Finite Group Databasebeta

← #510905

Subgroups — #510905

Order 512.

565 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27929C211yesmaximal, characteristic
2#27792C211yesmaximal, characteristic, complemented
3#13531C211yesmaximal, characteristic, complemented
4#13481C211yesmaximal, characteristic, complemented
5#13449C211yesmaximal, characteristic, complemented
6#13381C211yesmaximal, characteristic
7#13363C211yesmaximal, characteristic, complemented
8#13305C211yesmaximal, characteristic
9#13282C211yesmaximal, characteristic, complemented
10#13237C211yesmaximal, characteristic, complemented
11#13177C211yesmaximal, characteristic, complemented
12#13131C211yesmaximal, characteristic, complemented
13#13120C211yesmaximal, characteristic, complemented
14#12993C211yesmaximal, characteristic, complemented
15#12920C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#480—21no—
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2501—21no—
23#2492—21no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#2093—21no—
38#2093—21no—
39#2036—21no—
40#2024—21no—
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1928—21no—
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#123—41no—
64C2×#123—41no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C22×C4⋊C4—21no—
68C2×SD32—22no—
69C2×SD32—41no—
70C2×SD32—41no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#500—22no—
76#500—41no—
77#480—44no—
78#467—22no—
79#467—41no—
80#466—41no—
81#466—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—41no—
87C2×#107—41no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#399—41no—
105#399—41no—
106#393—41no—
107#391—21no—
108#391—21no—
109#391—41no—
110#390—21no—
111#390—21no—
112#388—41no—
113#384—21no—
114#384—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—41no—
119#367—41no—
120#367—41no—
121C4⋊C16—22no—
122#358—41no—
123#358—41no—
124#358—41no—
125#357—22no—
126#357—41no—
127#357—41no—
128C22⋊C16—22no—
129C22⋊C16—41no—
130C4×C16—41no—
131#340—21no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—41no—
137C22×D4—21no—
138C22×D4—41no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2411yescharacteristic
163C4⋊Q8—42no—
164C4⋊Q8—81no—
165#123—84no—
166#123—84no—
167C22⋊Q8—42no—
168C22⋊Q8—81no—
169#121—42no—
170#121—42no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊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⋊C4—41no—
193C2×C22⋊C4D811yescharacteristic
194C2×C22⋊C4D811yescharacteristic
195C2×C42—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198SD32—44no—
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—21no—
209C2×C16—21no—
210C2×C16#4112yes—
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—42no—
223#107—42no—
224#107—42no—
225#107—42no—
226#107#4112yes—
227#103—42no—
228#103—42no—
229#103—81no—
230#102—42no—
231#102—42no—
232#102—42no—
233C22⋊C8—22no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95D811yescharacteristic
245#95D811yescharacteristic
246C24—41no—
247C24C2×D811yescharacteristic
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8—161no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C22×D411yescharacteristic
288Q16—41no—
289Q16—41no—
290Q16—41no—
291Q16—41no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8AGL(1,ℤ/8)12yes—
309C2×C8C2×#4111yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8H(5,𝔽2)11yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C82−1+411yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C22⋊C4—22no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—161no—
342C22⋊C4SD3214yescomplemented
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—161no—
348C16—22no—
349C16—41no—
350C16—41no—
351C16—41no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23C2×SD3212yes—
366C23C22×D811yescharacteristic
367Q8—81no—
368Q8—81no—
369Q8—81no—
370Q8—81no—
371Q8—164no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—164no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—164no—
415C2×C4C2×SD3212yes—
416C2×C4C2×SD3212yes—
417C2×C4#50912yes—
418C2×C4#53511yescharacteristic
419C2×C4#54611yescharacteristic
420C2×C4#54711yescharacteristic
421C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
422C2×C4#57711yescharacteristic
423C2×C4#57811yescharacteristic
424C8—22no—
425C8—41no—
426C8—41no—
427C8#53811yescharacteristic
428C8#54211yescharacteristic
429C8#55911yescharacteristic
430C8#56211yescharacteristic
431C22—22no—
432C22—22no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22#320511yescharacteristic
446C22#329411yescharacteristic
447C22#333411yescharacteristic
448C22C22×SD3211yescharacteristic
449C22C2×#50911yescharacteristic
450C22#341611yescharacteristic
451C22#341711yescharacteristic
452C4—22no—
453C4—22no—
454C4—22no—
455C4—41no—
456C4—41no—
457C4—82no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4#271011yescharacteristic
465C4#322511yescharacteristic
466C4#330811yescharacteristic
467C4#335411yescharacteristic
468C2—42no—
469C2—321no—
470C2#3204611yescharacteristic
471C2#3377411yescharacteristic
472C2#3379111yescharacteristic
473C2#3386511yescharacteristic
474C2#3387311yescharacteristic
475C2#3390211yescharacteristic
476C2#3392011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27792C2semidirect
3. #13531C2semidirect
4. #13481C2semidirect
5. #13449C2semidirect
7. #13363C2semidirect
9. #13282C2semidirect
10. #13237C2semidirect
11. #13177C2semidirect
12. #13131C2semidirect
13. #13120C2semidirect
14. #12993C2semidirect
15. #12920C2semidirect
342. C22⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
#279292
#277922
#135312
#134812
#134492
#133812
#133632
#133052
#132822
#132372
#131772
#131312
#131202
#129932
#129202