Finite Group Databasebeta

← #72286865

Subgroups — #72286865

Order 2048.

892 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72286838C211yesmaximal, characteristic, complemented
2#72286797C211yesmaximal, characteristic, complemented
3#72286779C211yesmaximal, characteristic, complemented
4#72286459C214yesmaximal, complemented
5C2×#122963C211yesmaximal, characteristic, complemented
6#72231929C211yesmaximal, characteristic, complemented
7#72225141C211yesmaximal, characteristic, complemented
8#72224639C214yesmaximal, complemented
9#72223673C211yesmaximal, characteristic
10C4×#2263C411yescharacteristic
11#483604—21no—
12C2×#12062—21no—
13#483499C2211yescharacteristic
14C2×#12045C2211yescharacteristic
15#151085C412yes—
16#151085C2212yes—
17#151083C412yes—
18#151083C2212yes—
19#151038C411yescharacteristic
20#150064C411yescharacteristic
21#150063C411yescharacteristic
22#149502C2211yescharacteristic
23#149495C2211yescharacteristic
24#122963—22no—
25#122963C2214yes—
26#122963C2212yescomplemented
27#122962C2212yescomplemented
28#122961—22no—
29#122919C2211yescharacteristic
30C2×#7099C2211yescharacteristic
31#122464—21no—
32C2×#7070—21no—
33C2×#7053C2211yescharacteristic
34#93877C2214yescomplemented
35#93758C2212yescomplemented
36#93757C2214yes—
37#93757C2212yescomplemented
38#93668C2214yescomplemented
39C2×#2263C811yescharacteristic
40C16×#41—41no—
41C4×#508C2×C411yescharacteristic
42C2×C4×D16C2×C411yescharacteristic
43C2×D4×C16—41no—
44C8×#135—21no—
45C2×C8×D8—21no—
46#13654C2×C412yes—
47#13652C2×C412yes—
48C4×SD64C812yes—
49C4×D32C812yes—
50#13634C811yescharacteristic
51#13617C811yescharacteristic
52#13616C811yescharacteristic
53C4×M64—21no—
54C4×M64C2×C411yescharacteristic
55C4×M64C2×C411yescharacteristic
56#12062—22no—
57#12062—24no—
58#12062—42no—
59#12059—22no—
60SD16⋊C16—42no—
61#12045—21no—
62#12045—21no—
63#12045C2×C412yes—
64#12045C2×C411yescharacteristic
65#12045C2312yes—
66#12045C2311yescharacteristic
67#12044C2×C411yescharacteristic
68#12044C2311yescharacteristic
69#12043—21no—
70#12043—21no—
71#12012—21no—
72#12008C2×C411yescharacteristic
73C2×#1372C2×C411yescharacteristic
74C2×#1367—21no—
75#11613—41no—
76#11591—21no—
77C2×#1331—21no—
78C2×D4⋊C16—41no—
79C2×#1312D411yescharacteristic
80C2×#1312D411yescharacteristic
81C2×#1311C2311yescharacteristic
82#7463C2×C412yes—
83#7463C2×C412yes—
84#7386C2×C412yes—
85#7385C2×C412yes—
86#7385C2×C412yes—
87#7385C2×C412yes—
88#7332—21no—
89#7332C2312yes—
90#7330C2×C412yes—
91#7330C2312yes—
92#7099—21no—
93#7099—21no—
94#7099C2312yes—
95#7099C2312yes—
96#7071—22no—
97#7070—22no—
98#7070—24no—
99#7053D412yes—
100#7053C2312yes—
101C2×#508C2×C811yescharacteristic
102C22×D16C2×C811yescharacteristic
103C23×C16—81no—
104C8×#41—41no—
105C4×#135—21no—
106C2×C4×D8—21no—
107C2×C8×D4—41no—
108#2263C1618yescomplemented
109C2×SD64C2×C812yes—
110C2×D32C2×C812yes—
111C2×M64—21no—
112C2×M64—41no—
113C2×M64C2×C811yescharacteristic
114C2×M64C2×C811yescharacteristic
115Q8×C16—42no—
116C4×Q32C2×C811yescharacteristic
117C4×Q32C22×C411yescharacteristic
118C4×SD32—21no—
119C4×SD32—21no—
120C4×D16—21no—
121C4×D16—21no—
122C4×D16C2×C811yescharacteristic
