Finite Group Databasebeta

← #34022734

Subgroups — #34022734

Order 18688.

1100 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626607C211yesmaximal, characteristic
2C2×C4×#11924037C211yesmaximal, characteristic
3#22626124C212yesmaximal, complemented
4#22626074C212yesmaximal, complemented
5#22625958C214yesmaximal
6C4×#14556652C214yesmaximal, complemented
7C2×#17989140C211yesmaximal, characteristic
8C22×#11924037C2212yes—
9#17989608—21no—
10#17989607—21no—
11#17989604—21no—
12#17989603—21no—
13#17989590C2212yes—
14C2×#14556652C2218yes—
15#17989557C2212yes—
16C4×#11924037—21no—
17C4×#11924037—21no—
18C4×#11924037C2212yes—
19C4×#11924037C2212yes—
20C4×#11924037C2212yes—
21C2×#14556646—21no—
22C2×C4×#10596833—21no—
23#17989467C2218yes—
24#17989140—21no—
25#17989140—21no—
26#17989140—21no—
27#17989140—21no—
28#17989140—21no—
29#17989140—21no—
30#17989140—21no—
31#17989140—21no—
32C2×#14556590C2211yescharacteristic
33C2×#14556590C2211yescharacteristic
34C2×#14556582C411yescharacteristic
35C2×#14556582C411yescharacteristic
36C2×#14556568C412yes—
37C2×#14556568C2212yes—
38C2×C4×Dic146C412yes—
39C2×C4×Dic146C2212yes—
40C2×#14556550C411yescharacteristic
41C22×C4×Dic73C2211yescharacteristic
42C2×#117×C73C411yescharacteristic
43C2×#11924037—28no—
44C2×#11924037—22no—
45C2×#11924037C2314yes—
46C2×#11924037C2314yes—
47C22×#10596833—22no—
48#14556658—22no—
49C4⋊#10596833—22no—
50#14556652—216no—
51C22⋊#10596833—22no—
52C22⋊#10596833—22no—
53#14556646—22no—
54#14556646—22no—
55#14556646—22no—
56C4×#10596833—22no—
57C4×#10596833—22no—
58C4×#10596833—22no—
59C22×Dic146C2×C412yes—
60C22×Dic146C2312yes—
61C23×Dic73C2×C411yescharacteristic
62C23×Dic73C2311yescharacteristic
63#14556590—21no—
64#14556590—21no—
65#14556590—21no—
66#14556590—21no—
67#14556590—21no—
68#14556590—21no—
69#14556590—21no—
70#14556590—21no—
71#14556582—21no—
72#14556582—21no—
73#14556582—21no—
74#14556582—21no—
75#14556582—21no—
76#14556582—21no—
77#14556582—21no—
78#14556582—21no—
79#14556568—22no—
80#14556568—22no—
81#14556568—22no—
82#14556568—22no—
83#14556568—22no—
84#14556568—22no—
85#14556568—22no—
86#14556568—22no—
87C4×Dic146—22no—
88C4×Dic146—22no—
89C4×Dic146—22no—
90C4×Dic146—22no—
91C4×Dic146—22no—
92C4×Dic146—22no—
93C4×Dic146—22no—
94C4×Dic146—22no—
95#14556550—21no—
96#14556550—21no—
97#14556550—21no—
98#14556550—21no—
99C2×C4⋊Dic73C2×C411yescharacteristic
100C2×C4⋊Dic73C2×C411yescharacteristic
101C2×C4⋊Dic73C2×C412yes—
102C2×C4⋊Dic73C2×C412yes—
103C2×#11924007C2×C412yes—
104C2×#11924007C2×C412yes—
105C2×#11924007C2×C412yes—
106C2×#11924007C2312yes—
107C2×C22⋊Dic73C2×C412yes—
108C2×C4×Dic73C2×C412yes—
109C2×C4×Dic73C2×C411yescharacteristic
110C2×C4×Dic73C2×C411yescharacteristic
111C2×C4×Dic73D411yescharacteristic
112C2×C4×Dic73D411yescharacteristic
113C2×C4×Dic73D411yescharacteristic
114C2×C4×Dic73D411yescharacteristic
115C2×C4×Dic73D411yescharacteristic
116C2×C4×Dic73D411yescharacteristic
117C2×C4×Dic73D411yescharacteristic
118C2×C4×Dic73D411yescharacteristic
119C2×C4×Dic73C2311yescharacteristic
120C2×C4×Dic73C2311yescharacteristic
121C23×C4×C73C2×C411yescharacteristic
122#117×C73—21no—
123#117×C73—21no—
124#117×C73—21no—
125#117×C73—21no—
