Finite Group Databasebeta

← #31912446

Subgroups — #31912446

Order 18176.

752 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985983C211yesmaximal, characteristic
2#22596785C211yesmaximal, characteristic
3#22586468C214yesmaximal, complemented
4C2×#410×C71C211yesmaximal, characteristic
5C22×#11765849—21no—
6C22×#11765849C411yescharacteristic, complemented
7C22×#11765849C411yescharacteristic, complemented
8#17985983—21no—
9#17985983—21no—
10#17985983C412yescomplemented
11#17985983C2212yes—
12C2×#14556047C412yescomplemented
13C2×#14556045—21no—
14#17985966—21no—
15#17985966—21no—
16#17985966C412yescomplemented
17#17985966C2212yes—
18#17985963—21no—
19#17985963—21no—
20C22×#41×C71C2211yescharacteristic
21C2×#117×C71—21no—
22#410×C71—21no—
23#410×C71—21no—
24#410×C71C412yes—
25#410×C71C2212yes—
26C2×#99×C71—21no—
27C2×#99×C71C412yes—
28C2×#11765849—21no—
29C2×#11765849—21no—
30C2×#11765849—21no—
31C2×#11765849—22no—
32C2×#11765849—22no—
33C2×#11765849—22no—
34C2×#11765849—41no—
35C2×#11765849—41no—
36C2×#11765849—41no—
37C2×#11765849—41no—
38C2×#11765849—41no—
39C2×#11765849—41no—
40C2×#11765849—41no—
41C2×#11765849C2×C412yescomplemented
42C22×C71⋊C8—21no—
43C22×C71⋊C8—41no—
44C22×C71⋊C8C2×C412yescomplemented
45#14556047—22no—
46#14556047C2×C414yescomplemented
47#14556045—22no—
48#14556045—22no—
49C22⋊(C71⋊C8)—22no—
50C22⋊(C71⋊C8)—22no—
51C22⋊(C71⋊C8)—22no—
52C22⋊(C71⋊C8)—22no—
53C2×#41×C71—41no—
54C2×#41×C71—41no—
55C2×#41×C71—41no—
56C2×#41×C71—41no—
57C2×#41×C71C2×C411yescharacteristic
58C2×#41×C71C2×C411yescharacteristic
59C2×#41×C71C2×C411yescharacteristic
60C2×#41×C71D411yescharacteristic
61C2×#41×C71D411yescharacteristic
62C2×#41×C71D411yescharacteristic
63C2×#41×C71Q811yescharacteristic
64C2×#41×C71C2311yescharacteristic
65C22×Q8×C71C2×C411yescharacteristic
66C22×D4×C71C2×C411yescharacteristic
67C22×D4×C71D411yescharacteristic
68C22×D4×C71Q811yescharacteristic
69C23×C4×C71C2×C411yescharacteristic
70C23×C4×C71D411yescharacteristic
71C23×C4×C71D411yescharacteristic
72#117×C71—22no—
73#117×C71—22no—
74#117×C71—41no—
75#117×C71—41no—
76C2×C4⋊C4×C71—41no—
77C2×C22⋊C4×C71—22no—
78C2×C42×C71—41no—
79#99×C71—22no—
80#99×C71—22no—
81#99×C71—22no—
82#99×C71C2×C414yes—
83#11765849—24no—
84#11765849—42no—
85#11765849—42no—
86#11765849—42no—
87#11765849—42no—
88#11765849—41no—
89#11765849—41no—
90#11765849—41no—
91#11765849—41no—
92#11765849—42no—
93C2×C71⋊C8—22no—
94C2×C71⋊C8—22no—
95C2×C71⋊C8—42no—
96C2×C71⋊C8—42no—
97C2×C71⋊C8—42no—
98C2×C71⋊C8—42no—
99C2×C71⋊C8—42no—
100C24×C71C22⋊C411yescharacteristic
101C24×C71C4⋊C411yescharacteristic
102C24×C71C4⋊C411yescharacteristic
103#41×C71—41no—
104#41×C71—41no—
105#41×C71—41no—
106#41×C71—41no—
107#41×C71—41no—
108#41×C71—41no—
109#41×C71—41no—
110#41×C71—41no—
111#41×C71—41no—
112#41×C71—41no—
113#41×C71—41no—
114#41×C71—41no—
115#41×C71—41no—
116#41×C71—41no—
117#41×C71—41no—
118#41×C71—41no—
119C2×Q8×C71—41no—
120C2×Q8×C71—41no—
121C2×Q8×C71C4211yescharacteristic
122C2×Q8×C71C22⋊C411yescharacteristic
123C2×Q8×C71C4⋊C411yescharacteristic
