Finite Group Databasebeta

← #497922

Subgroups — #497922

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2168C211yesmaximal, characteristic
2#22658C211yesmaximal, characteristic
3#22656C211yesmaximal, characteristic
4#13178C211yesmaximal, characteristic, complemented
5#13176C211yesmaximal, characteristic, complemented
6#12921C211yesmaximal, characteristic, complemented
7#12921C211yesmaximal, characteristic, complemented
8#12921C211yesmaximal, characteristic, complemented
9#12919C211yesmaximal, characteristic, complemented
10#12890C211yesmaximal, characteristic, complemented
11#12836C211yesmaximal, characteristic, complemented
12#12834C211yesmaximal, characteristic, complemented
13#12674C211yesmaximal, characteristic, complemented
14#12557C211yesmaximal, characteristic, complemented
15#12556C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C22×#107C2211yescharacteristic
23#2354—21no—
24#2352—21no—
25#2168C418yescomplemented
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#1949—21no—
37#1948—21no—
38#1854—21no—
39#1854—21no—
40#1854—21no—
41#1853—21no—
42#1846C2211yescharacteristic
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C2×M32—41no—
68C2×M32—41no—
69C2×M32C2×C414yes—
70C22×C16—41no—
71C22×C16—41no—
72C22×C16C2×C414yescomplemented
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C8×D4C2×C414yes—
78C8×D4Q814yes—
79#427—22no—
80#427—22no—
81#427—22no—
82#427—41no—
83#427—41no—
84#427—41no—
85C2×#107—22no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—41no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—21no—
100C2×#106—21no—
101C2×C4⋊C8D411yescharacteristic
102C2×C22⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#390—41no—
106#388—41no—
107#388—41no—
108#388—41no—
109#387—41no—
110#386—41no—
111#384—21no—
112#382—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—41no—
120C4×C22⋊C4—41no—
121#367—22no—
122#367—41no—
123#366—22no—
124#366—41no—
125#366—41no—
126#366—41no—
127C4⋊C16C2×C414yescomplemented
128C22⋊C16C2×C414yes—
129C22⋊C16C2×C414yescomplemented
130#346C2×C414yescomplemented
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137C22×D4D811yescharacteristic
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C22×C412yes—
150C22×C8C22×C412yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×Q812yes—
155C22×C8C2×Q812yes—
156C22×C8C2411yescharacteristic
157C4×D4C4⋊C414yes—
158C4×D4Q1614yes—
159#117—42no—
160#117—42no—
161#117—42no—
162#117—81no—
163#117—81no—
164#117—81no—
165C2×C4⋊C4—21no—
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—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—81no—
178C2×C4⋊C4D811yescharacteristic
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4D811yescharacteristic
187C2×C22⋊C4D811yescharacteristic
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194M32—44no—
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×C16—44no—
204C2×C16C22×C412yes—
205C2×C16C22×C412yes—
206C2×C16C22×C412yes—
207C2×C16C22×C412yes—
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—21no—
220#107—21no—
221#107—21no—
222#107—42no—
223#107—81no—
224#107#4112yes—
225#107#4112yes—
226#106—42no—
227#106—42no—
228#106—42no—
229#106—81no—
230C4⋊C8C4⋊C414yes—
231C22⋊C8C4⋊C414yes—
232C22⋊C8C4⋊C414yes—
233C4×C8C22×C412yes—
234C4×C8C2×Q812yes—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243C24C2×D811yescharacteristic
244C24C2×D811yescharacteristic
245C2×D4—41no—
246C2×D4—41no—
247C2×D4#10714yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
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—161no—
267C22×C4C2×C4⋊C412yes—
268C22×C4C2×C4⋊C412yes—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×Q1612yes—
274C22×C4C2×Q1612yes—
275C22×C4C22×D411yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8C2×C4⋊C412yes—
291C2×C8C2×C4⋊C412yes—
292C2×C8C2×C4⋊C412yes—
293C2×C8C23×C411yescharacteristic
294C2×C8C22×Q811yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C2×C8H(5,𝔽2)11yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
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—82no—
316C4⋊C4—82no—
317C4⋊C4—161no—
318C4⋊C4#10714yes—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C22⋊C4—161no—
326C22⋊C4#10714yes—
327C22⋊C4#10714yes—
328C42—82no—
329C42—82no—
330C42—82no—
331C42—82no—
332C42—161no—
333C42C2×C4⋊C412yes—
334C42C2×Q1612yes—
335C16—22no—
336C16—22no—
337C16—22no—
338C16—22no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23C2×#10712yes—
352C23C2×#10712yes—
353C23C22×D811yescharacteristic
354D4—44no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
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—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—164no—
396C2×C4C2×#10712yes—
397C2×C4C2×#10712yes—
398C2×C4C2×#10712yes—
399C2×C4C2×#10712yes—
400C2×C4C22×C4⋊C411yescharacteristic
401C2×C4#54611yescharacteristic
402C2×C4#54611yescharacteristic
403C2×C4C22×Q1611yescharacteristic
404C2×C4#57711yescharacteristic
405C2×C4#57711yescharacteristic
406C8—22no—
407C8—22no—
408C8—22no—
409C8C4×#4111yescharacteristic
410C8#51811yescharacteristic
411C8#55411yescharacteristic
412C8#55411yescharacteristic
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22C22×#10711yescharacteristic
428C22#329411yescharacteristic
429C22#329411yescharacteristic
430C22#341411yescharacteristic
431C22#341411yescharacteristic
432C22#341511yescharacteristic
433C22#341711yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—82no—
439C4—82no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4#234911yescharacteristic
447C4#291211yescharacteristic
448C4#329911yescharacteristic
449C4#329911yescharacteristic
450C2—42no—
451C2—42no—
452C2#2270511yescharacteristic
453C2#3340911yescharacteristic
454C2#3341011yescharacteristic
455C2#3387611yescharacteristic
456C2#3388011yescharacteristic
457C2#3388211yescharacteristic
458C2#3388611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13178C2semidirect
5. #13176C2semidirect
6. #12921C2semidirect
7. #12921C2semidirect
8. #12921C2semidirect
9. #12919C2semidirect
10. #12890C2semidirect
11. #12836C2semidirect
12. #12834C2semidirect
13. #12674C2semidirect
14. #12557C2semidirect
15. #12556C2semidirect
25. #2168C4semidirect
72. C22×C16C2×C4semidirect
127. C4⋊C16C2×C4semidirect
129. C22⋊C16C2×C4semidirect
130. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21682
#226582
#226562
#131782
#131762
#129212
#129192
#128902
#128362
#128342
#126742
#125572
#125562