44no—
78C4×Q16—22no—
79C4×Q16—22no—
80C4×Q16—22no—
81C4×Q16—41no—
82C4×Q16—41no—
83C4×Q16D414yes—
84C4×D8—22no—
85C4×D8—22no—
86C2×#107—21no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#396—41no—
107#396—41no—
108#390—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#389—41no—
113#389—41no—
114#385—41no—
115#385—41no—
116#384—21no—
117#382—21no—
118#382—21no—
119#382—41no—
120C4×C4⋊C4—41no—
121C4×C4⋊C4D411yescharacteristic
122#367—22no—
123#367—22no—
124#366—22no—
125C4⋊C16—22no—
126C4⋊C16—41no—
127C4⋊C16—41no—
128#358—22no—
129#358—22no—
130#358—22no—
131#357—22no—
132#357—41no—
133#357—41no—
134C4×C16—22no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×Q8—21no—
140C22×D4—21no—
141C23×C4—41no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16C2×D412yes—
146C2×Q16C2×Q812yes—
147C2×Q16#4112yes—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×Q812yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158#125—42no—
159#125—42no—
160C4×Q8—24no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—81no—
165C4×Q8—81no—
166C4×D4—42no—
167C4×D4—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
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⋊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×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196SD32—28no—
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—
207C2×C16—21no—
208C2×C16C2×Q812yes—
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—21no—
218#107—21no—
219#107—21no—
220#107—42no—
221#107C2×D412yes—
222#106—42no—
223#106—42no—
224#106—81no—
225C4⋊C8—22no—
226#103—24no—
227#103—42no—
228#103—42no—
229#102—42no—
230#102—42no—
231#102—42no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8C2×D412yes—
237C4×C8#4112yes—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95D811yescharacteristic
245#95D811yescharacteristic
246C24—41no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—161no—
268C22×C4—161no—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C22×D411yescharacteristic
276Q16—22no—
277Q16—22no—
278Q16—22no—
279Q16C22⋊Q814yes—
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—21no—
292C2×C8#12112yes—
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—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—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—161no—
326C22⋊C4—161no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—44no—
332C42—82no—
333C42—82no—
334C42#12112yes—
335C42AGL(1,ℤ/8)12yes—
336C16—22no—
337C16—41no—
338C16—41no—
339C16—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23C22×D811yescharacteristic
348Q8—24no—
349Q8—42no—
350Q8—42no—
351Q8—42no—
352D4—42no—
353D4—42no—
354D4—81no—
355D4—81no—
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—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
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—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—164no—
393C2×C4#45912yes—
394C2×C4#45912yes—
395C2×C4#50812yes—
396C2×C4C2×#12111yescharacteristic
397C2×C4#54611yescharacteristic
398C2×C4D4×Q811yescharacteristic
399C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
400C2×C4#57711yescharacteristic
401C2×C4#57811yescharacteristic
402C8—22no—
403C8—41no—
404C8—41no—
405C8C2×C22⋊Q811yescharacteristic
406C8#53711yescharacteristic
407C8#55211yescharacteristic
408C8#55411yescharacteristic
409C22—82no—
410C22—82no—
411C22—82no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22C2×#45911yescharacteristic
417C22#329411yescharacteristic
418C22Q8×D811yescharacteristic
419C22C2×#50811yescharacteristic
420C22#341411yescharacteristic
421C22#341611yescharacteristic
422C22#341611yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—44no—
430C4—82no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#241711yescharacteristic
437C4#304611yescharacteristic
438C4#333011yescharacteristic
439C4#338411yescharacteristic
440C2—162no—
441C2#2481111yescharacteristic
442C2#3357911yescharacteristic
443C2#3358711yescharacteristic
444C2#3386811yescharacteristic
445C2#3389311yescharacteristic
446C2#3394811yescharacteristic
447C2#3395511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2229C2semidirect
4. #22685C2semidirect
5. #13507C2semidirect
6. #13506C2semidirect
9. #13281C2semidirect
10. #13159C2semidirect
13. #12814C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×C4×SD322
#285342
#226852
#135072
#135062
#134762
#134742
#132812
#131592
#130132
#128142
#128132
#128122