124C2×Q8×C71C22×C411yescharacteristic
125C2×D4×C71—41no—
126C2×D4×C71—41no—
127C2×D4×C71—41no—
128C2×D4×C71—41no—
129C2×D4×C71—41no—
130C2×D4×C71—41no—
131C2×D4×C71C4211yescharacteristic
132C2×D4×C71C22⋊C411yescharacteristic
133C2×D4×C71C22⋊C411yescharacteristic
134C2×D4×C71C22⋊C411yescharacteristic
135C2×D4×C71C4⋊C411yescharacteristic
136C2×D4×C71C4⋊C411yescharacteristic
137C2×D4×C71C4⋊C411yescharacteristic
138C2×D4×C71C4⋊C411yescharacteristic
139C2×D4×C71C4⋊C411yescharacteristic
140C2×D4×C71C22×C411yescharacteristic
141C2×D4×C71C2×D411yescharacteristic
142C2×D4×C71C2×Q811yescharacteristic
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—21no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—42no—
156C22×C4×C71C4211yescharacteristic
157C22×C4×C71C4211yescharacteristic
158C22×C4×C71C22⋊C411yescharacteristic
159C22×C4×C71C22⋊C411yescharacteristic
160C22×C4×C71C22⋊C411yescharacteristic
161C22×C4×C71C22⋊C411yescharacteristic
162C22×C4×C71C22⋊C411yescharacteristic
163C22×C4×C71C22⋊C411yescharacteristic
164C22×C4×C71C22⋊C411yescharacteristic
165C22×C4×C71C4⋊C411yescharacteristic
166C22×C4×C71C4⋊C411yescharacteristic
167C22×C4×C71C4⋊C411yescharacteristic
168C22×C4×C71C4⋊C411yescharacteristic
169C22×C4×C71C22×C411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C4⋊C4×C71—42no—
173C4⋊C4×C71—42no—
174C22⋊C4×C71—24no—
175C22⋊C4×C71—42no—
176C42×C71—42no—
177C42×C71—42no—
178C23×C71—21no—
179C23×C71—21no—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71—21no—
183C23×C71—21no—
184C23×C71—81no—
185C23×C71—81no—
186C23×C71—81no—
187C23×C71#9511yescharacteristic
188C23×C71#9511yescharacteristic
189C23×C71#9511yescharacteristic
190C23×C71#9511yescharacteristic
191C23×C71C2×C22⋊C411yescharacteristic
192C23×C71C2×C4⋊C411yescharacteristic
193C23×C71C2×C4⋊C411yescharacteristic
194Q8×C71—41no—
195Q8×C71—41no—
196Q8×C71—41no—
197Q8×C71—41no—
198D4×C71—41no—
199D4×C71—41no—
200D4×C71—41no—
201D4×C71—41no—
202D4×C71—41no—
203D4×C71—41no—
204D4×C71—41no—
205D4×C71—41no—
206D4×C71—41no—
207D4×C71—41no—
208D4×C71—41no—
209D4×C71—41no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—21no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—21no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
240C2×C4×C71—44no—
241C2×C4×C71—84no—
242C2×C4×C71#9511yescharacteristic
243C2×C4×C71#9511yescharacteristic
244C2×C4×C71#9511yescharacteristic
245C2×C4×C71#9511yescharacteristic
246C2×C4×C71#9911yescharacteristic
247C2×C4×C71#9911yescharacteristic
248C2×C4×C71#9911yescharacteristic
249C2×C4×C71#9911yescharacteristic
250C2×C4×C71C2×C4211yescharacteristic
251C2×C4×C71C2×C22⋊C411yescharacteristic
252C2×C4×C71C2×C22⋊C411yescharacteristic
253C2×C4×C71C2×C4⋊C411yescharacteristic
254C71⋊C8—44no—
255C71⋊C8—42no—
256C71⋊C8—42no—
257C22×C71—21no—
258C22×C71—21no—
259C22×C71—21no—
260C22×C71—21no—
261C22×C71—21no—
262C22×C71—21no—
263C22×C71—41no—
264C22×C71—41no—
265C22×C71—41no—
266C22×C71—41no—
267C22×C71—41no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71—42no—
271C22×C71—42no—
272C22×C71—81no—
273C22×C71—81no—
274C22×C71—81no—
