Finite Group Databasebeta

← #32033817

Subgroups — #32033817

Order 18176.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986276C211yesmaximal, characteristic, complemented
2C2×C4×#11765850C211yesmaximal, characteristic
3#22593258C211yesmaximal, characteristic, complemented
4#22589665C212yesmaximal, complemented
5#22589473C212yesmaximal, complemented
6#22589302C211yesmaximal, characteristic, complemented
7#22589277C211yesmaximal, characteristic, complemented
8#22588688C212yesmaximal
9#22588587C212yesmaximal, complemented
10#22588340C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C71C211yesmaximal, characteristic, complemented
12C22×#11765850C2211yescharacteristic
13C2×#14556104—21no—
14C2×C4×D284C2211yescharacteristic
15#17986276—24no—
16C4×#11765850—24no—
17C2×#14556061C2212yes—
18C2×#14556061C2211yescharacteristic
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×#14556049C2212yes—
21C2×#14556049C2211yescharacteristic
22C2×#14556048C2212yes—
23C2×#14556048C2211yescharacteristic
24C2×C4×C71⋊C8C2211yescharacteristic
25#17985921—22no—
26#17985901C2212yes—
27#17985897C2212yes—
28#17985869C2211yescharacteristic
29#17985856—21no—
30#17985761C2212yes—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985754C2212yes—
34#17985754C2212yes—
35C23×Q8×C71—21no—
36D4×Q8×C71—24no—
37D4×Q8×C71—24no—
38C2×C4⋊Q8×C71C2212yes—
39C2×C4⋊Q8×C71C2211yescharacteristic
40C2×C22⋊Q8×C71—21no—
41C2×C22⋊Q8×C71—22no—
42C2×C4×Q8×C71C2211yescharacteristic
43C2×C4×D4×C71C2212yes—
44C2×C4×D4×C71C2211yescharacteristic
45C22×D284C2311yescharacteristic
46C22×C4×D71—21no—
47C2×#11765850—22no—
48C2×#11765850—22no—
49C2×#11765850Q814yescomplemented
50C22×C71⋊C8C2311yescharacteristic
51C22×C71⋊C8C2312yes—
52C22×C71⋊C8C2311yescharacteristic
53#14556104—44no—
54C4×D284—22no—
55C4×D284—22no—
56C2×C4⋊Dic71—21no—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71C2312yes—
59C2×C4⋊Dic71C2311yescharacteristic
60C2×Dic71⋊C4—22no—
61C2×(C2×D71)⋊C4—22no—
62C2×(C2×D71)⋊C4—21no—
63C2×(C2×D71)⋊C4—22no—
64#14556061—22no—
65#14556061—24no—
66C4⋊(C71⋊C8)—22no—
67#14556049—22no—
68#14556049—24no—
69#14556048—22no—
70#14556048—24no—
71C4×C71⋊C8—22no—
72C71⋊#95—22no—
73C22×Q8×C71—28no—
74C22×Q8×C71—22no—
75C22×Q8×C71—21no—
76C22×Q8×C71—44no—
77C22×Q8×C71—44no—
78C22×Q8×C71D411yescharacteristic
79C22×D4×C71C2311yescharacteristic
80C23×C4×C71—22no—
81C23×C4×C71—21no—
82C4⋊Q8×C71—24no—
83C4⋊Q8×C71—24no—
84C4⋊Q8×C71—22no—
85C4⋊Q8×C71D414yes—
86C22⋊Q8×C71—28no—
87C22⋊Q8×C71—216no—
88C4×Q8×C71—22no—
89C4×Q8×C71—22no—
90C4×D4×C71—24no—
91C4×D4×C71—24no—
92C4×D4×C71—22no—
93C4×D4×C71D414yes—
94C2×C4⋊C4×C71—22no—
95C2×C4⋊C4×C71—21no—
96C2×C4⋊C4×C71D411yescharacteristic
97C2×C4⋊C4×C71D412yes—
98C2×C4⋊C4×C71C2312yes—
99C2×C4⋊C4×C71C2311yescharacteristic
100C2×C22⋊C4×C71—22no—
101C2×C22⋊C4×C71—21no—
102C2×C42×C71C2312yes—
103C2×C42×C71C2311yescharacteristic
104C23×D71—21no—
105C2×D284—21no—
106C2×D284—21no—
107C2×D284—21no—
108C2×D284—21no—
109C2×D284—21no—
110C2×D284C2×Q812yescomplemented
111C22×Dic71—21no—
112C22×Dic71—22no—
113C2×C4×D71—42no—
114C2×C4×D71—42no—
115C2×C4×D71—42no—
116#11765850—28no—
117C2×C71⋊C8—21no—
118C2×C71⋊C8—21no—
119C2×C71⋊C8—21no—
120C2×C71⋊C8—22no—
121C2×C71⋊C8—22no—
