Finite Group Databasebeta

← #32008690

Subgroups — #32008690

Order 18176.

1212 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986149C211yesmaximal, characteristic
2C2×#17985901C211yesmaximal, characteristic
3#22589224C212yesmaximal, complemented
4#22589036C212yesmaximal, complemented
5#22588960C212yesmaximal, complemented
6#22588695C212yesmaximal, complemented
7#22588648C211yesmaximal, characteristic, complemented
8#22588527C211yesmaximal, characteristic, complemented
9#22588270C211yesmaximal, characteristic
10#22588102C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C71C211yesmaximal, characteristic, complemented
12C23×D284C2211yescharacteristic
13C22×#11765852C2212yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×(C2×D71)⋊C4—21no—
16C22×(C2×D71)⋊C4—21no—
17#17986149C418yescomplemented
18C2×#14556061C2212yescomplemented
19C2×#14556052C2212yescomplemented
20C2×#14556048C2212yes—
21C2×#14556048C2212yes—
22C2×#14556048C2212yes—
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25C2×#14556039C2212yescomplemented
26#17985901—21no—
27#17985901—21no—
28#17985901C2212yes—
29#17985901C2212yes—
30#17985761C2212yescomplemented
31#17985761C2212yescomplemented
32#17985754C2212yescomplemented
33C2×C22⋊Q8×C71C2211yescharacteristic
34C2×C4×Q8×C71C2211yescharacteristic, complemented
35C22×C42×C71C2211yescharacteristic
36#391×C71C2211yescharacteristic, complemented
37#389×C71C2211yescharacteristic, complemented
38#387×C71—21no—
39#385×C71—21no—
40#380×C71C2211yescharacteristic
41C4×C4⋊C4×C71—21no—
42C4×C4⋊C4×C71C2211yescharacteristic, complemented
43C4×C22⋊C4×C71—21no—
44C24×D71—21no—
45C22×D284—21no—
46C22×D284—22no—
47C22×D284—42no—
48C22×D284—42no—
49C22×D284C2×C414yes—
50C22×D284C2312yes—
51C23×Dic71—21no—
52C2×#11765852—42no—
53C2×#11765852—42no—
54C2×#11765852C2×C418yescomplemented
55C22×C71⋊C8C2312yes—
56C22×C71⋊C8C2312yes—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—42no—
59C2×C4⋊Dic71C2312yes—
60C2×(C2×D71)⋊C4—22no—
61C2×(C2×D71)⋊C4—22no—
62C2×(C2×D71)⋊C4—22no—
63C2×(C2×D71)⋊C4—22no—
64C2×(C2×D71)⋊C4—44no—
65C2×(C2×D71)⋊C4—44no—
66#14556061—42no—
67#14556052—42no—
68#14556048—42no—
69#14556048—42no—
70#14556048C2×C418yescomplemented
71C22⋊(C71⋊C8)—22no—
72C22⋊(C71⋊C8)C2×C414yes—
73#14556039—42no—
74C22×Q8×C71C2311yescharacteristic
75C23×C4×C71D411yescharacteristic
76C23×C4×C71D411yescharacteristic
77C23×C4×C71C2311yescharacteristic
78C22⋊Q8×C71—22no—
79C22⋊Q8×C71C2×C414yes—
80C4×Q8×C71—42no—
81C2×C4⋊C4×C71—21no—
82C2×C4⋊C4×C71—21no—
83C2×C4⋊C4×C71—21no—
84C2×C4⋊C4×C71C2311yescharacteristic
85C2×C4⋊C4×C71C2311yescharacteristic
86C2×C4⋊C4×C71C2311yescharacteristic
87C2×C22⋊C4×C71—21no—
88C2×C22⋊C4×C71—21no—
89C2×C42×C71—21no—
90C2×C42×C71—21no—
91C2×C42×C71—42no—
92C2×C42×C71D411yescharacteristic
93C2×C42×C71D411yescharacteristic
94C2×C42×C71C2311yescharacteristic
95C2×C42×C71C2311yescharacteristic
96#95×C71—21no—
97#95×C71—21no—
98#95×C71—21no—
99#95×C71—21no—
100#95×C71—21no—
101C23×D71—24no—
102C23×D71—22no—
103C23×D71—42no—
104C23×D71—42no—
105C23×D71—42no—
106C23×D71—42no—
107C23×D71—42no—
108C23×D71—42no—
109C2×D284—22no—
110C2×D284—22no—
111C2×D284—22no—
112C2×D284—22no—
113C2×D284—24no—
114C2×D284—22no—
115C2×D284—48no—
116C2×D284—42no—
117C2×D284—42no—
118C2×D284—48no—
119C2×D284C22×C414yes—
120C22×Dic71—22no—
121C22×Dic71—42no—
122C22×Dic71—42no—
123C22×Dic71—42no—
124#11765852—48no—
125C2×C71⋊C8—22no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—22no—
129C2×C71⋊C8—22no—
130C2×C71⋊C8C22×C414yes—
131C4⋊Dic71—22no—
132C4⋊Dic71—22no—
133C4⋊Dic71—42no—
134(C2×D71)⋊C4—44no—
135(C2×D71)⋊C4—44no—
136(C2×D71)⋊C4—44no—
137(C2×D71)⋊C4—44no—
138C24×C71C2×D411yescharacteristic
139C2×Q8×C71—21no—
140C2×Q8×C71—22no—
141C2×Q8×C71—22no—
142C2×Q8×C71C22×C412yes—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—21no—
148C22×C4×C71—21no—
149C22×C4×C71—22no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—41no—
153C22×C4×C71—41no—
154C22×C4×C71—41no—
155C22×C4×C71C22×C412yes—
