Finite Group Databasebeta

← #497032

Subgroups — #497032

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33606C211yesmaximal, characteristic
2#21546C211yesmaximal, characteristic
3#21527C211yesmaximal, characteristic, complemented
4#13143C211yesmaximal, characteristic, complemented
5#13141C211yesmaximal, characteristic, complemented
6#13119C211yesmaximal, characteristic, complemented
7#13118C211yesmaximal, characteristic, complemented
8#13066C211yesmaximal, characteristic, complemented
9#13065C211yesmaximal, characteristic
10#13039C211yesmaximal, characteristic, complemented
11#13038C211yesmaximal, characteristic, complemented
12#12957C211yesmaximal, characteristic, complemented
13#12955C211yesmaximal, characteristic, complemented
14#12831C211yesmaximal, characteristic, complemented
15C2×#1375C211yesmaximal, characteristic, complemented
16C2×#510C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19#3050C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#444C2211yescharacteristic
22C2×#441C2211yescharacteristic
23#2929—21no—
24C22×#97C2211yescharacteristic
25C4×C22⋊Q8—21no—
26#2220C412yescomplemented
27#2219C412yescomplemented
28#2217C412yescomplemented
29#2216C412yescomplemented
30#2159C2211yescharacteristic
31C2×#365C2211yescharacteristic
32C2×#365C2211yescharacteristic
33C2×#362C2211yescharacteristic
34C2×#362C2211yescharacteristic
35C2×#360C2211yescharacteristic
36C2×#360C2211yescharacteristic
37#2141C2211yescharacteristic
38#2139C2211yescharacteristic
39#1975C2211yescharacteristic
40#1971C2211yescharacteristic
41#1946C2211yescharacteristic
42#1943C2211yescharacteristic
43#1938—21no—
44#1935—21no—
45#1896—21no—
46#1892—21no—
47#1810C2211yescharacteristic
48#1807C2211yescharacteristic
49#1776C2211yescharacteristic
50#1773C2211yescharacteristic
51#1765—21no—
52#1763—21no—
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1383C2211yescharacteristic
58#1383C2211yescharacteristic
59#1375C2211yescharacteristic
60#1375C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C2×#135C2311yescharacteristic
64C22×Q16C2311yescharacteristic
65C2×#131—21no—
66C23×C8D411yescharacteristic
67C23×C8C2311yescharacteristic
68#525—21no—
69C2×C22⋊Q8—21no—
70C2×C4×Q8—21no—
71C22×C42D411yescharacteristic
72#510—41no—
73#510—41no—
74C2×Q32C2×C412yescomplemented
75C2×SD32C2×C412yescomplemented
76C2×M32—21no—
77C2×M32—41no—
78C2×M32—41no—
79C2×M32—41no—
80C2×M32—41no—
81C2×M32C2×C412yes—
82C2×M32C2311yescharacteristic
83C2×M32C2311yescharacteristic
84C2×M32C2311yescharacteristic
85C2×M32C2311yescharacteristic
86C22×C16—21no—
87C22×C16C2×C412yes—
88#467—21no—
89#467—21no—
90#467C2×C411yescharacteristic
91#467C2×C412yes—
92#467C2×C411yescharacteristic
93#467C2×C412yes—
94#466C2×C411yescharacteristic
95#466C2×C411yescharacteristic
96#465—21no—
97#465—21no—
98#444—41no—
99#444—41no—
100#441—41no—
101#441—41no—
102#435—41no—
103#435—41no—
104C8×D4—41no—
105C8×D4—41no—
106C2×#108C2311yescharacteristic
107C2×#108C2311yescharacteristic
108#427C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#106C2311yescharacteristic
111C2×#106C2311yescharacteristic
112#424—21no—
113C2×#104—21no—
114C2×#104—21no—
115#416—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×#103—21no—
119#408—21no—
120#407—21no—
121#405—21no—
122C2×#97—41no—
123C2×#97—41no—
124C2×#97D411yescharacteristic
125C2×#97D411yescharacteristic
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128#391—21no—
129#389—21no—
130#387—41no—
131#385—41no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—41no—
135C4×C22⋊C4—41no—
136#367C2×C412yescomplemented
137#366C2×C412yescomplemented
138#365—41no—
139#365—41no—
140#362—41no—
141#362—41no—
142#360—41no—
143#360—41no—
144#358C2×C412yescomplemented
145#358C2×C412yescomplemented
146#358C2×C412yescomplemented
147#357C2×C412yescomplemented
148#341—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#339—21no—
153#339—21no—
154C2×#41—21no—
155C22×Q8—21no—
156C23×C4—21no—
157C23×C4C2×D411yescharacteristic
158#135—21no—
159#135—21no—
160#135—21no—
161#135—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16C22×C412yes—
168C2×Q16C22×C411yescharacteristic
169C2×SD16—21no—
170C2×D8C22×C411yescharacteristic
171#131—42no—
172#131—42no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—41no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—41no—
184C22×C8—41no—
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8#4111yescharacteristic