122C2×C71⋊C8—21no—
123C2×C71⋊C8—21no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8C2×Q812yescomplemented
126C4⋊Dic71—21no—
127C4⋊Dic71—21no—
128C4⋊Dic71—22no—
129C4⋊Dic71—22no—
130C4⋊Dic71—42no—
131C4⋊Dic71—81no—
132Dic71⋊C4—44no—
133(C2×D71)⋊C4—42no—
134(C2×D71)⋊C4—44no—
135(C2×D71)⋊C4—82no—
136C24×C71—21no—
137C2×Q8×C71—28no—
138C2×Q8×C71—22no—
139C2×Q8×C71—24no—
140C2×Q8×C71—22no—
141C2×Q8×C71—44no—
142C2×Q8×C71—432no—
143C2×Q8×C71—44no—
144C2×Q8×C71—42no—
145C2×Q8×C71—42no—
146C2×Q8×C71D814yescomplemented
147C2×D4×C71—21no—
148C2×D4×C71—21no—
149C2×D4×C71—21no—
150C2×D4×C71C2×D412yes—
151C2×D4×C71C2×Q812yes—
152C2×D4×C71#4112yes—
153C22×C4×C71—28no—
154C22×C4×C71—22no—
155C22×C4×C71—21no—
156C22×C4×C71—24no—
157C22×C4×C71—44no—
158C22×C4×C71—44no—
159C22×C4×C71—42no—
160C22×C4×C71—42no—
161C22×C4×C71C2×D412yes—
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D412yes—
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2411yescharacteristic
166C4⋊C4×C71—28no—
167C4⋊C4×C71—24no—
168C4⋊C4×C71—24no—
169C4⋊C4×C71—22no—
170C4⋊C4×C71—22no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71C2×D414yes—
173C4⋊C4×C71C2×D412yes—
174C22⋊C4×C71—28no—
175C22⋊C4×C71—24no—
176C42×C71—22no—
177C42×C71—22no—
178C42×C71C2×D412yes—
179C42×C71C2×D412yes—
180C23×C71—22no—
181C23×C71—22no—
182C23×C71—22no—
183C23×C71—41no—
184C23×C71—41no—
185C23×C71—41no—
186C23×C71—41no—
187C23×C71C22×D411yescharacteristic
188C22×D71—22no—
189C22×D71—41no—
190C22×D71—41no—
191C22×D71—41no—
192C22×D71—41no—
193C22×D71—41no—
194C22×D71—41no—
195Q8×C71—28no—
196Q8×C71—416no—
197Q8×C71—48no—
198D4×C71—22no—
199D4×C71—22no—
200D4×C71—22no—
201D4×C71C22⋊Q814yescomplemented
202C2×C4×C71—22no—
203C2×C4×C71—22no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—22no—
207C2×C4×C71—24no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—416no—
212C2×C4×C71—42no—
213C2×C4×C71—41no—
214C2×C4×C71—42no—
215C2×C4×C71—41no—
216C2×C4×C71—48no—
217C2×C4×C71#12012yes—
218C2×C4×C71#12012yes—
219C2×C4×C71C2×D814yes—
220C2×C4×C71C2×D812yes—
221C2×C4×C71AGL(1,ℤ/8)12yes—
222C2×C4×C71C22×D411yescharacteristic
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×Q811yescharacteristic
225C2×C4×C71C2×#4111yescharacteristic
226C2×C4×C712−1+411yescharacteristic
227C2×C4×C712−1+411yescharacteristic
228C2×Dic71—41no—
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71—41no—
232C2×Dic71—41no—
233C2×Dic71—42no—
234D284—22no—
235D284—22no—
236D284—41no—
237D284—41no—
238C4×D71—84no—
239C71⋊C8—22no—
240C71⋊C8—41no—
241C71⋊C8—42no—
242C22×C71—24no—
243C22×C71—42no—
244C22×C71—42no—
245C22×C71—42no—
246C22×C71—42no—
247C22×C71—42no—
248C22×C71—42no—
249C22×C71C2×#12011yescharacteristic
250C22×C71#54811yescharacteristic
251C22×C71D4×Q811yescharacteristic
252C22×C71C22×D811yescharacteristic
253C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C71#57711yescharacteristic
255C22×C71#57811yescharacteristic
256C2×D71—42no—
257C2×D71—42no—
258C2×D71—42no—
259C2×D71—81no—
260C2×D71—81no—
261C2×D71—81no—
262C2×D71—81no—
263C4×C71—24no—
264C4×C71—22no—
265C4×C71—44no—
266C4×C71—42no—
267C4×C71—42no—
268C4×C71#44714yes—
269C4×C71C2×C22⋊Q811yescharacteristic
270C4×C71#53611yescharacteristic