126C2×C4⋊C4×C73C2×C411yescharacteristic
127C2×C4⋊C4×C73C2×C411yescharacteristic
128C2×C22⋊C4×C73C2×C412yes—
129C2×C42×C73C2×C412yes—
130#11924037—28no—
131#11924037—24no—
132#11924037—28no—
133#11924037#4118yes—
134C2×#10596833—28no—
135C2×#10596833—24no—
136C2×#10596833—24no—
137C2×Dic146—22no—
138C2×Dic146—22no—
139C2×Dic146—22no—
140C2×Dic146—22no—
141C2×Dic146—28no—
142C2×Dic146—44no—
143C22×Dic73—24no—
144C22×Dic73—42no—
145C22×Dic73C22⋊C411yescharacteristic
146C22×Dic73C22⋊C411yescharacteristic
147C22×Dic73C22⋊C411yescharacteristic
148C22×Dic73C22⋊C411yescharacteristic
149C22×Dic73C22×C412yes—
150C22×Dic73C22×C411yescharacteristic
151C22×Dic73C22×C411yescharacteristic
152C22×Dic73C22×C412yes—
153C22×Dic73C22×C411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155C22×Dic73C2×D411yescharacteristic
156C22×Dic73C2×D411yescharacteristic
157C22×Dic73C2×D411yescharacteristic
158C22×Dic73C2411yescharacteristic
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—22no—
164C4⋊Dic73—22no—
165C4⋊Dic73—22no—
166C4⋊Dic73—22no—
167#11924007—22no—
168#11924007—22no—
169#11924007—22no—
170#11924007—22no—
171#11924007—22no—
172#11924007—22no—
173#11924007—22no—
174#11924007—22no—
175C22⋊Dic73—22no—
176C22⋊Dic73—22no—
177C4×Dic73—22no—
178C4×Dic73—22no—
179C4×Dic73C22⋊C411yescharacteristic
180C4×Dic73C22⋊C411yescharacteristic
181C4×Dic73C22⋊C411yescharacteristic
182C4×Dic73C22⋊C411yescharacteristic
183C4×Dic73C22⋊C411yescharacteristic
184C4×Dic73C22⋊C411yescharacteristic
185C4×Dic73C22⋊C411yescharacteristic
186C4×Dic73C22⋊C411yescharacteristic
187C4×Dic73C2×D411yescharacteristic
188C4×Dic73C2×D411yescharacteristic
189C4×Dic73C2×D411yescharacteristic
190C4×Dic73C2×D411yescharacteristic
191C4×Dic73C2×D411yescharacteristic
192C4×Dic73C2×D411yescharacteristic
193C4×Dic73C2×D411yescharacteristic
194C4×Dic73C2×D411yescharacteristic
195C24×C73C22×C411yescharacteristic
196C22×C4×C73—24no—
197C22×C4×C73C22⋊C411yescharacteristic
198C22×C4×C73C22⋊C411yescharacteristic
199C22×C4×C73C22⋊C411yescharacteristic
200C22×C4×C73C22⋊C411yescharacteristic
201C22×C4×C73C22×C412yes—
202C22×C4×C73C22×C411yescharacteristic
203C22×C4×C73C22×C411yescharacteristic
204C22×C4×C73C22×C412yes—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C22⋊C4×C73—22no—
210C22⋊C4×C73—22no—
211C42×C73—22no—
212C42×C73—22no—
213C23×C73—22no—
214C23×C73—22no—
215C23×C73—41no—
216C23×C73C2×C22⋊C411yescharacteristic
217C23×C73C2×C22⋊C411yescharacteristic
218C23×C73C23×C411yescharacteristic
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—24no—
227C2×C4×C73—42no—
228C2×C4×C73#10114yescomplemented
229C2×C4×C73C2×C22⋊C411yescharacteristic
230C2×C4×C73C2×C22⋊C411yescharacteristic
231C2×C4×C73C2×C22⋊C411yescharacteristic
232C2×C4×C73C2×C22⋊C411yescharacteristic
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—24no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—42no—
248C2×Dic73C2×C22⋊C411yescharacteristic
249C2×Dic73C2×C22⋊C411yescharacteristic
250C2×Dic73C2×C22⋊C411yescharacteristic
251C2×Dic73C2×C22⋊C411yescharacteristic
252C2×Dic73C2×C22⋊C411yescharacteristic
253C2×Dic73C2×C22⋊C411yescharacteristic
254C2×Dic73C22×D411yescharacteristic