191C22×C8#4111yescharacteristic
192C22×C8C2411yescharacteristic
193#123—41no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200#121—21no—
201#121—21no—
202#120—41no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×D4—41no—
206C4×D4—41no—
207#117—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223Q32—42no—
224SD32—42no—
225M32—21no—
226M32—21no—
227M32—21no—
228M32—21no—
229M32—21no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—42no—
234M32—42no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—42no—
238C2×C16—42no—
239C2×C16C22×C412yes—
240C2×C16C22×C412yes—
241#108—21no—
242#108—21no—
243#108—21no—
244#108—21no—
245#107—21no—
246#107C22×C412yes—
247#107C22×C411yescharacteristic
248#107C22×C411yescharacteristic
249#106—21no—
250#106—21no—
251#106—21no—
252#106—21no—
253#106—21no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256#104—42no—
257#104—42no—
258#103—22no—
259#103—22no—
260#103—21no—
261#103—21no—
262#103—42no—
263#103—42no—
264#102—21no—
265#102—21no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269C22⋊C8—41no—
270#97—21no—
271#97—21no—
272#97—21no—
273#97—21no—
274#97—41no—
275#97—41no—
276#97—41no—
277C4×C8—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283C24AGL(1,ℤ/8)11yescharacteristic
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—81no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4AGL(1,ℤ/8)11yescharacteristic
316C22×C4AGL(1,ℤ/8)11yescharacteristic
317C22×C4AGL(1,ℤ/8)11yescharacteristic
318C22×C4C22×D411yescharacteristic
319Q16—22no—
320Q16—21no—
321Q16—22no—
322Q16—21no—
323Q16—41no—
324Q16—41no—
325SD16—41no—
326SD16—41no—
327D8—21no—
328D8—21no—
329M16—41no—
330M16—41no—
331M16—41no—
332M16—41no—
333M16—82no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—82no—
354C2×C8C4×D411yescharacteristic
355C2×C8C4×D411yescharacteristic
356C2×C8#13512yes—
357C2×C8C23×C411yescharacteristic
358C2×C8C2×#4111yescharacteristic
359C2×C8H(5,𝔽2)11yescharacteristic
360C2×C8H(5,𝔽2)11yescharacteristic
361C2×C8H(5,𝔽2)11yescharacteristic
362C2×C82−1+411yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—82no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—42no—
389C42—41no—
390C42—82no—
391C42—82no—
392C42#13512yes—
393C16—22no—
394C16—22no—
395C16—41no—
396C16—41no—
397C23—21no—
398C23—22no—
399C23—21no—
400C23—41no—
401C23—41no—
402C23#44211yescharacteristic
403C23#44211yescharacteristic
404C23C2×AGL(1,ℤ/8)11yescharacteristic
405Q8—41no—
406Q8—42no—
407Q8—42no—
408Q8—41no—
409Q8—81no—
410Q8—81no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—81no—
415D4—82no—
416D4—81no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—84no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4#44211yescharacteristic
452C2×C4#44211yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4#53411yescharacteristic
455C2×C4#53711yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C8—21no—
460C8—21no—
461C8—22no—
462C8—41no—
463C8—81no—
464C8—81no—
465C8#51811yescharacteristic
466C8#52011yescharacteristic
467C8#53811yescharacteristic
468C8#53911yescharacteristic
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22#218012yes—
482C22C2×#44211yescharacteristic
483C22#318811yescharacteristic
484C22#324711yescharacteristic
485C4—21no—
486C4—21no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—82no—
491C4—81no—
492C4—82no—
493C4—81no—
494C4—161no—
495C4—161no—
496C4#232511yescharacteristic
497C4C4×#13511yescharacteristic
498C4#318811yescharacteristic
499C4#324711yescharacteristic
500C2—21no—
501C2—21no—
502C2—22no—
503C2—41no—
504C2—161no—
505C2#2218411yescharacteristic
506C2C2×#218011yescharacteristic
507C2#3346811yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21527C2semidirect
4. #13143C2semidirect
5. #13141C2semidirect
6. #13119C2semidirect
7. #13118C2semidirect
8. #13066C2semidirect
10. #13039C2semidirect
11. #13038C2semidirect
12. #12957C2semidirect
13. #12955C2semidirect
14. #12831C2semidirect
15. C2×#1375C2semidirect
26. #2220C4semidirect
27. #2219C4semidirect
28. #2217C4semidirect
29. #2216C4semidirect
74. C2×Q32C2×C4semidirect
75. C2×SD32C2×C4semidirect
136. #367C2×C4semidirect
137. #366C2×C4semidirect
144. #358C2×C4semidirect
145. #358C2×C4semidirect
146. #358C2×C4semidirect
147. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
#336062
#215462
#215272
#131432
#131412
#131192
#131182
#130662
#130652
#130392
#130382
#129572
#129552
#128312
C2×#13752