Finite Group Databasebeta

← #32055617

Subgroups — #32055617

Order 18176.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22593361C211yesmaximal, characteristic, complemented
2#22589728C211yesmaximal, characteristic
3#22589714C211yesmaximal, characteristic
4#22589486C211yesmaximal, characteristic
5#22589476C211yesmaximal, characteristic
6#22589297C211yesmaximal, characteristic, complemented
7#22589292C211yesmaximal, characteristic, complemented
8#22589106C211yesmaximal, characteristic, complemented
9#22589059C211yesmaximal, characteristic, complemented
10#22588932C211yesmaximal, characteristic, complemented
11#22588925C211yesmaximal, characteristic, complemented
12#22588470C211yesmaximal, characteristic, complemented
13#22588350C211yesmaximal, characteristic
14C4×#14556049C211yesmaximal, characteristic
15C71×#2357C211yesmaximal, characteristic
16C2×#14556128C2211yescharacteristic
17C2×#14556120C2211yescharacteristic
18C2×#14556061C2211yescharacteristic
19C2×#14556059C2211yescharacteristic
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×C4⋊(C71⋊C8)C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×#14556049C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×C4×C71⋊C8C2211yescharacteristic
28#17985932—21no—
29#17985919—21no—
30#17985903—21no—
31#17985897C2211yescharacteristic
32#17985897C2211yescharacteristic
33#17985896—21no—
34#17985865C2211yescharacteristic
35C4×C4⋊Dic71C2211yescharacteristic
36#17985761C2211yescharacteristic
37#17985761C2211yescharacteristic
38#17985761C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985754C2211yescharacteristic
47#17985754C2211yescharacteristic
48C2×C4×D4×C71C2211yescharacteristic
49#390×C71C2211yescharacteristic
50#390×C71C2211yescharacteristic
51#388×C71—21no—
52#388×C71—21no—
53#382×C71C2211yescharacteristic
54#381×C71—21no—
55C4×C4⋊C4×C71C2211yescharacteristic
56C4×C4⋊C4×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C22⋊C4×C71—21no—
59C22×Dic142—21no—
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556128—22no—
65#14556128—41no—
66#14556120—22no—
67#14556120—41no—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×Dic71⋊C4—21no—
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×C4×Dic71—21no—
78#14556061—22no—
79#14556059—22no—
80C4⋊(C71⋊C8)—22no—
81C4⋊(C71⋊C8)—41no—
82C4⋊(C71⋊C8)—41no—
83#14556049—22no—
84#14556049—22no—
85#14556049—41no—
86#14556049—41no—
87C4×C71⋊C8—22no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C22×D4×C71C2311yescharacteristic
94C23×C4×C71—21no—
95C4×D4×C71—22no—
96C4×D4×C71Q814yescomplemented
97C2×C4⋊C4×C71D411yescharacteristic
98C2×C4⋊C4×C71D411yescharacteristic
99C2×C4⋊C4×C71C2311yescharacteristic
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C22⋊C4×C71—21no—
103C2×C22⋊C4×C71—21no—
104C2×C22⋊C4×C71—21no—
105C2×C22⋊C4×C71—21no—
106C2×C22⋊C4×C71—21no—
107C2×C42×C71—21no—
108C2×C42×C71—21no—
109C2×C42×C71C2311yescharacteristic
110C2×C42×C71C2311yescharacteristic
111C2×C42×C71C2311yescharacteristic
112#95×C71—21no—
113#95×C71—21no—
114#95×C71—21no—
115#95×C71—21no—
116#95×C71D411yescharacteristic
117#95×C71D411yescharacteristic
118C2×Dic142—42no—
119C2×Dic142—42no—
120C2×Dic142—81no—
121C22×Dic71—21no—
122C22×Dic71—21no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C2×C71⋊C8—21no—
126C2×C71⋊C8—21no—
127C2×C71⋊C8—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C4⋊Dic71—21no—
138C4⋊Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—42no—
144C4⋊Dic71#4112yes—
145C4⋊Dic71#4112yes—
146Dic71⋊C4—42no—
147Dic71⋊C4—42no—
148Dic71⋊C4—42no—
149Dic71⋊C4—81no—
150C4×Dic71—81no—
151C24×C71—21no—
152C2×D4×C71—21no—
153C2×D4×C71—21no—
154C2×D4×C71—21no—
155C2×D4×C71—21no—
156C2×D4×C71—21no—
157C2×D4×C71C2×Q812yes—
158C22×C4×C71—21no—
