Finite Group Databasebeta

← #72234383

Subgroups — #72234383

Order 1024.

512 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#512795C211yesmaximal, characteristic
2#483474C211yesmaximal, characteristic
3#483471C211yesmaximal, characteristic
4#151069C211yesmaximal, characteristic, complemented
5#151069C211yesmaximal, characteristic, complemented
6#151068C211yesmaximal, characteristic, complemented
7#151067C211yesmaximal, characteristic, complemented
8#151066C211yesmaximal, characteristic, complemented
9#151066C211yesmaximal, characteristic, complemented
10#151056C211yesmaximal, characteristic, complemented
11#151054C211yesmaximal, characteristic, complemented
12#151054C211yesmaximal, characteristic, complemented
13#151038C211yesmaximal, characteristic, complemented
14#151037C211yesmaximal, characteristic, complemented
15C2×#7463C211yesmaximal, characteristic, complemented
16C16×#41C2211yescharacteristic
17#17836—21no—
18#17829—21no—
19#13660C411yescharacteristic
20#13660C411yescharacteristic
21#13645C411yescharacteristic
22#13645C411yescharacteristic
23#13645C411yescharacteristic
24#13645C411yescharacteristic
25D4×C32C411yescharacteristic
26D4×C32C411yescharacteristic
27#13640C2211yescharacteristic
28#13640C2211yescharacteristic
29#13640C2211yescharacteristic
30C4×M64C2211yescharacteristic
31C2×#1397C2211yescharacteristic
32C2×#1397C2211yescharacteristic
33#12036—21no—
34#12035—21no—
35#12035—21no—
36#12034—21no—
37#12034—21no—
38#12032—21no—
39#12024C2211yescharacteristic
40#12024C2211yescharacteristic
41#12024C2211yescharacteristic
42#12024C2211yescharacteristic
43#12023C2211yescharacteristic
44#12023C2211yescharacteristic
45#12008C2211yescharacteristic
46#12008C2211yescharacteristic
47#12008C2211yescharacteristic
48C2×#1372C2211yescharacteristic
49C2×#1372C2211yescharacteristic
50C2×#1372C2211yescharacteristic
51#7463C2211yescharacteristic
52#7463C2211yescharacteristic
53#7463C2211yescharacteristic
54#7463C2211yescharacteristic
55#7463C2211yescharacteristic
56#7463C2211yescharacteristic
57#7463C2211yescharacteristic
58#7463C2211yescharacteristic
59#7463C2211yescharacteristic
60#7463C2211yescharacteristic
61#7463C2211yescharacteristic
62#7463C2211yescharacteristic
63#7463C2211yescharacteristic
64#7463C2211yescharacteristic
65#7463C2211yescharacteristic
66#7463C2211yescharacteristic
67#2990—41no—
68C8×#41D411yescharacteristic
69#2925—41no—
70C2×M64C2×C411yescharacteristic
71C2×M64C2×C411yescharacteristic
72C22×C32C2×C411yescharacteristic
73C22×C32C2×C411yescharacteristic
74C22×C32C2×C411yescharacteristic
75C22×C32C2×C411yescharacteristic
76Q8×C16C2×C411yescharacteristic
77Q8×C16Q811yescharacteristic
78D4×C16C2×C411yescharacteristic
79D4×C16C2×C411yescharacteristic
80D4×C16C2×C411yescharacteristic
81D4×C16Q811yescharacteristic
82D4×C16Q811yescharacteristic
83D4×C16Q811yescharacteristic
84#2162D411yescharacteristic
85#2162D411yescharacteristic
86#2162D411yescharacteristic
87C2×C4×C16C2311yescharacteristic
88C2×C4×C16C2311yescharacteristic
89C2×C4×C16C2311yescharacteristic
90#1574—41no—
91#1573—41no—
92#1573—41no—
93#1572—41no—
94#1572—41no—
95#1570—41no—
96#1568—21no—
97#1568—21no—
98#1568—21no—
99#1568—21no—
100#1567—21no—
101#1567—21no—
102#1564—21no—
103#1564—21no—
104#1564—21no—
105C2×#335—21no—
106C2×#335—21no—
