Finite Group Databasebeta

← #33098071

Subgroups — #33098071

Order 18688.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×Dic73C211yesmaximal, characteristic, complemented
2#22612396C211yesmaximal, characteristic
3#22610889C211yesmaximal, characteristic, complemented
4#22610701C211yesmaximal, characteristic, complemented
5#22610665C211yesmaximal, characteristic, complemented
6#22610660C211yesmaximal, characteristic, complemented
7#22610616C211yesmaximal, characteristic
8#22610616C211yesmaximal, characteristic
9#22610560C211yesmaximal, characteristic
10#22610553C211yesmaximal, characteristic
11#22610525C211yesmaximal, characteristic, complemented
12#22610518C211yesmaximal, characteristic, complemented
13#22609941C211yesmaximal, characteristic
14C4⋊C8×Dic73C211yesmaximal, characteristic, complemented
15C2×#478×C73C211yesmaximal, characteristic
16C2×#14556550—21no—
17C2×#14556525C2211yescharacteristic
18C2×#14556523C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556512C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×C8×Dic73C2211yescharacteristic
23C2×C8×Dic73C2211yescharacteristic
24#17988490C2211yescharacteristic
25#17988490C2211yescharacteristic
26#17988486C2211yescharacteristic
27#17988486C2211yescharacteristic
28#17988486C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988480—21no—
31#17988480—21no—
32#17988473—21no—
33#17988473—21no—
34#17988453C2211yescharacteristic
35C4⋊C4×Dic73—21no—
36C4⋊C4×Dic73—21no—
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C4×C22⋊Dic73—21no—
41C42×Dic73C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#125×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54#478×C73C418yescomplemented
55C2×#107×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×C4⋊C8×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59C2×#102×C73C2211yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556550—44no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C22⋊Dic73—21no—
72C2×C22⋊Dic73—21no—
73C2×C22⋊Dic73—21no—
74C2×C22⋊Dic73—21no—
75C2×C4×Dic73—21no—
76C2×C4×Dic73—21no—
77C2×C4×Dic73—21no—
78C2×C4×Dic73D411yescharacteristic
79C2×C4×Dic73D411yescharacteristic
80C2×C4×Dic73D411yescharacteristic
81C2×C4×Dic73D411yescharacteristic
82C2×C4×Dic73C2311yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84#14556525—41no—
85#14556523—41no—
86C4⋊#10596831—22no—
87#14556512—41no—
88#14556512—41no—
89C8×Dic73—41no—
90C8×Dic73—41no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100C22×D4×C73C2311yescharacteristic
101C23×C4×C73—21no—
102C22×C8×C73C2311yescharacteristic
103C22×C8×C73C2311yescharacteristic
104#125×C73—22no—
105#125×C73C2×C414yes—
106C4×D4×C73—22no—
107C4×D4×C73C2×C414yes—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C42×C73C2311yescharacteristic
115#107×C73C2×C414yes—
116#106×C73C2×C414yes—
117C4⋊C8×C73C2×C414yes—
118#102×C73C2×C414yes—
119#102×C73C2×C414yes—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—42no—
143C4⋊Dic73—42no—
144C4⋊Dic73—42no—
145C4⋊Dic73—42no—
146C22⋊Dic73—42no—
147C22⋊Dic73—42no—
148C22⋊Dic73—42no—
149C22⋊Dic73—42no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73—21no—
153C4×Dic73—21no—
154C4×Dic73—22no—
155C4×Dic73—22no—
156C4×Dic73—22no—
157C4×Dic73—41no—
158C4×Dic73—42no—
159C4×Dic73—42no—
160C4×Dic73—42no—
161C24×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73C22×C412yes—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—24no—
172C22×C4×C73—42no—
173C22×C4×C73—42no—
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2411yescharacteristic
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73C22×C412yes—
182C2×C8×C73C22×C412yes—
183C4⋊C4×C73—24no—
184C4⋊C4×C73—24no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73C22×C412yes—
189C4⋊C4×C73C22×C412yes—
190C4⋊C4×C73C22×C412yes—
191C22⋊C4×C73—24no—
192C42×C73C22×C412yes—
193C42×C73#4112yes—
194C23×C73—22no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73C22×D411yescharacteristic
202D4×C73—22no—
203D4×C73—22no—
204D4×C73—41no—
205D4×C73—41no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—48no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73C4×D412yes—
221C2×C4×C73C4×D412yes—
222C2×C4×C73C23×C411yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73H(5,𝔽2)11yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×C4×C732−1+411yescharacteristic
227C2×C4×C732−1+411yescharacteristic
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73#13512yes—
250C2×Dic73#13512yes—
251C2×Dic73AGL(1,ℤ/8)12yes—
252C8×C73—22no—
253C8×C73—22no—
254#10596831—41no—
255#10596831—41no—
256C22×C73—42no—
257C22×C73—42no—
258C22×C73—42no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73C2×C4×D411yescharacteristic
264C22×C73#53611yescharacteristic
265C22×C73#53711yescharacteristic
266C22×C73C2×#13511yescharacteristic
267C22×C73C2×#13511yescharacteristic
268C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C73#57511yescharacteristic
270C4×C73—22no—
271C4×C73—22no—
272C4×C73—42no—
273C4×C73—42no—
274C4×C73—42no—
275C4×C73#51911yescharacteristic
276C4×C73#52011yescharacteristic
277C4×C73#54111yescharacteristic
278C4×C73#54211yescharacteristic