255C2×Dic73C22×D411yescharacteristic
256C2×Dic73C2×#4114yes—
257Dic146—44no—
258Dic146—44no—
259Dic146—44no—
260Dic146—44no—
261#10596833—28no—
262#10596833—44no—
263C22×C73—21no—
264C22×C73—21no—
265C22×C73—21no—
266C22×C73—21no—
267C22×C73—21no—
268C22×C73—21no—
269C22×C73—21no—
270C22×C73—44no—
271C22×C73C2×#10112yes—
272C22×C73C22×C22⋊C411yescharacteristic
273C22×C73C4×#4112yes—
274C4×C73—21no—
275C4×C73—21no—
276C4×C73—42no—
277C4×C73—42no—
278C4×C73C2×#10112yes—
279C4×C73C2×#10112yes—
280Dic73—21no—
281Dic73—21no—
282Dic73—21no—
283Dic73—21no—
284Dic73—21no—
285Dic73—42no—
286Dic73—42no—
287Dic73#41312yes—
288Dic73#52512yes—
289Dic73#53212yes—
290#21425—731nomaximal, Sylow
291C2×C73—22no—
292C2×C73—22no—
293C2×C73—41no—
294C2×C73#229711yescharacteristic
295C2×C73C22×#10111yescharacteristic
296C2×C73C2×#41311yescharacteristic
297C2×#533—731no—
298#2920—731no—
299C2×C4×M16—731no—
300#1997—732no—
301#1966—732no—
302#1901—734no—
303C4×#101—734no—
304C73#2142511yescharacteristic, complemented, Sylow
305C22×M16—732no—
306#533—1461no—
307#533—1461no—
308#533—1461no—
309#533—1461no—
310#533—1461no—
311#533—1461no—
312#533—1461no—
313#533—1461no—
314C2×C4⋊Q8—731no—
315C2×C4⋊Q8—731no—
316C2×#125—731no—
317C2×#125—731no—
318C2×C22⋊Q8—732no—
319C2×C22⋊Q8—732no—
320C2×C4×Q8—732no—
321C2×C4×Q8—732no—
322C2×#117—731no—
323C2×#117#6440811yescharacteristic
324C22×C42—731no—
325#432—1461no—
326#432—1461no—
327#431—1461no—
328#431—1461no—
329#423—732no—
330C2×#101—738no—
331#407—732no—
332C4×M16—732no—
333C4×M16—732no—
334C4×M16—732no—
335C4×M16—1461no—
336C4×M16—1461no—
337C2×#97—1461no—
338C2×C4×C8—1461no—
339#343—738no—
340C22×Q8—732no—
341C22×Q8—732no—
342C23×C4—731no—
343C23×C4—731no—
344C23×C4C2×#6440811yescharacteristic
345C2×M16—734no—
346C2×M16—734no—
347C2×M16—1468no—
348C2×M16—1462no—
349C22×C8—1462no—
350C4⋊Q8—1461no—
351C4⋊Q8—1461no—
352C4⋊Q8—1461no—
353C4⋊Q8—1461no—
354C4⋊Q8—1461no—
355C4⋊Q8—1461no—
356C4⋊Q8—1461no—
357C4⋊Q8—1461no—
358#125—1461no—
359#125—1461no—
360#125—1461no—
361#125—1461no—
362#125—1461no—
363#125—1461no—
364#125—1461no—
365#125—1461no—
366C22⋊Q8—1462no—
367C22⋊Q8—1462no—
368C22⋊Q8—1462no—
369C22⋊Q8—1462no—
370C22⋊Q8—1462no—
371C22⋊Q8—1462no—
372C22⋊Q8—1462no—
373C22⋊Q8—1462no—
374C4×Q8—1462no—
375C4×Q8—1462no—
376C4×Q8—1462no—
377C4×Q8—1462no—
378C4×Q8—1462no—
379C4×Q8—1462no—
380C4×Q8—1462no—
381C4×Q8—1462no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—1461no—
387#117—1461no—
388#117—1461no—
389#117—1461no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—732no—
393C2×C4⋊C4—732no—
394C2×C4⋊C4—732no—
395C2×C4⋊C4—732no—
396C2×C4⋊C4—732no—
397C2×C4⋊C4—732no—
398C2×C4⋊C4C2×#6440811yescharacteristic
399C2×C4⋊C4C2×#6440811yescharacteristic
400C2×C22⋊C4—732no—
401C2×C22⋊C4C2×#6440812yes—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—731no—
407C2×C42—731no—
408C2×C42—731no—
409C2×C42—731no—
410C2×C42—731no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—731no—
414C2×C42—732no—
415C2×C42C2×#6440812yes—
416C4⋊C8—1462no—
417C4⋊C8—1462no—
418#101—14616no—
419C22⋊C8—1462no—
420C22⋊C8—1462no—
421#97—1462no—
422#97—1462no—
423#97—1462no—
424C4×C8—1462no—
425C4×C8—1462no—
426C4×C8—1462no—
