Finite Group Databasebeta

← #504133

Subgroups — #504133

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×C4×D16C211yesmaximal, characteristic
3#26482C211yesmaximal, characteristic, complemented
4#22682C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic, complemented
6#13507C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13165C211yesmaximal, characteristic, complemented
10#13104C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#13012C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic, complemented
14#12937C211yesmaximal, characteristic, complemented
15#12935C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2454—21no—
23#2349—21no—
24#2228—24no—
25C4×D16—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#2065—21no—
40#2025—21no—
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1864—21no—
48#1864—21no—
49#1864—21no—
50#1846C2211yescharacteristic
51#1804—21no—
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×D16—22no—
69C2×D16—22no—
70C2×D16Q814yes—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—41no—
77C4×D8—22no—
78C4×D8—22no—
79C4×D8—22no—
80C4×D8—22no—
81C4×D8—22no—
82C4×D8—41no—
83C4×D8—41no—
84C4×D8—41no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#399—41no—
104#396—41no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#385—41no—
112#385—41no—
113#385—41no—
114#382—21no—
115#380—41no—
116#380—41no—
117C4×C4⋊C4D411yescharacteristic
118C2×#95—41no—
119#367—22no—
120#367—41no—
121#366—22no—
122#366—22no—
123#366—22no—
124C4⋊C16—22no—
125#357—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130#357D414yes—
131C4×C16—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×D4—21no—
139C22×D4—21no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
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×D8C2×D412yes—
153C2×D8C2×Q812yes—
154C2×D8C2×Q812yes—
155C2×D8#4112yes—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2411yescharacteristic
160#125—42no—
161#125—81no—
162C4×D4—24no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—81no—
168C4×D4—81no—
169C4×D4—81no—
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⋊C4D811yescharacteristic
183C2×C4⋊C4D811yescharacteristic
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×C22⋊C4—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195D16—28no—
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×C16C2×D412yes—
207C2×C16C2×Q812yes—
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#107C2×D412yes—
219#107#4112yes—
220#106—42no—
221#106—81no—
222C4⋊C8—41no—
223#102—24no—
224#102—42no—
225#102—42no—
226#102—42no—
227#102—42no—
228#102—81no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95D811yescharacteristic
242#95D811yescharacteristic
243C24—41no—
244C24—41no—
245C2×D4—22no—
246C2×D4—22no—
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—
257C22×C4—44no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—161no—
271C22×C4—161no—
272C22×C4—161no—
273C22×C4—161no—
274C22×C4—161no—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C22×D411yescharacteristic
282D8—22no—
283D8—22no—
284D8—22no—
285D8—22no—
286D8—22no—
287D8—41no—
288D8—41no—
289D8C22⋊Q814yes—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8C22×D411yescharacteristic
300C2×C8C22×Q811yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C82−1+411yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
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—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—161no—
322C4⋊C4—161no—
323C4⋊C4—161no—
324C4⋊C4—161no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—161no—
331C22⋊C4—161no—
332C22⋊C4—161no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C16—22no—
340C16—22no—
341C16—41no—
342C16—41no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23C22×D811yescharacteristic
358D4—24no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—42no—
363D4—42no—
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—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
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—81no—
400C2×C4—81no—
401C2×C4—164no—
402C2×C4—164no—
403C2×C4—164no—
404C2×C4—164no—
405C2×C4#50812yes—
406C2×C4#52511yescharacteristic
407C2×C4D4211yescharacteristic
408C2×C4D4×Q811yescharacteristic
409C2×C4#57511yescharacteristic
410C2×C4#57611yescharacteristic
411C2×C4#57811yescharacteristic
412C8—41no—
413C8—41no—
414C8—41no—
415C8C2×C22⋊Q811yescharacteristic
416C8#53911yescharacteristic
417C8#54711yescharacteristic
418C8#55411yescharacteristic
419C22—44no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22#310011yescharacteristic
434C22D4×D811yescharacteristic
435C22Q8×D811yescharacteristic
436C22C2×#50811yescharacteristic
437C22#341411yescharacteristic
438C22#341511yescharacteristic
439C22#341511yescharacteristic
440C4—22no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—82no—
447C4—82no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4#244911yescharacteristic
453C4#312411yescharacteristic
454C4#326911yescharacteristic
455C4#338911yescharacteristic
456C2—84no—
457C2—162no—
458C2#2636811yescharacteristic
459C2#3366211yescharacteristic
460C2#3368211yescharacteristic
461C2#3383411yescharacteristic
462C2#3384611yescharacteristic
463C2#3394711yescharacteristic
464C2#3395611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
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×#2228C2semidirect
3. #26482C2semidirect
4. #22682C2semidirect
5. #13528C2semidirect
6. #13507C2semidirect
7. #13474C2semidirect
8. #13453C2semidirect
9. #13165C2semidirect
10. #13104C2semidirect
13. #12937C2semidirect
14. #12937C2semidirect
15. #12935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×C4×D162
#264822
#226822
#135282
#135072
#134742
#134532
#131652
#131042
#130122
#129372
#129352