275C22×C71—81no—
276C22×C71#34211yescharacteristic
277C22×C71#34211yescharacteristic
278C22×C71C2×#9511yescharacteristic
279C4×C71—21no—
280C4×C71—21no—
281C4×C71—21no—
282C4×C71—21no—
283C4×C71—21no—
284C4×C71—21no—
285C4×C71—84no—
286C4×C71#34211yescharacteristic
287C4×C71#34211yescharacteristic
288C4×C71C2×#9911yescharacteristic
289C4×C71C2×#9911yescharacteristic
290#12589—711nomaximal, Sylow
291C2×C71—21no—
292C2×C71—21no—
293C2×C71—21no—
294C2×C71—41no—
295C2×C71—41no—
296C2×C71—41no—
297C2×C71—41no—
298C2×C71—41no—
299C2×C71—41no—
300C2×C71C2×#34211yescharacteristic
301C2×#413—711no—
302C2×#410D7111yescharacteristic
303#2879—711no—
304#1395—714no—
305C71#1258911yescharacteristic, complemented, Sylow
306C22×#41C2×D7111yescharacteristic
307C22×M16—711no—
308C22×M16—711no—
309C22×M16—1421no—
310C2×#117—21no—
311#413—712no—
312#413—712no—
313#413—1421no—
314#413—1421no—
315C2×#101—712no—
316C2×#100—1421no—
317#410—21no—
318#410—21no—
319#410Dic7112yes—
320#410C2×D7112yes—
321C2×#99—21no—
322C2×#99Dic7112yes—
323#408—712no—
324#408—712no—
325#408—1421no—
326#408—1421no—
327#407—1421no—
328#407—1421no—
329C2×#41—41no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—41no—
333C2×#41Dic14211yescharacteristic
334C2×#41C4×D7111yescharacteristic
335C2×#41C4×D7111yescharacteristic
336C2×#41D28411yescharacteristic
337C2×#41C2×Dic7111yescharacteristic
338C2×#41C22⋊D7111yescharacteristic
339C2×#41C22⋊D7111yescharacteristic
340C2×#41C22×D7111yescharacteristic
341C22×Q8C4×D7111yescharacteristic
342C22×D4Dic14211yescharacteristic
343C22×D4C2×Dic7111yescharacteristic
344C22×D4C22⋊D7111yescharacteristic
345C23×C4C4×D7111yescharacteristic
346C23×C4D28411yescharacteristic
347C23×C4C22⋊D7111yescharacteristic
348C2×M16—2841no—
349C2×M16—2841no—
350C2×M16—2841no—
351C2×M16—2841no—
352C2×M16—2841no—
353C2×M16—2841no—
354C2×M16—2841no—
355C2×M16—712no—
356C2×M16—1422no—
357C2×M16—1422no—
358C2×M16—1422no—
359C2×M16—1421no—
360C2×M16—1421no—
361C2×M16—1421no—
362C22×C8—2841no—
363C22×C8—712no—
364C22×C8—1421no—
365#117—22no—
366#117—22no—
367#117—41no—
368#117—41no—
369C2×C4⋊C4—41no—
370C2×C22⋊C4—22no—
371C2×C42—41no—
372#101—714no—
373#101—1422no—
374#100—1422no—
375#100—1422no—
376#99—22no—
377#99—22no—
378#99—22no—
379#99C2×Dic7114yes—
380C22⋊C8—1422no—
381C22⋊C8—1422no—
382C22⋊C8—1422no—
383C22⋊C8—1422no—
384C24C22⋊Dic7111yescharacteristic
385C24Dic71⋊C411yescharacteristic
386C24C4⋊Dic7111yescharacteristic
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8C4×Dic7111yescharacteristic
406C2×Q8(C2×D71)⋊C411yescharacteristic
407C2×Q8Dic71⋊C411yescharacteristic
408C2×Q8C2×C4×D7111yescharacteristic
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C4×Dic7111yescharacteristic
416C2×D4(C2×D71)⋊C411yescharacteristic
417C2×D4C22⋊Dic7111yescharacteristic
418C2×D4C22⋊Dic7111yescharacteristic
419C2×D4Dic71⋊C411yescharacteristic
420C2×D4Dic71⋊C411yescharacteristic
421C2×D4Dic71⋊C411yescharacteristic
422C2×D4C4⋊Dic7111yescharacteristic
423C2×D4C4⋊Dic7111yescharacteristic
424C2×D4C22×Dic7111yescharacteristic