271C4×C71#54811yescharacteristic
272C4×C71D4×Q811yescharacteristic
273Dic71—82no—
274Dic71—81no—
275#24326—711nomaximal, Sylow
276C2×C71—44no—
277C2×C71#240911yescharacteristic
278C2×C71C2×#44711yescharacteristic
279C2×C71#301311yescharacteristic
280C2×C71#335811yescharacteristic
281C2×C71#336711yescharacteristic
282C2×C71Q8×D811yescharacteristic
283C2×C71#338411yescharacteristic
284D71—82no—
285C2×D4×Q8D7111yescharacteristic, complemented
286C2×#474—711no—
287C2×C4×D8—711no—
288#2538—711no—
289#2057—712no—
290#2023—712no—
291#1990—711no—
292#1984—711no—
293#1879—712no—
294#1862—712no—
295#1802—711no—
296C71#2432611yescharacteristic, complemented, Sylow
297C23×Q8—21no—
298C22×D8—711no—
299D4×Q8—24no—
300D4×Q8—24no—
301C2×C4⋊Q8C2×D7112yes—
302C2×C4⋊Q8C2×D7111yescharacteristic
303C2×C22⋊Q8—21no—
304C2×C22⋊Q8—22no—
305C2×C22⋊Q8—1421no—
306C2×C4×Q8C2×D7111yescharacteristic
307C2×C4×D4—711no—
308C2×C4×D4C2×D7112yes—
309C2×C4×D4C2×D7111yescharacteristic
310#474—1424no—
311C4×D8—1424no—
312C2×#107—711no—
313C2×#107—712no—
314C2×C4⋊C8—711no—
315C2×#102—712no—
316C2×#102—711no—
317C2×#102—712no—
318C2×#102—711no—
319C2×C4×C8—711no—
320#396—1422no—
321#390—712no—
322#389—712no—
323#384—711no—
324#380—1421no—
325#340—712no—
326#340—712no—
327#340—712no—
328#336—712no—
329#336—712no—
330C22×Q8—28no—
331C22×Q8—22no—
332C22×Q8—21no—
333C22×Q8—44no—
334C22×Q8—44no—
335C22×Q8C22⋊D7111yescharacteristic
336C22×D4—711no—
337C22×D4C22×D7111yescharacteristic
338C23×C4—22no—
339C23×C4—21no—
340C23×C4—1421no—
341C2×D8—714no—
342C2×D8—1422no—
343C2×D8—1422no—
344C22×C8—711no—
345C22×C8—711no—
346C22×C8—712no—
347C4⋊Q8—24no—
348C4⋊Q8—24no—
349C4⋊Q8—22no—
350C4⋊Q8C22⋊D7114yes—
351C22⋊Q8—28no—
352C22⋊Q8—216no—
353C22⋊Q8—2844no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×D4—24no—
357C4×D4—24no—
358C4×D4—22no—
359C4×D4—1422no—
360C4×D4—1422no—
361C4×D4C22⋊D7114yes—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—711no—
365C2×C4⋊C4—712no—
366C2×C4⋊C4—1421no—
367C2×C4⋊C4—1422no—
368C2×C4⋊C4—1421no—
369C2×C4⋊C4C22⋊D7111yescharacteristic
370C2×C4⋊C4C22⋊D7112yes—
371C2×C4⋊C4C22×D7112yes—
372C2×C4⋊C4C22×D7111yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1421no—
376C2×C22⋊C4—1422no—
377C2×C22⋊C4—1422no—
378C2×C42C22×D7112yes—
379C2×C42C22×D7111yescharacteristic
380#107—1422no—
381#107—1424no—
382C4⋊C8—1422no—
383#102—1422no—
384#102—1424no—
385#102—1422no—
386#102—1424no—
387C4×C8—1422no—
388#95—1422no—
389C24—21no—
390C24—1421no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—432no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8#1176585014yescomplemented
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—712no—
405C2×D4—1421no—
406C2×D4—1421no—
407C2×D4—1421no—
408C2×D4—1421no—
409C2×D4—1421no—
410C2×D4C2×C22⋊D7112yes—
411C2×D4Q8×D7112yes—
412C2×D4#1176586612yes—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—2842no—
422C22×C4—2842no—
423C22×C4—2842no—
424C22×C4—1421no—
425C22×C4—1422no—
426C22×C4C2×C22⋊D7112yes—
427C22×C4C2×C22⋊D7111yescharacteristic
428C22×C4C2×C22⋊D7112yes—
429C22×C4C2×C22⋊D7111yescharacteristic
430C22×C4C23×D7111yescharacteristic
431D8—1428no—
432C2×C8—712no—
433C2×C8—1421no—
434C2×C8—1422no—
435C2×C8—1422no—
436C2×C8—1421no—