107C2×#335—21no—
108C4⋊C32C2×C411yescharacteristic
109C4⋊C32C2×C411yescharacteristic
110C4⋊C32C2×C411yescharacteristic
111C4⋊C32C2×C411yescharacteristic
112C4⋊C32C2×C411yescharacteristic
113C4⋊C32C2×C411yescharacteristic
114C22⋊C32C2×C411yescharacteristic
115C22⋊C32C2×C411yescharacteristic
116C22⋊C32C2×C411yescharacteristic
117C22⋊C32C2×C411yescharacteristic
118C22⋊C32C2×C411yescharacteristic
119C22⋊C32C2×C411yescharacteristic
120#1397C2×C411yescharacteristic
121#1397C2×C411yescharacteristic
122#1397C2×C411yescharacteristic
123#1397C2×C411yescharacteristic
124#1397C2311yescharacteristic
125#1397C2311yescharacteristic
126#1397C2311yescharacteristic
127#1397C2311yescharacteristic
128C4×C32C2×C411yescharacteristic
129C4×C32C2×C411yescharacteristic
130#1372—21no—
131#1372—21no—
132#1372—21no—
133#1372—21no—
134#1372—21no—
135#1372C2311yescharacteristic
136#1372C2311yescharacteristic
137#1372C2311yescharacteristic
138#1372C2311yescharacteristic
139#1372C2311yescharacteristic
140#1372C2311yescharacteristic
141#1372C2311yescharacteristic
142#1372C2311yescharacteristic
143#1371—21no—
144#1371—21no—
145#1371—21no—
146#1371—21no—
147#1371—21no—
148#1371—21no—
149#1371—21no—
150C4×#41D811yescharacteristic
151C22×C16C22×C411yescharacteristic
152C22×C16C22×C411yescharacteristic
153C22×C16C22×C411yescharacteristic
154C22×C16C2×Q811yescharacteristic
155C22×C16C2×Q811yescharacteristic
156C22×C16C2×Q811yescharacteristic
157#446—81no—
158C8×Q8C4⋊C411yescharacteristic
159C8×Q8Q1611yescharacteristic
160#435—81no—
161#435—81no—
162C8×D4—81no—
163C8×D4C4⋊C411yescharacteristic
164C8×D4C4⋊C411yescharacteristic
165C8×D4C4⋊C411yescharacteristic
166C8×D4Q1611yescharacteristic
167C8×D4Q1611yescharacteristic
168C8×D4Q1611yescharacteristic
169#433—81no—
170#433—81no—
171#432—41no—
172#432—41no—
173#432—41no—
174#432—41no—
175#431—41no—
176#431—41no—
177#431—41no—
178#431D811yescharacteristic
179#431D811yescharacteristic
180#431D811yescharacteristic
181#423—41no—
182#423—41no—
183C2×C4⋊C8—41no—
184C2×C4⋊C8—41no—
185C2×C4⋊C8—41no—
186#408—81no—
187#405—81no—
188C2×C4×C8C2×D411yescharacteristic
189C2×C4×C8C2×D411yescharacteristic
190C2×C4×C8C2×D411yescharacteristic
191M64—22no—
192M64—41no—
193C2×C32—22no—
194C2×C32—22no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32C22×C411yescharacteristic
198C2×C32C22×C411yescharacteristic
199C2×C32C22×C411yescharacteristic
200C2×C32C22×C411yescharacteristic
201C2×C32C22×C411yescharacteristic
202C2×C32C22×C411yescharacteristic
203C2×C32C22×C411yescharacteristic
204C2×C32C22×C411yescharacteristic
205C4⋊C16C4⋊C411yescharacteristic
206C4⋊C16C4⋊C411yescharacteristic
207C4⋊C16C4⋊C411yescharacteristic
208C4⋊C16C4⋊C411yescharacteristic
209C4⋊C16C4⋊C411yescharacteristic
210C4⋊C16C4⋊C411yescharacteristic
211C22⋊C16C4⋊C411yescharacteristic
212C22⋊C16C4⋊C411yescharacteristic
213C22⋊C16C4⋊C411yescharacteristic
214C22⋊C16C4⋊C411yescharacteristic
215C22⋊C16C4⋊C411yescharacteristic
216C22⋊C16C4⋊C411yescharacteristic
217C4×C16C22×C411yescharacteristic
218C4×C16C22×C411yescharacteristic
219C4×C16C22×C411yescharacteristic
220C4×C16C2×D411yescharacteristic
221C4×C16C2×D411yescharacteristic
222C4×C16C2×D411yescharacteristic
223C4×C16C2×Q811yescharacteristic
224C4×C16C2×Q811yescharacteristic
225C4×C16C2×Q811yescharacteristic