425C2×D4C2×C22⋊D7111yescharacteristic
426C2×D4C2×Dic14211yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4C4×Dic7111yescharacteristic
441C22×C4C4×Dic7111yescharacteristic
442C22×C4(C2×D71)⋊C411yescharacteristic
443C22×C4(C2×D71)⋊C411yescharacteristic
444C22×C4(C2×D71)⋊C411yescharacteristic
445C22×C4(C2×D71)⋊C411yescharacteristic
446C22×C4(C2×D71)⋊C411yescharacteristic
447C22×C4(C2×D71)⋊C411yescharacteristic
448C22×C4C22⋊Dic7111yescharacteristic
449C22×C4Dic71⋊C411yescharacteristic
450C22×C4Dic71⋊C411yescharacteristic
451C22×C4Dic71⋊C411yescharacteristic
452C22×C4C4⋊Dic7111yescharacteristic
453C22×C4C2×C4×D7111yescharacteristic
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C2×D28411yescharacteristic
456M16—2842no—
457M16—2842no—
458M16—2842no—
459M16—2842no—
460M16—2841no—
461M16—2841no—
462M16—2841no—
463M16—2841no—
464M16—2842no—
465M16—1424no—
466C2×C8—2842no—
467C2×C8—2842no—
468C2×C8—2842no—
469C2×C8—2842no—
470C2×C8—2842no—
471C2×C8—1422no—
472C2×C8—1422no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C42—42no—
478C42—42no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23C71⋊#9511yescharacteristic
489C23C71⋊#9511yescharacteristic
490C23C71⋊#9511yescharacteristic
491C23C71⋊#9511yescharacteristic
492C23C2×C22⋊Dic7111yescharacteristic
493C23C2×Dic71⋊C411yescharacteristic
494C23C2×C4⋊Dic7111yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—84no—
543C2×C4C71⋊#9511yescharacteristic
544C2×C4C71⋊#9511yescharacteristic
545C2×C4C71⋊#9511yescharacteristic
546C2×C4C71⋊#9511yescharacteristic
547C2×C4#1455604211yescharacteristic
548C2×C4#1455604211yescharacteristic
549C2×C4#1455604211yescharacteristic
550C2×C4#1455604211yescharacteristic
551C2×C4C2×C4×Dic7111yescharacteristic
552C2×C4C2×(C2×D71)⋊C411yescharacteristic
553C2×C4C2×(C2×D71)⋊C411yescharacteristic
554C2×C4C2×Dic71⋊C411yescharacteristic
555C8—2844no—
556C8—2842no—
557C8—2842no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#1798576511yescharacteristic
578C22#1798576511yescharacteristic
579C22C2×C71⋊#9511yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—84no—
587C4#1798576511yescharacteristic
588C4#1798576511yescharacteristic
589C4C2×#1455604211yescharacteristic
590C4C2×#1455604211yescharacteristic
591C2—21no—
592C2—21no—
593C2—21no—
594C2—41no—
595C2—41no—
596C2—41no—
597C2—41no—
598C2—41no—
599C2—41no—
600C2C2×#1798576511yescharacteristic
subgroupG/H
centerC4C2×#14556042
commutatorC2×C4×C71C2×C42
FrattiniC2×#41C22×D71
FittingC2×#410×C71C2
socleC2×C71C2×#342
prime radicalC22×#41×C71C22
cosocleC2×#41×C71C23
hypercenterC2×#410D71
hypocenterC71#12589
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #22586468C2semidirect
6. C22×#11765849C4semidirect
7. C22×#11765849C4semidirect
10. #17985983C4semidirect
12. C2×#14556047C4semidirect
16. #17985966C4semidirect
41. C2×#11765849C2×C4semidirect
44. C22×C71⋊C8C2×C4semidirect
46. #14556047C2×C4semidirect
305. C71#12589semidirect

Maximal subgroups

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

subgroupindex
C2×#179859832
#225967852
#225864682
C2×#410×C712
#1258971