159C22×C4×C71—24no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—81no—
163C22×C4×C71—81no—
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2411yescharacteristic
171C4⋊C4×C71—22no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71—21no—
175C4⋊C4×C71—21no—
176C4⋊C4×C71—21no—
177C4⋊C4×C71—41no—
178C4⋊C4×C71C2×Q812yes—
179C22⋊C4×C71—24no—
180C22⋊C4×C71—42no—
181C22⋊C4×C71—42no—
182C22⋊C4×C71—81no—
183C22⋊C4×C71—81no—
184C42×C71—21no—
185C42×C71—21no—
186C42×C71—21no—
187C42×C71—21no—
188C42×C71—81no—
189C42×C71—81no—
190C42×C71C2×Q812yes—
191C42×C71#4112yes—
192C23×C71—22no—
193C23×C71—41no—
194C23×C71—41no—
195C23×C71—41no—
196C23×C71—41no—
197C23×C71—41no—
198C23×C71—41no—
199C23×C71C22×D411yescharacteristic
200D4×C71—22no—
201D4×C71—22no—
202D4×C71—41no—
203D4×C71—41no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—22no—
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—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—84no—
229C2×C4×C71C22⋊Q812yes—
230C2×C4×C71C22⋊Q812yes—
231C2×C4×C71#13512yes—
232C2×C4×C71C22×Q811yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71H(5,𝔽2)11yescharacteristic
237C2×C4×C712−1+411yescharacteristic
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250Dic142—84no—
251C71⋊C8—41no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C22×C71—42no—
256C22×C71—42no—
257C22×C71—42no—
258C22×C71—81no—
259C22×C71—81no—
260C22×C71—81no—
261C22×C71—81no—
262C22×C71C2×C22⋊Q811yescharacteristic
263C22×C71#54611yescharacteristic
264C22×C71#54811yescharacteristic
265C22×C71C2×#13511yescharacteristic
266C22×C71#57511yescharacteristic
267C22×C71#57611yescharacteristic
268C22×C71#57711yescharacteristic
269C4×C71—22no—
270C4×C71—22no—
271C4×C71—22no—
272C4×C71—42no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—81no—
277C4×C71—81no—
278C4×C71#53311yescharacteristic
279C4×C71#54411yescharacteristic
280C4×C71#55511yescharacteristic
281C4×C71#55611yescharacteristic
282Dic71—42no—
283Dic71—42no—
284Dic71—81no—
285Dic71—81no—
286#26028—711nomaximal, Sylow
287C2×C71—82no—
288C2×C71#243911yescharacteristic
289C2×C71#307611yescharacteristic
290C2×C71#308111yescharacteristic
291C2×C71#329711yescharacteristic
292C2×C71#332611yescharacteristic
293C2×C71#336511yescharacteristic
294C2×C71#337011yescharacteristic
295#2550—711no—
296#2357D7111yescharacteristic
297#2067—711no—
298#2065—711no—
299#2025—711no—
300#2023—711no—
301#1989—711no—
302#1988—711no—
303#1949—711no—
304#1939—711no—
305#1921—711no—
306#1920—711no—
307#1835—711no—
308#1804—711no—
309C4×#102—711no—
310C71#2602811yescharacteristic, complemented, Sylow
311C2×C4⋊Q8—711no—
312C2×#125—711no—
313C2×C4×D4C2×D7111yescharacteristic
314C2×#107—711no—
315C2×#106—711no—
316C2×C4⋊C8—711no—
317C2×C4⋊C8—711no—
318C2×C4⋊C8—711no—
319C2×#102—711no—
320C2×#102—711no—
321C2×#102—711no—
322C2×#102—711no—
323C2×C4×C8—711no—
324#398—1421no—
325#395—1421no—
326#391—1421no—
327#390C2×D7111yescharacteristic
328#390C2×D7111yescharacteristic
329#389—711no—
330#389—711no—
331#389—1421no—
332#388—21no—
333#388—21no—
334#382—711no—
335#382C2×D7111yescharacteristic
336#381—21no—
337C4×C4⋊C4—711no—
338C4×C4⋊C4C2×D7111yescharacteristic
339C4×C4⋊C4C2×D7111yescharacteristic
340C4×C4⋊C4C2×D7111yescharacteristic
341C4×C22⋊C4—21no—
342#340—711no—
343#340—711no—
344#340—711no—
345#340—711no—
346#340—711no—
347#340—711no—
348#340—711no—
349#340—711no—
350#340—711no—
351#340—711no—
352#336—711no—
353#336—711no—
354C22×Q8—1421no—
355C22×D4C22×D7111yescharacteristic
356C23×C4—21no—
357C22×C8—711no—
358C22×C8—711no—
359C22×C8—711no—
360C22×C8—711no—
361C4⋊Q8—2841no—
362C4⋊Q8—1422no—
363#125—2841no—
364#125—1422no—
365C4×D4—22no—
366C4×D4Dic14214yescomplemented
367C2×C4⋊C4—711no—
368C2×C4⋊C4—711no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—1421no—