226C4×C16C2411yescharacteristic
227#335—21no—
228#335—21no—
229#335—21no—
230#335—21no—
231#335—21no—
232#335—21no—
233#335—21no—
234#335—21no—
235#335—41no—
236#335—41no—
237#335—41no—
238#335—41no—
239#335—41no—
240#334—41no—
241#334—41no—
242#334—41no—
243#334—41no—
244#334—41no—
245#334—41no—
246#334—41no—
247C2×#41SD3211yescharacteristic
248C2×M16—81no—
249C2×M16—81no—
250C22×C8—81no—
251C22×C8—81no—
252C22×C8—81no—
253C22×C8C2×C4⋊C411yescharacteristic
254C22×C8C2×C4⋊C411yescharacteristic
255C22×C8C2×C4⋊C411yescharacteristic
256C22×C8C2×Q1611yescharacteristic
257C22×C8C2×Q1611yescharacteristic
258C22×C8C2×Q1611yescharacteristic
259C4×Q8#10711yescharacteristic
260C4×Q8SD3211yescharacteristic
261C4×D4#10711yescharacteristic
262C4×D4#10711yescharacteristic
263C4×D4#10711yescharacteristic
264C4×D4SD3211yescharacteristic
265C4×D4SD3211yescharacteristic
266C4×D4SD3211yescharacteristic
267#117SD3211yescharacteristic
268#117SD3211yescharacteristic
269#117SD3211yescharacteristic
270C2×C42C2×D811yescharacteristic
271C2×C42C2×D811yescharacteristic
272C2×C42C2×D811yescharacteristic
273C2×C16—21no—
274C2×C16—21no—
275C2×C16—21no—
276C2×C16—21no—
277C2×C16—21no—
278C2×C16—21no—
279C2×C16—41no—
280C2×C16—41no—
281C2×C16—41no—
282C2×C16C2×C4⋊C411yescharacteristic
283C2×C16C2×C4⋊C411yescharacteristic
284C2×C16C2×C4⋊C411yescharacteristic
285C2×C16C2×C4⋊C411yescharacteristic
286C2×C16C2×C4⋊C411yescharacteristic
287C2×C16C2×C4⋊C411yescharacteristic
288C2×C16C23×C411yescharacteristic
289C2×C16C22×Q811yescharacteristic
290C4⋊C8—41no—
291C4⋊C8—41no—
292C4⋊C8—41no—
293C4⋊C8—41no—
294C4⋊C8—41no—
295C4⋊C8—41no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298C4⋊C8—81no—
299C4⋊C8—81no—
300C4⋊C8—81no—
301C4⋊C8—81no—
302C4⋊C8—81no—
303C4⋊C8#10711yescharacteristic
304C4⋊C8#10711yescharacteristic
305C4⋊C8#10711yescharacteristic
306C4⋊C8#10711yescharacteristic
307C4⋊C8#10711yescharacteristic
308C4⋊C8#10711yescharacteristic
309C22⋊C8—81no—
310C22⋊C8—81no—
311C22⋊C8—81no—
312C22⋊C8—81no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315C22⋊C8—81no—
316C22⋊C8#10711yescharacteristic
317C22⋊C8#10711yescharacteristic
318C22⋊C8#10711yescharacteristic
319C22⋊C8#10711yescharacteristic
320C22⋊C8#10711yescharacteristic
321C22⋊C8#10711yescharacteristic
322#97—81no—
323#97—81no—
324#97—81no—
325#97—81no—
326C4×C8—81no—
327C4×C8—81no—
328C4×C8—81no—
329C4×C8C2×C4⋊C411yescharacteristic
330C4×C8C2×C4⋊C411yescharacteristic
331C4×C8C2×C4⋊C411yescharacteristic
332C4×C8C2×D811yescharacteristic
333C4×C8C2×D811yescharacteristic
334C4×C8C2×D811yescharacteristic
335C4×C8C2×Q1611yescharacteristic
336C4×C8C2×Q1611yescharacteristic
337C4×C8C2×Q1611yescharacteristic
338C4×C8C22×D411yescharacteristic
339C32—21no—
340C32—21no—
341C32—21no—
342C32—21no—
343C32—21no—
344C32—21no—
345C32—21no—
346C32#13112yes—
347#41—42no—
348C2×Q8#36711yescharacteristic
349C2×D4#36711yescharacteristic
350C2×D4#36711yescharacteristic
351C2×D4#36711yescharacteristic
352C22×C4C2×#10711yescharacteristic
353C22×C4C2×#10711yescharacteristic
354C22×C4C2×#10711yescharacteristic
355C22×C4C2×SD3211yescharacteristic
356C22×C4C2×SD3211yescharacteristic
357C22×C4C2×SD3211yescharacteristic
358C22×C4C2×SD3211yescharacteristic
359C22×C4C2×SD3211yescharacteristic
360C22×C4C2×SD3211yescharacteristic
361M16—162no—