437C2×C8—1421no—
438C2×C8—1422no—
439C2×C8—1421no—
440C2×C8—1421no—
441C4⋊C4—28no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—2844no—
448C4⋊C4—2842no—
449C4⋊C4—5681no—
450C4⋊C4—1421no—
451C4⋊C4—1421no—
452C4⋊C4—1422no—
453C4⋊C4—1422no—
454C4⋊C4C2×C22⋊D7114yes—
455C4⋊C4C2×C22⋊D7112yes—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—2842no—
459C22⋊C4—2844no—
460C22⋊C4—5682no—
461C42—22no—
462C42—22no—
463C42C2×C22⋊D7112yes—
464C42C2×C22⋊D7112yes—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—2841no—
473C23—2841no—
474C23—2841no—
475C23—2841no—
476C23—2841no—
477C23—2841no—
478C23—1422no—
479C23C22×C22⋊D7111yescharacteristic
480Q8—28no—
481Q8—416no—
482Q8—48no—
483D4—22no—
484D4—22no—
485D4—22no—
486D4—2841no—
487D4—2841no—
488D4—1422no—
489D4—1422no—
490D4#1455610414yescomplemented
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—48no—
506C2×C4—2842no—
507C2×C4—2841no—
508C2×C4—2842no—
509C2×C4—2841no—
510C2×C4—2841no—
511C2×C4—2842no—
512C2×C4—5684no—
513C2×C4#1455609712yes—
514C2×C4#1455609712yes—
515C2×C4C2×#1176585014yes—
516C2×C4C2×#1176585012yes—
517C2×C4#1455615412yes—
518C2×C4C22×C22⋊D7111yescharacteristic
519C2×C4C22×C22⋊D7111yescharacteristic
520C2×C4C2×Q8×D7111yescharacteristic
521C2×C4C2×#1176586611yescharacteristic
522C2×C4#1455617611yescharacteristic
523C2×C4#1455617611yescharacteristic
524C8—2841no—
525C8—2842no—
526C8—1422no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—2842no—
535C22—2842no—
536C22—2842no—
537C22—5681no—
538C22—5681no—
539C22—5681no—
540C22—5681no—
541C22C2×#1455609711yescharacteristic
542C22#1798666011yescharacteristic
543C22Q8×C22⋊D7111yescharacteristic
544C22C22×#1176585011yescharacteristic
545C22C2×#1455615411yescharacteristic
546C22#1798680711yescharacteristic
547C22#1798681411yescharacteristic
548C4—22no—
549C4—24no—
550C4—42no—
551C4—44no—
552C4—42no—
553C4—5682no—
554C4—5681no—
555C4#1798613314yes—
556C4C2×#1455610411yescharacteristic
557C4#1798657311yescharacteristic
558C4#1798666011yescharacteristic
559C4Q8×C22⋊D7111yescharacteristic
560C2—44no—
561C2—5682no—
562C2#2259169911yescharacteristic
563C2C2×#1798613311yescharacteristic
564C2#2259765511yescharacteristic
565C2#2260097511yescharacteristic
566C2#2260105811yescharacteristic
567C2Q8×#1176585011yescharacteristic
568C2#2260123511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×D4×Q8×C71C2
socleC23×C71C22×D4
prime radicalC22×#11765850C22
hypercenterC2×D4×Q8D71
hypocenterC71#24326
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17986276C2semidirect
3. #22593258C2semidirect
4. #22589665C2semidirect
5. #22589473C2semidirect
6. #22589302C2semidirect
7. #22589277C2semidirect
9. #22588587C2semidirect
10. #22588340C2semidirect
11. C2×D4×Q8×C71C2semidirect
49. C2×#11765850Q8semidirect
110. C2×D284C2×Q8semidirect
125. C2×C71⋊C8C2×Q8semidirect
146. C2×Q8×C71D8semidirect
201. D4×C71C22⋊Q8semidirect
285. C2×D4×Q8D71semidirect
296. C71#24326semidirect
400. C2×Q8#11765850semidirect
490. D4#14556104semidirect

Maximal subgroups

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

subgroupindex
C2×#179862762
C2×C4×#117658502
#225932582
#225896652
#225894732
#225893022
#225892772
#225886882
#225885872
#225883402
C2×D4×Q8×C712
#2432671