427C24C22×#6440811yescharacteristic
428C2×Q8—2924no—
429C2×Q8—1468no—
430C2×Q8—1462no—
431C2×Q8—1462no—
432C2×Q8—1462no—
433C2×Q8—1462no—
434C22×C4—24no—
435C22×C4—2922no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—732no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—731no—
448C22×C4—731no—
449C22×C4—732no—
450C22×C4—1464no—
451C22×C4C22⋊#6440811yescharacteristic
452C22×C4C22⋊#6440811yescharacteristic
453C22×C4C22⋊#6440811yescharacteristic
454C22×C4C22⋊#6440811yescharacteristic
455C22×C4C22×#6440812yes—
456C22×C4C22×#6440811yescharacteristic
457C22×C4C22×#6440811yescharacteristic
458C22×C4C22×#6440812yes—
459M16—738no—
460M16—1468no—
461M16—1464no—
462M16—1468no—
463C2×C8—1464no—
464C2×C8—1464no—
465C2×C8—1468no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1462no—
471C4⋊C4—1462no—
472C4⋊C4—1462no—
473C4⋊C4—1462no—
474C4⋊C4—1462no—
475C4⋊C4—1462no—
476C4⋊C4—1462no—
477C4⋊C4—1462no—
478C4⋊C4—1462no—
479C4⋊C4—1462no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—1462no—
489C22⋊C4—1462no—
490C42—22no—
491C42—22no—
492C42—731no—
493C42—731no—
494C42—731no—
495C42—731no—
496C42—731no—
497C42—731no—
498C42—731no—
499C42—731no—
500C42—731no—
501C42—731no—
502C42—731no—
503C42—731no—
504C42—731no—
505C42—731no—
506C42—731no—
507C42—731no—
508C42—1462no—
509C42—1462no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23C2×C22⋊#6440811yescharacteristic
514C23C2×C22⋊#6440811yescharacteristic
515C23C23×#6440811yescharacteristic
516Q8—2924no—
517Q8—2924no—
518Q8—2924no—
519Q8—2924no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—2922no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—731no—
534C2×C4—731no—
535C2×C4—731no—
536C2×C4—734no—
537C2×C4—731no—
538C2×C4—731no—
539C2×C4—731no—
540C2×C4—731no—
541C2×C4—731no—
542C2×C4—1462no—
543C2×C4—1462no—
544C2×C4—1464no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4—1462no—
553C2×C4#1455665214yescomplemented
554C2×C4C2×C22⋊#6440811yescharacteristic
555C2×C4C2×C22⋊#6440811yescharacteristic
556C2×C4C2×C22⋊#6440811yescharacteristic
557C2×C4C2×C22⋊#6440811yescharacteristic
558C8—2924no—
559C8—1468no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—44no—
568C22C2×#1455665212yes—
569C22C22×C22⋊#6440811yescharacteristic
570C22#41×#6440812yes—
571C4—21no—
572C4—21no—
573C4—42no—
574C4—42no—
575C4—2922no—
576C4—2922no—
577C4—732no—
578C4—732no—
579C4—732no—
580C4—1461no—
581C4—1461no—
582C4—1461no—
583C4—1461no—
584C4—1461no—
585C4C2×#1455665212yes—
586C4C2×#1455665212yes—
587C2—22no—
588C2—22no—
589C2—41no—
590C2#2262639211yescharacteristic
591C2C22×#1455665211yescharacteristic
592C2D73⋊#41311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#117×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenterC2×#117#64408
hypocenterC73#21425
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #22626124C2semidirect
4. #22626074C2semidirect
6. C4×#14556652C2semidirect
228. C2×C4×C73#101semidirect
304. C73#21425semidirect
553. C2×C4#14556652semidirect

Maximal subgroups

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

subgroupindex
#226266072
C2×C4×#119240372
#226261242
#226260742
#226259582
C4×#145566522
C2×#179891402
#2142573