Finite Group Databasebeta

← #33103731

Subgroups — #33103731

Order 18688.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612824C211yesmaximal, characteristic
2#22612800C211yesmaximal, characteristic
3#22610670C211yesmaximal, characteristic, complemented
4#22610665C211yesmaximal, characteristic, complemented
5#22610525C211yesmaximal, characteristic, complemented
6#22610518C211yesmaximal, characteristic, complemented
7#22610511C211yesmaximal, characteristic, complemented
8#22610504C211yesmaximal, characteristic, complemented
9#22610026C211yesmaximal, characteristic
10#22610021C211yesmaximal, characteristic
11#22609964C211yesmaximal, characteristic
12#22609831C211yesmaximal, characteristic, complemented
13#106×Dic73C211yesmaximal, characteristic, complemented
14#22609712C211yesmaximal, characteristic
15C2×#428×C73C211yesmaximal, characteristic
16C2×#14556550—21no—
17C2×#14556525C2211yescharacteristic
18C2×#14556523C2211yescharacteristic
19C2×C22⋊#10596831C2211yescharacteristic
20C2×#14556501C2211yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22#17988486C2211yescharacteristic
23#17988486C2211yescharacteristic
24#17988486C2211yescharacteristic
25#17988486C2211yescharacteristic
26#17988480—21no—
27#17988480—21no—
28#17988473—21no—
29#17988473—21no—
30#17988455—21no—
31#17988450—21no—
32#17988447C2211yescharacteristic
33#17988447C2211yescharacteristic
34C4⋊C4×Dic73C2211yescharacteristic
35C4⋊C4×Dic73C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4×C4⋊Dic73—21no—
39C4×C22⋊Dic73C2211yescharacteristic
40#17988350C2211yescharacteristic
41#17988350C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C22×M16×C73C2211yescharacteristic
53C2×#117×C73C2211yescharacteristic
54C22×C4⋊C4×C73C2211yescharacteristic
55#428×C73C418yescomplemented
56C2×#107×C73C2211yescharacteristic
57C2×#107×C73C2211yescharacteristic
58C2×#106×C73C2211yescharacteristic
59C2×#106×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⋊Dic73D411yescharacteristic
74C2×C22⋊Dic73D411yescharacteristic
75C2×C4×Dic73—21no—
76C2×C4×Dic73—21no—
77C2×C4×Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73C2311yescharacteristic
80C2×C4×Dic73C2311yescharacteristic
81#14556525—22no—
82#14556523—22no—
83C22⋊#10596831—22no—
84#14556501—22no—
85C8×Dic73—22no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498D411yescharacteristic
97#14556498D411yescharacteristic
98C23×C4×C73—21no—
99C23×C4×C73C2311yescharacteristic
100C2×M16×C73—22no—
101C2×M16×C73—22no—
102C2×M16×C73C2×C414yes—
103C22×C8×C73C2311yescharacteristic
104C22×C8×C73C2311yescharacteristic
105#117×C73—22no—
106#117×C73C2×C414yes—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—22no—
109C2×C4⋊C4×C73C2×C414yes—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C22⋊C4×C73—21no—
115C2×C42×C73—21no—
116#107×C73C2×C414yes—
117#107×C73C2×C414yes—
118#106×C73C2×C414yes—
119#106×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—22no—
147C22⋊Dic73—41no—
148C22⋊Dic73—42no—
149C22⋊Dic73—81no—
150C4×Dic73—21no—
151C4×Dic73—21no—
152C4×Dic73—21no—
153C4×Dic73—21no—
154C4×Dic73—42no—
155C4×Dic73—42no—
156C4×Dic73—81no—
157C4×Dic73—81no—
158C24×C73C2×D411yescharacteristic
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—24no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—42no—
169C22×C4×C73—42no—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73#4112yes—
173C22×C4×C73C2411yescharacteristic
174M16×C73—28no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73C22×C412yes—
178C2×C8×C73C22×C412yes—
179C2×C8×C73#4112yes—
180C2×C8×C73#4112yes—
181C4⋊C4×C73—24no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73C22×C412yes—
187C4⋊C4×C73C22×C412yes—
188C4⋊C4×C73C22×C412yes—
189C4⋊C4×C73C22×C412yes—
190C22⋊C4×C73—24no—
191C42×C73—24no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73C4×D412yes—
199C23×C73C22×D411yescharacteristic
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73—48no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73C4×D412yes—
220C2×C4×C73C23×C411yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×Dic73—21no—
227C2×Dic73—21no—
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×Dic73AGL(1,ℤ/8)12yes—
250C8×C73—22no—
251C8×C73#11714yescomplemented
252#10596831—41no—
253#10596831—41no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73C2×C4×D411yescharacteristic
262C22×C73#54711yescharacteristic
263C22×C73#54711yescharacteristic
264C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C73#57511yescharacteristic
266C22×C73#57711yescharacteristic
267C22×C73#57811yescharacteristic
268C4×C73—22no—
269C4×C73—42no—
270C4×C73—42no—
271C4×C73—42no—
272C4×C73—42no—
273C4×C73C2×#11711yescharacteristic
274C4×C73#51911yescharacteristic
275C4×C73#55211yescharacteristic
276C4×C73#55211yescharacteristic
277Dic73—22no—
278Dic73—41no—
279Dic73—41no—
280Dic73—41no—
281Dic73—81no—
282Dic73—81no—
283Dic73—81no—
284Dic73—81no—
285#22622—731nomaximal, Sylow
286C2×C73—42no—
287C2×C73#234411yescharacteristic
288C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
289C2×C73#294711yescharacteristic