279Dic73—22no—
280Dic73—22no—
281Dic73—22no—
282Dic73—41no—
283Dic73—81no—
284Dic73—81no—
285Dic73—81no—
286Dic73—81no—
287#22114—731nomaximal, Sylow
288C2×C73—82no—
289C2×C73#232311yescharacteristic
290C2×C73C4×#13511yescharacteristic
291C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
292C2×C73#322811yescharacteristic
293C2×C73#323211yescharacteristic
294C2×C73#324711yescharacteristic
295C2×C73#325011yescharacteristic
296C2×#478D7311yescharacteristic
297C4×#125—731no—
298#2293—731no—
299#1989—731no—
300#1949—731no—
301#1942—731no—
302#1941—731no—
303#1934—731no—
304#1934—731no—
305#1926—731no—
306#1925—731no—
307#1921—731no—
308#1920—731no—
309#1804—731no—
310C4×C4⋊C8—731no—
311C73#2211411yescharacteristic, complemented, Sylow
312C2×#125C2×D7311yescharacteristic
313C2×C4×D4C2×D7311yescharacteristic
314C2×#117—1461no—
315#478Dic7318yescomplemented
316C2×#107—731no—
317C2×#107C2×D7311yescharacteristic
318C2×#106—731no—
319C2×#106C2×D7311yescharacteristic
320C2×C4⋊C8—731no—
321C2×C4⋊C8C2×D7311yescharacteristic
322C2×#102—731no—
323C2×#102—731no—
324C2×#102C2×D7311yescharacteristic
325C2×#102C2×D7311yescharacteristic
326C2×C4×C8—731no—
327C2×C4×C8—731no—
328#390—731no—
329#390—731no—
330#389—731no—
331#389—731no—
332#389—731no—
333#389—731no—
334#388—1461no—
335#388—1461no—
336#387—1461no—
337#387—1461no—
338#382—731no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—1461no—
343C4×C4⋊C4—1461no—
344C4×C22⋊C4—1461no—
345C43—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#336—731no—
355#336—731no—
356C22×D4C22×D7311yescharacteristic
357C23×C4—21no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362#125—22no—
363#125C2×Dic7314yes—
364C4×D4—22no—
365C4×D4C2×Dic7314yes—
366#117—2924no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C42—731no—
386C2×C42—731no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—731no—
390C2×C42—731no—
391C2×C42—1461no—
392C2×C42—1461no—
393C2×C42—1461no—
394C2×C42C22×D7311yescharacteristic
395#107—2921no—
396#107C2×Dic7314yes—
397#106—2921no—
398#106C2×Dic7314yes—
399C4⋊C8—1462no—
400C4⋊C8C2×Dic7314yes—
401#102—2921no—
402#102—2921no—
403#102C2×Dic7314yes—
404#102C2×Dic7314yes—
405C4×C8—2921no—
406C4×C8—2921no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95—1461no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C22×Dic7312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4D4×D7311yescharacteristic
438C22×C4D4×D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8C22×Dic7312yes—
451C2×C8C22×Dic7312yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—2922no—
458C4⋊C4—2922no—
459C4⋊C4—2922no—
460C4⋊C4—2922no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4C22×Dic7312yes—
470C4⋊C4C22×Dic7312yes—
471C4⋊C4C22×Dic7312yes—
472C22⋊C4—24no—
473C22⋊C4—2922no—
474C22⋊C4—2922no—
475C22⋊C4—2922no—
476C22⋊C4—2922no—
477C42—2922no—
478C42—2921no—
479C42—2922no—
480C42—2922no—
481C42—1462no—
482C42—1462no—
483C42—1462no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42C22×Dic7312yes—
489C42#1192402712yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D7311yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—48no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—732no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4D4×Dic7312yes—
541C2×C4D4×Dic7312yes—
542C2×C4C23×Dic7311yescharacteristic
543C2×C4C2×#1192402711yescharacteristic
544C2×C4#1455663711yescharacteristic
545C2×C4#1455663811yescharacteristic
546C2×C4#1455663911yescharacteristic
547C2×C4#1455663911yescharacteristic
548C8—22no—
549C8—22no—
550C8—2921no—
551C8—2921no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×D4×Dic7311yescharacteristic
560C22#1798916011yescharacteristic
561C22#1798916711yescharacteristic
562C22#135×D7311yescharacteristic
563C22#135×D7311yescharacteristic
564C22AGL(1,ℤ/8)×D7311yescharacteristic
565C22#1798938311yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—2921no—
572C4—5841no—
573C4—5841no—
574C4—5841no—
575C4—5841no—
576C4—1462no—
577C4—1462no—
578C4—1462no—
579C4#1798906511yescharacteristic
580C4#1798907011yescharacteristic
581C4#1798919711yescharacteristic
582C4#1798921011yescharacteristic
583C2—82no—
584C2#2261261511yescharacteristic
585C2#135×Dic7311yescharacteristic
586C2AGL(1,ℤ/8)×Dic7311yescharacteristic
587C2#2262106611yescharacteristic
588C2#2262109711yescharacteristic
589C2#2262125611yescharacteristic
590C2#2262128211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#478×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×#478D73
hypocenterC73#22114
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #125×Dic73C2semidirect
3. #22610889C2semidirect
4. #22610701C2semidirect
5. #22610665C2semidirect
6. #22610660C2semidirect
11. #22610525C2semidirect
12. #22610518C2semidirect
14. C4⋊C8×Dic73C2semidirect
54. #478×C73C4semidirect
311. C73#22114semidirect
315. #478Dic73semidirect

Maximal subgroups

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

subgroupindex
#125×Dic732
#226123962
#226108892
#226107012
#226106652
#226106602
#226106162
#226105602
#226105532
#226105252
#226105182
#226099412
C4⋊C8×Dic732
C2×#478×C732
#2211473