123C4×D16C2×C812yes—
124C4×D16C22×C412yes—
125C4×D16C22×C411yescharacteristic
126D4×C16—42no—
127D4×C16—44no—
128D4×C16—82no—
129D4×C16—82no—
130#2162—41no—
131#2162—81no—
132#2158C2×C811yescharacteristic
133C2×#366C2×C811yescharacteristic
134C2×C4⋊C16—41no—
135#2138—21no—
136C2×#357—21no—
137C2×C22⋊C16—81no—
138C2×C4×C16—21no—
139C2×C4×C16—21no—
140C2×C4×C16—81no—
141C2×C4×C16C22×C411yescharacteristic
142C2×C4×C16#4111yescharacteristic
143C8×Q16—21no—
144C8×Q16—21no—
145C8×SD16—41no—
146C8×SD16—41no—
147C8×D8—21no—
148C8×D8—22no—
149C8×D8—21no—
150C8×D8—22no—
151C8×D8—41no—
152C8×D8—41no—
153#1564—21no—
154C2×#335—21no—
155#1476—41no—
156C2×D4⋊C8—41no—
157C2×C82C2×D411yescharacteristic
158#1424C2×C812yes—
159#1423C2×C812yes—
160#1416C2×C812yes—
161#1415C2×C812yes—
162#1415C2×C812yes—
163#1415C2×C812yes—
164#1397—22no—
165#1397C22×C412yes—
166#1397C22×C412yes—
167C4×C32—21no—
168C4×C32—41no—
169C4×C32C2×C812yes—
170#1372—21no—
171#1372C22×C411yescharacteristic
172#1372C22×C411yescharacteristic
173#1372C22×C412yes—
174#1371—21no—
175#1370—41no—
176#1367—22no—
177#1367—22no—
178#1367—41no—
179Q8⋊C16—42no—
180#1332—21no—
181#1332—21no—
182#1331—21no—
183#1331—22no—
184#1331—21no—
185#1331—22no—
186D4⋊C16—42no—
187D4⋊C16—44no—
188#1312—21no—
189#1312—21no—
190#1312C2×D412yes—
191#1312C2×D412yes—
192#1311C22×C411yescharacteristic
193#1311C2×D411yescharacteristic
194#1311#4111yescharacteristic
195#1311C2411yescharacteristic
196C2×#135—21no—
197C22×D8—21no—
198C23×C8—81no—
199C4×#41—41no—
200C2×C4×D4—41no—
201#508—22no—
202#508C2×C1614yescomplemented
203C2×Q32C22×C811yescharacteristic
204C2×SD32—21no—
205C2×D16—21no—
206C2×D16—22no—
207C2×D16C2×C1614yescomplemented
208C2×D16C22×C812yes—
209C2×D16C22×C811yescharacteristic
210C22×C16—21no—
211C22×C16—41no—
212C22×C16—41no—
213C22×C16—82no—
214C22×C16—84no—
215C22×C16—81no—
216C22×C16—164no—
217C22×C16C22×C811yescharacteristic
218C22×C16#13111yescharacteristic
219C8×Q8—41no—
220C8×Q8—41no—
221C4×Q16—21no—
222C4×Q16—21no—
223C4×SD16—41no—
224C4×SD16—41no—
225C4×D8—22no—
226C4×D8—21no—
227C4×D8—22no—
228C4×D8—21no—
229C4×D8—41no—
230C4×D8—41no—
231C8×D4—42no—
232C8×D4—41no—
233C8×D4—42no—
234C8×D4—41no—
235C8×D4—81no—
236C8×D4—81no—
237C8×D4—81no—
238C8×D4—81no—
239#431—41no—
240#431—81no—
241#427—21no—
242C2×#107—21no—
243C2×C4⋊C8—41no—
244#416—41no—
245C2×#102—41no—
246C2×C22⋊C8—81no—
247C2×C4×C8—81no—
248C2×C4×C8C4×D411yescharacteristic
249C2×C4×C8AGL(1,ℤ/8)11yescharacteristic
250C2×C4×C8AGL(1,ℤ/8)11yescharacteristic
251SD64C2×C1618yescomplemented
252D32C2×C1618yescomplemented
253M64—22no—
254M64—42no—
255M64—82no—
256M64C2×C1614yescomplemented
257C2×C32—22no—
258C2×C32—42no—
259C2×C32C22×C812yes—
260C2×C32C22×C812yes—
261#367—21no—
262#366—21no—
263#366C22×C812yes—
264#366C22×C811yescharacteristic
265#366C22×C811yescharacteristic
266C4⋊C16—42no—
267C4⋊C16—42no—
268C4⋊C16—81no—
269C4⋊C16—82no—
270#358—21no—
271#358—21no—
272#357—22no—
273#357—21no—
274#357—22no—
275#357—21no—
276C22⋊C16—84no—
277C22⋊C16—82no—
278C22⋊C16—81no—
279C4×C16—22no—
280C4×C16—22no—
281C4×C16—21no—
282C4×C16—41no—
283C4×C16—41no—
284C4×C16—81no—