372C2×C4⋊C4—1421no—
373C2×C4⋊C4—1421no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4C22⋊D7111yescharacteristic
377C2×C4⋊C4C22⋊D7111yescharacteristic
378C2×C4⋊C4C22×D7111yescharacteristic
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—1421no—
389C2×C42C22×D7111yescharacteristic
390C2×C42C22×D7111yescharacteristic
391C2×C42C22×D7111yescharacteristic
392#107—1422no—
393#106—1422no—
394C4⋊C8—2841no—
395C4⋊C8—2841no—
396C4⋊C8—1422no—
397#102—2841no—
398#102—2841no—
399#102—1422no—
400#102—1422no—
401C4×C8—1422no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—1421no—
407#95—1421no—
408#95—1421no—
409#95—1421no—
410#95—1421no—
411#95C22⋊D7111yescharacteristic
412#95C22⋊D7111yescharacteristic
413C24—21no—
414C2×Q8—2842no—
415C2×Q8—2842no—
416C2×Q8—5681no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic14212yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—1421no—
430C22×C4—1421no—
431C22×C4—1421no—
432C22×C4—1421no—
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4C23×D7111yescharacteristic
440C2×C8—1421no—
441C2×C8—1421no—
442C2×C8—1421no—
443C2×C8—1421no—
444C2×C8—1421no—
445C2×C8—1421no—
446C2×C8—1421no—
447C2×C8—1421no—
448C2×C8—1421no—
449C2×C8—1421no—
450C2×C8—1421no—
451C2×C8—1421no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—2842no—
460C4⋊C4—2842no—
461C4⋊C4—2842no—
462C4⋊C4—2842no—
463C4⋊C4—5681no—
464C4⋊C4—712no—
465C4⋊C4—712no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4C2×Dic14212yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—81no—
484C42—5681no—
485C42C2×Dic14212yes—
486C42#1176586512yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C22×C22⋊D7111yescharacteristic
495Q8—5684no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
525C2×C4—2841no—
526C2×C4—2841no—
527C2×C4—2841no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—1422no—
536C2×C4—1422no—
537C2×C4#1455610712yes—
538C2×C4#1455610712yes—
539C2×C4#1455614812yes—
540C2×C4C22×Dic14211yescharacteristic
541C2×C4#41×D7111yescharacteristic
542C2×C4#41×D7111yescharacteristic
543C2×C4C2×#1176586511yescharacteristic
544C2×C4#1455617411yescharacteristic
545C2×C4#1455617511yescharacteristic
546C8—2841no—
547C8—2841no—
548C8—2841no—
549C8—2841no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1455610711yescharacteristic
558C22#1798663711yescharacteristic
559C22#1798665611yescharacteristic
560C22C2×#1455614811yescharacteristic
561C22#1798679711yescharacteristic
562C22#1798680211yescharacteristic
563C22#1798681011yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—2842no—
574C4—2842no—
575C4—5681no—
576C4—5681no—
577C4#1798655211yescharacteristic
578C4#1798662711yescharacteristic
579C4#1798671211yescharacteristic
580C4#1798671711yescharacteristic
581C2—82no—
582C2#2259202511yescharacteristic
583C2#2259812811yescharacteristic
584C2#2259817111yescharacteristic
585C2#2260018411yescharacteristic
586C2#2260058711yescharacteristic
587C2#2260103211yescharacteristic
588C2#2260108711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2357C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
hypercenter#2357D71
hypocenterC71#26028
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #22593361C2semidirect
6. #22589297C2semidirect
7. #22589292C2semidirect
8. #22589106C2semidirect
9. #22589059C2semidirect
10. #22588932C2semidirect
11. #22588925C2semidirect
12. #22588470C2semidirect
96. C4×D4×C71Q8semidirect
310. C71#26028semidirect
366. C4×D4Dic142semidirect

Maximal subgroups

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

subgroupindex
#225933612
#225897282
#225897142
#225894862
#225894762
#225892972
#225892922
#225891062
#225890592
#225889322
#225889252
#225884702
#225883502
C4×#145560492
C71×#23572
#2602871