Finite Group Databasebeta

← #506436

Subgroups — #506436

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2183C211yesmaximal, characteristic, complemented
2C2×#2177C211yesmaximal, characteristic
3#28267C211yesmaximal, characteristic, complemented
4#28263C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13239C211yesmaximal, characteristic
10#13235C211yesmaximal, characteristic
11#13168C211yesmaximal, characteristic, complemented
12#13166C211yesmaximal, characteristic, complemented
13#12908C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2580—21no—
21#2538—21no—
22#2183—24no—
23#2177—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#2067—21no—
42#2067—21no—
43#2065—21no—
44#2065—21no—
45#1970C2211yescharacteristic
46#1970C2211yescharacteristic
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1942C2211yescharacteristic
52#1942C2211yescharacteristic
53#1846C2211yescharacteristic
54#1845C2211yescharacteristic
55#1804—21no—
56#1802—21no—
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×C22⋊Q8—41no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×D16—22no—
68C2×D16—22no—
69C2×D16Q814yescomplemented
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C8×Q8—22no—
75C8×Q8—22no—
76C4×D8—22no—
77C4×D8—22no—
78C4×D8—22no—
79C4×D8—22no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#106—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#399—41no—
101#396—41no—
102#396—41no—
103#393—41no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#385—41no—
113#385—41no—
114#384—21no—
115#382—21no—
116#380—41no—
117#367—22no—
118#367—22no—
119#367—41no—
120#366—22no—
121#366—41no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125#357—22no—
126#357—22no—
127#357—41no—
128#357—41no—
129#357—41no—
130#357—41no—
131#346—22no—
132#346D414yes—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138C22×Q8D811yescharacteristic
139C22×D4—21no—
140C22×D4—21no—
141C23×C4—41no—
142C23×C4—41no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×Q812yes—
154C2×D8C2×Q812yes—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2411yescharacteristic
159C22⋊Q8—84no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4D811yescharacteristic
179C2×C4⋊C4D811yescharacteristic
180C2×C4⋊C4D811yescharacteristic
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
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×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194D16—28no—
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×C16C2×D412yes—
204C2×C16C2×D412yes—
205C2×C16C2×Q812yes—
206C2×C16#4112yes—
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—21no—
218#107—21no—
219#107—81no—
220#106—81no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224#102—42no—
225#102—42no—
226#102—81no—
227#102—81no—
228#102—81no—
229#102—81no—
230C4×C8—21no—
231C4×C8—21no—
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—
242C24—41no—
243C24—41no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
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—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—161no—
270C22×C4—161no—
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C22×D411yescharacteristic
278D8—22no—
279D8—22no—
280D8—22no—
281D8—22no—
282D8—41no—
283D8—41no—
284D8—41no—
285D8—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8#12112yes—
295C2×C8C22×D411yescharacteristic
296C2×C8C22×Q811yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8H(5,𝔽2)11yescharacteristic
300C2×C82−1+411yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
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—82no—
319C4⋊C4—161no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—161no—
329C22⋊C4—161no—
330C22⋊C4—161no—
331C22⋊C4—161no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42#12112yes—
337C42#13512yes—
338C16—22no—
339C16—41no—
340C16—41no—
341C16C22⋊Q814yescomplemented
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23C22×D811yescharacteristic
357Q8—44no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
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—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
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—81no—
399C2×C4—164no—
400C2×C4—164no—
401C2×C4#46612yes—
402C2×C4#46612yes—
403C2×C4#50912yes—
404C2×C4C2×#12111yescharacteristic
405C2×C4D4×Q811yescharacteristic
406C2×C4#55011yescharacteristic
407C2×C4C2×#13511yescharacteristic
408C2×C4#57611yescharacteristic
409C2×C4#57811yescharacteristic
410C8—22no—
411C8—41no—
412C8—41no—
413C8C2×C22⋊Q811yescharacteristic
414C8#53711yescharacteristic
415C8#55411yescharacteristic
416C8#55611yescharacteristic
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22C2×#46611yescharacteristic
432C22Q8×D811yescharacteristic
433C22#339111yescharacteristic
434C22C2×#50911yescharacteristic
435C22#341411yescharacteristic
436C22#341511yescharacteristic
437C22#341611yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4#253811yescharacteristic
451C4#306111yescharacteristic
452C4#338011yescharacteristic
453C4#339311yescharacteristic
454C2—162no—
455C2—162no—
456C2#2826311yescharacteristic
457C2#3360511yescharacteristic
458C2#3361211yescharacteristic
459C2#3395111yescharacteristic
460C2#3395611yescharacteristic
461C2#3396611yescharacteristic
462C2#3396811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×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
1. C2×#2183C2semidirect
3. #28267C2semidirect
4. #28263C2semidirect
5. #13476C2semidirect
6. #13476C2semidirect
7. #13474C2semidirect
8. #13474C2semidirect
11. #13168C2semidirect
12. #13166C2semidirect
13. #12908C2semidirect
14. #12813C2semidirect
15. #12811C2semidirect
69. C2×D16Q8semidirect
341. C16C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#21832
C2×#21772
#282672
#282632
#134762
#134742
#132392
#132352
#131682
#131662
#129082
#128132
#128112