362M16—321no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—81no—
373C2×C8—81no—
374C2×C8—81no—
375C2×C8—81no—
376C2×C8—81no—
377C2×C8—81no—
378C2×C8—81no—
379C2×C8—81no—
380C2×C8—162no—
381C2×C8—321no—
382C2×C8—321no—
383C2×C8C2×#10711yescharacteristic
384C2×C8C2×#10711yescharacteristic
385C2×C8C2×#10711yescharacteristic
386C2×C8C2×#10711yescharacteristic
387C2×C8C2×#10711yescharacteristic
388C2×C8C2×#10711yescharacteristic
389C2×C8C22×C4⋊C411yescharacteristic
390C2×C8C22×Q1611yescharacteristic
391C4⋊C4#36711yescharacteristic
392C4⋊C4#36711yescharacteristic
393C4⋊C4#36711yescharacteristic
394C4⋊C4#36711yescharacteristic
395C4⋊C4#36711yescharacteristic
396C4⋊C4#36711yescharacteristic
397C22⋊C4#36711yescharacteristic
398C22⋊C4#36711yescharacteristic
399C22⋊C4#36711yescharacteristic
400C22⋊C4#36711yescharacteristic
401C22⋊C4#36711yescharacteristic
402C22⋊C4#36711yescharacteristic
403C42C2×#10711yescharacteristic
404C42C2×#10711yescharacteristic
405C42C2×#10711yescharacteristic
406C42C2×SD3211yescharacteristic
407C42C2×SD3211yescharacteristic
408C42C2×SD3211yescharacteristic
409C42C2×SD3211yescharacteristic
410C42C2×SD3211yescharacteristic
411C42C2×SD3211yescharacteristic
412C42C22×D811yescharacteristic
413C16—21no—
414C16—21no—
415C16—21no—
416C16—21no—
417C16—21no—
418C16—21no—
419C16—21no—
420C16C2×#13111yescharacteristic
421C16#56811yescharacteristic
422C23C2×#36711yescharacteristic
423C23C2×#36711yescharacteristic
424C23C2×#36711yescharacteristic
425Q8—41no—
426D4—22no—
427D4—22no—
428D4—41no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4C2×#36711yescharacteristic
442C2×C4C2×#36711yescharacteristic
443C2×C4C2×#36711yescharacteristic
444C2×C4C2×#36711yescharacteristic
445C2×C4C2×#36711yescharacteristic
446C2×C4C2×#36711yescharacteristic
447C2×C4C2×#36711yescharacteristic
448C2×C4C2×#36711yescharacteristic
449C2×C4C2×#36711yescharacteristic
450C2×C4C22×#10711yescharacteristic
451C2×C4C22×SD3211yescharacteristic
452C2×C4C22×SD3211yescharacteristic
453C8—21no—
454C8—21no—
455C8—21no—
456C8—21no—
457C8—21no—
458C8—21no—
459C8—21no—
460C8—161no—
461C8—161no—
462C8—161no—
463C8—161no—
464C8—161no—
465C8—161no—
466C8—161no—
467C8—161no—
468C8#290611yescharacteristic
469C8#291611yescharacteristic
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22#1363812yes—
479C22C22×#36711yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4#1782811yescharacteristic
492C4#3341411yescharacteristic
493C2—22no—
494C2—41no—
495C2—41no—
496C2#48347011yescharacteristic
497C2C2×#1363811yescharacteristic
498C2#51272411yescharacteristic
subgroupG/H
center, socleC22C22×#367
commutatorC2×C16C23×C4
Frattini, cosocleC4×C16C24
prime radicalC2×#41SD32
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ C16×#41 ◁ G
LECSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #151069C2semidirect
5. #151069C2semidirect
6. #151068C2semidirect
7. #151067C2semidirect
8. #151066C2semidirect
9. #151066C2semidirect
10. #151056C2semidirect
11. #151054C2semidirect
12. #151054C2semidirect
13. #151038C2semidirect
14. #151037C2semidirect
15. C2×#7463C2semidirect

Maximal subgroups

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

subgroupindex
#5127952
#4834742
#4834712
#1510692
#1510682
#1510672
#1510662
#1510562
#1510542
#1510382
#1510372
C2×#74632