290C2×C73#334511yescharacteristic
291C2×C73#334911yescharacteristic
292C2×C73#335111yescharacteristic
293C2×C73#335311yescharacteristic
294C2×#428D7311yescharacteristic
295#2347—731no—
296#2344—731no—
297#1943—731no—
298#1942—731no—
299#1921—731no—
300#1920—731no—
301#1919—731no—
302#1918—731no—
303#1825—731no—
304#1824—731no—
305#1809—731no—
306#1776—731no—
307C4×#106—731no—
308#1753—731no—
309C73#2262211yescharacteristic, complemented, Sylow
310C22×M16C2×D7311yescharacteristic
311C2×#117—1461no—
312C2×#117C2×D7311yescharacteristic
313C22×C4⋊C4C2×D7311yescharacteristic
314#428Dic7318yescomplemented
315C2×#107—731no—
316C2×#107C2×D7311yescharacteristic
317C2×#107C2×D7311yescharacteristic
318C2×#106—731no—
319C2×#106C2×D7311yescharacteristic
320C2×#106C2×D7311yescharacteristic
321C2×C22⋊C8—731no—
322C2×#97—731no—
323C2×C4×C8—731no—
324#389—731no—
325#389—731no—
326#389—731no—
327#389—731no—
328#388—1461no—
329#388—1461no—
330#387—1461no—
331#387—1461no—
332#383—1461no—
333#381—1461no—
334#380—731no—
335#380—731no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4—731no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—1461no—
341C4×C22⋊C4—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#336—731no—
353#336—731no—
354C23×C4—21no—
355C23×C4C22×D7311yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C2×Dic7314yes—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8C22×D7311yescharacteristic
362C22×C8C22×D7311yescharacteristic
363#117—22no—
364#117—2924no—
365#117C2×Dic7314yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4C2×Dic7314yes—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—731no—
383C2×C22⋊C4—731no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C42—21no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42—1461no—
392C2×C42—1461no—
393#107—1462no—
394#107C2×Dic7314yes—
395#107C2×Dic7314yes—
396#106—1462no—
397#106C2×Dic7314yes—
398#106C2×Dic7314yes—
399C22⋊C8—1462no—
400#97—1462no—
401C4×C8—1462no—
402#95—731no—
403#95—731no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414C24D4×D7311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4C22×Dic7312yes—
435C22×C4D4×D7311yescharacteristic
436C22×C4#1192402712yes—
437C22×C4C23×D7311yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—1461no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8C22×Dic7312yes—
448C2×C8C22×Dic7312yes—
449C2×C8#1192402812yes—
450C2×C8#1192402812yes—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—2922no—
457C4⋊C4—2922no—
458C4⋊C4—2922no—
459C4⋊C4—2922no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4C22×Dic7312yes—
469C4⋊C4C22×Dic7312yes—
470C4⋊C4C22×Dic7312yes—
471C4⋊C4C22×Dic7312yes—
472C22⋊C4—24no—
473C22⋊C4—2922no—
474C22⋊C4—2921no—
475C22⋊C4—5841no—
476C22⋊C4—1462no—
477C42—24no—
478C42—2922no—
479C42—2922no—
480C42—5841no—
481C42—5841no—
482C42—1461no—
483C42—1461no—
484C42—1461no—
485C42—1461no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23D4×Dic7312yes—
493C23C2×D4×D7311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
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—732no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4D4×Dic7312yes—
538C2×C4C23×Dic7311yescharacteristic
539C2×C4C2×#1192402711yescharacteristic
540C2×C4C2×#1192402811yescharacteristic
541C2×C4C2×#1192402811yescharacteristic
542C2×C4#1455663911yescharacteristic
543C2×C4#1455663911yescharacteristic
544C8—22no—
545C8—2921no—
546C8—2921no—
547C8#1455655014yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×D4×Dic7311yescharacteristic
556C22#1798923811yescharacteristic
557C22#1798923811yescharacteristic
558C22AGL(1,ℤ/8)×D7311yescharacteristic
559C22#1798938311yescharacteristic
560C22#1798939611yescharacteristic
561C22#1798940311yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—2921no—
568C4—2921no—
569C4—2921no—
570C4—5841no—
571C4—5841no—
572C4—5841no—
573C4—5841no—
574C4—1462no—
575C4C2×#1455655011yescharacteristic
576C4#1798906511yescharacteristic
577C4#1798927211yescharacteristic
578C4#1798927211yescharacteristic
579C2—42no—
580C2#2261280011yescharacteristic
581C2AGL(1,ℤ/8)×Dic7311yescharacteristic
582C2#2261877511yescharacteristic
583C2#2262240211yescharacteristic
584C2#2262246211yescharacteristic
585C2#2262248711yescharacteristic
586C2#2262251111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#428×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterC2×#428D73
hypocenterC73#22622
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #22610670C2semidirect
4. #22610665C2semidirect
5. #22610525C2semidirect
6. #22610518C2semidirect
7. #22610511C2semidirect
8. #22610504C2semidirect
12. #22609831C2semidirect
13. #106×Dic73C2semidirect
55. #428×C73C4semidirect
251. C8×C73#117semidirect
309. C73#22622semidirect
314. #428Dic73semidirect
547. C8#14556550semidirect

Maximal subgroups

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

subgroupindex
#226128242
#226128002
#226106702
#226106652
#226105252
#226105182
#226105112
#226105042
#226100262
#226100212
#226099642
#226098312
#106×Dic732
#226097122
C2×#428×C732
#2262273