Finite Group Databasebeta

← #504769

Subgroups — #504769

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2225C211yesmaximal, characteristic, complemented
2C2×C4×SD32C211yesmaximal, characteristic
3#26484C211yesmaximal, characteristic
4#23184C211yesmaximal, characteristic, complemented
5#13449C211yesmaximal, characteristic
6#13409C211yesmaximal, characteristic, complemented
7#13364C211yesmaximal, characteristic, complemented
8#13327C211yesmaximal, characteristic
9#13148C211yesmaximal, characteristic, complemented
10#13131C211yesmaximal, characteristic
11#13110C211yesmaximal, characteristic, complemented
12#13104C211yesmaximal, characteristic, complemented
13#13014C211yesmaximal, characteristic
14#13013C211yesmaximal, characteristic
15#12938C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2452—21no—
24#2365—21no—
25#2225—24no—
26C4×SD32—24no—
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#2067—21no—
40#2023—21no—
41#1981—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1878—21no—
49#1878—21no—
50#1862—21no—
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16—21no—
62C22×Q16C2311yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C22⋊Q8—41no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32—22no—
75C2×SD32—22no—
76C2×SD32—41no—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#500—22no—
82#500—41no—
83#494—22no—
84#494—22no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—41no—
88C4×Q16—41no—
89C4×D8—22no—
90C4×D8—41no—
91C2×#107—21no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#396—41no—
109#393—41no—
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#389—41no—
117#387—41no—
118#387—41no—
119#385—41no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—41no—
122C4×C4⋊C4D411yescharacteristic
123#367—22no—
124#367D414yes—
125C4⋊C16—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#358—22no—
130#358—22no—
131#358—41no—
132#357—22no—
133#357—22no—
134C4×C16—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—41no—
143C22×D4—21no—
144C23×C4—41no—
145C2×Q16—24no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—81no—
157C2×Q16—81no—
158C2×Q16#4112yes—
159C2×Q16#4112yes—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8#4112yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170C4⋊Q8—24no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—81no—
174C22⋊Q8—84no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—81no—
178C4×Q8—81no—
179C4×D4—42no—
180C4×D4—81no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4D811yescharacteristic
194C2×C4⋊C4D811yescharacteristic
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206SD32—28no—
207SD32—44no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16C2×D412yes—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—42no—
229#107C2×D412yes—
230#107C2×D412yes—
231C4⋊C8—41no—
232#103—42no—
233#103—42no—
234#103—81no—
235#103—81no—
236#102—42no—
237#102—42no—
238#102—81no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95D811yescharacteristic
253#95D811yescharacteristic
254C24—41no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—161no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—161no—
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C22×D411yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—22no—
296Q16—22no—
297Q16—41no—
298Q16—41no—
299Q16—41no—
300Q16—41no—
301Q16—84no—
302D8—22no—
303D8—22no—
304D8—41no—
305D8—41no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8C22×D411yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C82−1+411yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4SD3214yescomplemented
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—161no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—82no—
352C42—82no—
353C42—82no—
354C16—22no—
355C16—41no—
356C16—41no—
357C16—41no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23C22×D811yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—42no—
369Q8—42no—
370Q8—81no—
371Q8—81no—
372Q8—81no—
373Q8—81no—
374Q8—164no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
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—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
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—81no—
415C2×C4—164no—
416C2×C4C2×SD3212yes—
417C2×C4C2×SD3212yes—
418C2×C4C2×SD3212yes—
419C2×C4#52511yescharacteristic
420C2×C4D4211yescharacteristic
421C2×C4#54811yescharacteristic
422C2×C4#57511yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57711yescharacteristic
425C8—41no—
426C8—41no—
427C8—41no—
428C8#52511yescharacteristic
429C8#53911yescharacteristic
430C8#54811yescharacteristic
431C8#55111yescharacteristic
432C22—82no—
433C22—82no—
434C22—82no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22#310011yescharacteristic
440C22D4×D811yescharacteristic
441C22#335811yescharacteristic
442C22C22×SD3211yescharacteristic
443C22#341411yescharacteristic
444C22#341611yescharacteristic
445C22#341711yescharacteristic
446C4—22no—
447C4—22no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4#246311yescharacteristic
460C4#311811yescharacteristic
461C4#327111yescharacteristic
462C4#337011yescharacteristic
463C2—162no—
464C2#2688911yescharacteristic
465C2#3366311yescharacteristic
466C2#3368111yescharacteristic
467C2#3383511yescharacteristic
468C2D4×SD3211yescharacteristic
469C2#3392611yescharacteristic
470C2#3394311yescharacteristic
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×#2225C2semidirect
4. #23184C2semidirect
6. #13409C2semidirect
7. #13364C2semidirect
9. #13148C2semidirect
11. #13110C2semidirect
12. #13104C2semidirect
15. #12938C2semidirect
339. C4⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×#22252
C2×C4×SD322
#264842
#231842
#134492
#134092
#133642
#133272
#131482
#131312
#131102
#131042
#130142
#130132
#129382