285C4×C16—82no—
286C4×C16C4×D411yescharacteristic
287C4×C16C4×D411yescharacteristic
288C4×C16C22×C811yescharacteristic
289C4×C16#13111yescharacteristic
290C4×C16C23×C411yescharacteristic
291C4×C16C2×#4111yescharacteristic
292#335—22no—
293#335—21no—
294#335—21no—
295#335—41no—
296#334—41no—
297Q8⋊C8—41no—
298Q8⋊C8—41no—
299D4⋊C8—42no—
300D4⋊C8—41no—
301D4⋊C8—42no—
302D4⋊C8—41no—
303C82C4×D411yescharacteristic
304C82AGL(1,ℤ/8)11yescharacteristic
305C82AGL(1,ℤ/8)11yescharacteristic
306C82C22×D411yescharacteristic
307C2×#41—41no—
308C22×D4—41no—
309C23×C4—81no—
310#135—24no—
311#135—42no—
312C2×Q16—21no—
313C2×SD16—41no—
314C2×D8—22no—
315C2×D8—21no—
316C2×D8—24no—
317C2×D8—41no—
318C2×D8—42no—
319C22×C8—21no—
320C22×C8—21no—
321C22×C8—82no—
322C22×C8—82no—
323C22×C8—81no—
324C22×C8—81no—
325C22×C8—81no—
326C22×C8—162no—
327C22×C8—162no—
328C22×C8C8×D411yescharacteristic
329C22×C8#44411yescharacteristic
330C4×Q8—41no—
331C4×Q8—41no—
332C4×D4—42no—
333C4×D4—41no—
334C4×D4—42no—
335C4×D4—41no—
336C4×D4—81no—
337C4×D4—81no—
338C4×D4—81no—
339C4×D4—81no—
340#117—41no—
341#117—81no—
342C2×C4⋊C4—41no—
343C2×C22⋊C4—81no—
344C2×C42—81no—
345C2×C42#44211yescharacteristic
346Q32—21no—
347Q32C22×C1612yescomplemented
348SD32—24no—
349D16—22no—
350D16—21no—
351D16—24no—
352D16C22×C1614yes—
353D16C22×C1612yescomplemented
354C2×C16—22no—
355C2×C16—21no—
356C2×C16—21no—
357C2×C16—42no—
358C2×C16—42no—
359C2×C16—42no—
360C2×C16—82no—
361C2×C16—82no—
362C2×C16—82no—
363C2×C16—161no—
364C2×C16—164no—
365C2×C16—168no—
366C2×C16—162no—
367C2×C16C8×D411yescharacteristic
368C2×C16C8×D411yescharacteristic
369C2×C16C22×C1612yes—
370C2×C16#50412yes—
371C2×C16C23×C811yescharacteristic
372C2×C16C2×#13111yescharacteristic
373#107—22no—
374#107—21no—
375#107—21no—
376#107—41no—
377#106—41no—
378C4⋊C8—42no—
379C4⋊C8—41no—
380C4⋊C8—41no—
381C4⋊C8—81no—
382C4⋊C8—81no—
383C4⋊C8—81no—
384#103—41no—
385#103—41no—
386#102—42no—
387#102—41no—
388#102—42no—
389#102—41no—
390C22⋊C8—82no—
391C22⋊C8—82no—
392C22⋊C8—81no—
393C22⋊C8—81no—
394C22⋊C8—81no—
395C4×C8—21no—
396C4×C8—21no—
397C4×C8—81no—
398C4×C8—81no—
399C4×C8—81no—
400C4×C8C8×D411yescharacteristic
401C4×C8#44211yescharacteristic
402C4×C8#44211yescharacteristic
403C4×C8#44211yescharacteristic
404C4×C8#44411yescharacteristic
405C4×C8C2×C4×D411yescharacteristic
406C4×C8C2×AGL(1,ℤ/8)11yescharacteristic
407C4×C8C2×AGL(1,ℤ/8)11yescharacteristic
408C32—22no—
409C32—42no—
410C32—82no—
411C32C22×C1614yescomplemented
412C24—81no—
413#41—44no—
414#41—82no—
415C2×Q8—41no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—41no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—82no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—82no—
426C22×C4—81no—
427C22×C4—82no—
428C22×C4—162no—
429C22×C4—162no—
430C22×C4#158011yescharacteristic
431C22×C4AGL(1,ℤ/16)11yescharacteristic
432C22×C4AGL(1,ℤ/16)11yescharacteristic
433Q16—22no—
434Q16—41no—
435SD16—44no—
436D8—24no—
437D8—22no—
438D8—44no—
439D8—42no—
440D8—41no—
441C2×C8—22no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—42no—
447C2×C8—81no—
448C2×C8—81no—
449C2×C8—81no—
450C2×C8—82no—
451C2×C8—162no—
452C2×C8—162no—
453C2×C8—162no—
454C2×C8—162no—
455C2×C8—164no—
456C2×C8—162no—
457C2×C8#158011yescharacteristic
458C2×C8#158011yescharacteristic
459C2×C8D4×C1612yes—
460C2×C8#217112yes—
461C2×C8C2×C8×D411yescharacteristic
462C2×C8C2×#44411yescharacteristic
463C4⋊C4—42no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—82no—
471C22⋊C4—81no—
472C22⋊C4—82no—
473C22⋊C4—81no—
474C42—81no—
475C42—81no—
476C42—81no—
477C42#158011yescharacteristic
478C42AGL(1,ℤ/16)11yescharacteristic
479C42AGL(1,ℤ/16)11yescharacteristic
480C42C2×#44211yescharacteristic
481C16—21no—
482C16—21no—
483C16—41no—
484C16—41no—
485C16—82no—
486C16—82no—
487C16—162no—
488C16—162no—
489C16D4×C1612yes—
490C16C23×C1611yescharacteristic
491C16C2×#50411yescharacteristic
492C23—81no—
493C23—82no—
494C23—84no—
495C23—162no—
496C23—162no—
497C23#1206711yescharacteristic
498Q8—42no—
499Q8—81no—
500D4—44no—
501D4—42no—
502D4—84no—
503D4—82no—
504D4—81no—
505D4—84no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—84no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—162no—
514C2×C4—161no—
515C2×C4—161no—
516C2×C4—162no—
517C2×C4—162no—
518C2×C4—162no—
519C2×C4—164no—
520C2×C4#1206212yes—
521C2×C4#1206711yescharacteristic
522C2×C4#1206711yescharacteristic
523C2×C4#1206711yescharacteristic
524C2×C4#1206812yes—
525C2×C4C2×#158011yescharacteristic
526C2×C4C2×AGL(1,ℤ/16)11yescharacteristic
527C2×C4C2×AGL(1,ℤ/16)11yescharacteristic
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—161no—
535C8—161no—
536C8—162no—
537C8#1206212yes—
538C8C2×D4×C1611yescharacteristic
539C8C2×#217111yescharacteristic
540C22—21no—
541C22—21no—
542C22—82no—
543C22—84no—
544C22—168no—
545C22—161no—
546C22—162no—
547C22—162no—
548C22—161no—
549C22#12296312yes—
550C22C2×#1206711yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—82no—
555C4—161no—
556C4—161no—
557C4—162no—
558C4#12296312yes—
559C4AGL(1,ℤ/32)12yes—
560C4C2×#1206211yescharacteristic
561C4C2×#1206811yescharacteristic
562C2—22no—
563C2—162no—
564C2—164no—
565C2C2×AGL(1,ℤ/32)11yescharacteristic
566C2#7222364611yescharacteristic
567C2C2×#12296311yescharacteristic
subgroupG/H
centerC2×C4#12067
commutatorC16C23×C16
Frattini, cosocle#1311C24
socleC22C2×#12067
prime radicalC2×#2263C8
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ C2×#7053 ◁ G
LECSG ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72286838C2semidirect
2. #72286797C2semidirect
3. #72286779C2semidirect
4. #72286459C2semidirect
5. C2×#122963C2semidirect
6. #72231929C2semidirect
7. #72225141C2semidirect
8. #72224639C2semidirect
26. #122963C22semidirect
27. #122962C22semidirect
34. #93877C22semidirect
35. #93758C22semidirect
37. #93757C22semidirect
38. #93668C22semidirect
108. #2263C16semidirect
202. #508C2×C16semidirect
207. C2×D16C2×C16semidirect
251. SD64C2×C16semidirect
252. D32C2×C16semidirect
256. M64C2×C16semidirect
347. Q32C22×C16semidirect
353. D16C22×C16semidirect
411. C32C22×C16semidirect

Maximal subgroups

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

subgroupindex
#722868382
#722867972
#722867792
#722864592
C2×#1229632
#722319292
#722251412
#722246392
#722236732