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D411yescharacteristic
159C22×C4×C71C2×D411yescharacteristic
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71#4112yes—
162C22×C4×C71C2411yescharacteristic
163C4⋊C4×C71—24no—
164C4⋊C4×C71—21no—
165C4⋊C4×C71—22no—
166C4⋊C4×C71—22no—
167C4⋊C4×C71—42no—
168C4⋊C4×C71—42no—
169C4⋊C4×C71C22×C412yes—
170C22⋊C4×C71—24no—
171C22⋊C4×C71—42no—
172C42×C71—22no—
173C42×C71—22no—
174C42×C71—41no—
175C42×C71—41no—
176C42×C71—42no—
177C42×C71—42no—
178C23×C71—21no—
179C23×C71—21no—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71C4×D412yes—
183C23×C71AGL(1,ℤ/8)12yes—
184C23×C71#13712yes—
185C23×C71C22×D411yescharacteristic
186C22×D71—24no—
187C22×D71—48no—
188C22×D71—48no—
189C22×D71—42no—
190C22×D71—42no—
191C22×D71—42no—
192C22×D71—42no—
193C22×D71—42no—
194C22×D71—42no—
195C22×D71—48no—
196C22×D71—84no—
197C22×D71—84no—
198C22×D71—84no—
199C22×D71—84no—
200Q8×C71—24no—
201Q8×C71—42no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
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—44no—
220C2×C4×C71—44no—
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—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71C4×D412yes—
232C2×C4×C71AGL(1,ℤ/8)12yes—
233C2×C4×C71C23×C411yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71H(5,𝔽2)12yes—
236C2×C4×C71H(5,𝔽2)12yes—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—84no—
241D284—24no—
242D284—24no—
243D284—48no—
244D284—42no—
245D284—42no—
246C71⋊C8—24no—
247C71⋊C8—42no—
248C22×C71—22no—
249C22×C71—22no—
250C22×C71—22no—
251C22×C71—22no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71#44014yes—
255C22×C71C2×C4×D411yescharacteristic
256C22×C71#53411yescharacteristic
257C22×C71#53511yescharacteristic
258C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C71C2×#13711yescharacteristic
261C22×C71#57511yescharacteristic
262C2×D71—44no—
263C2×D71—44no—
264C2×D71—44no—
265C2×D71—816no—
266C2×D71—82no—
267C2×D71—82no—
268C2×D71—82no—
269C2×D71—82no—
270C4×C71—22no—
271C4×C71—22no—
272C4×C71—42no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—42no—
277C4×C71—81no—
278C4×C71—81no—
279C4×C71#51812yes—
280C4×C71#53812yes—
281Dic71—82no—
282#22049—711nomaximal, Sylow
283C2×C71—24no—
284C2×C71#232111yescharacteristic
285C2×C71C2×#44011yescharacteristic
286C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
287C2×C71#319311yescharacteristic
288C2×C71#319411yescharacteristic
289C2×C71#321511yescharacteristic
290C2×C71#321711yescharacteristic
291D71—84no—
292C2×#450—711no—
293C4×C22⋊Q8D7111yescharacteristic, complemented
294C2×#390—711no—
295#1973—712no—
296#1936—712no—
297#1925—712no—
298#1880—712no—
299#1872—711no—
300#1852—711no—
301#1786—711no—
302#1749—711no—
303C71#2204911yescharacteristic, complemented, Sylow
304C23×D4—711no—
305C22×SD16—712no—
306C2×C22⋊Q8C2×D7111yescharacteristic
307C2×C4×Q8C2×D7111yescharacteristic, complemented
308C22×C4⋊C4—711no—
309C22×C22⋊C4—1421no—
310C22×C22⋊C4—1421no—
311C22×C42C2×D7111yescharacteristic
312#450—718no—
313C2×#107—712no—
314C2×#103—712no—
315C2×#102—712no—
316C2×#102—712no—
317C2×#102—712no—
318C2×C22⋊C8—711no—
319C2×C22⋊C8—711no—
320C2×#97—712no—
321#391C2×D7111yescharacteristic, complemented
322#390—712no—
323#390—712no—
324#390—1421no—
325#390—1421no—
326#389C2×D7111yescharacteristic, complemented
327#387—21no—
328#385—21no—
329#380C2×D7111yescharacteristic
330C4×C4⋊C4—21no—
331C4×C4⋊C4C2×D7111yescharacteristic, complemented
332C4×C22⋊C4—21no—
333#340—712no—
334#340—712no—
335#336—712no—
336C25—1421no—
337C22×Q8C22×D7111yescharacteristic
338C22×D4—2842no—
339C22×D4—2842no—
340C22×D4—714no—
341C22×D4—712no—
342C22×D4—1421no—
343C22×D4—1422no—
344C23×C4—1421no—
345C23×C4C22⋊D7111yescharacteristic
346C23×C4C22⋊D7111yescharacteristic
347C23×C4C22×D7111yescharacteristic
348C2×SD16—2842no—
349C2×SD16—2842no—
350C2×SD16—718no—
351C22×C8—712no—
352C22×C8—712no—
353C22⋊Q8—22no—
354C22⋊Q8C4×D7114yes—
355C4×Q8—42no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—2842no—
360C2×C4⋊C4—712no—
361C2×C4⋊C4—1421no—
362C2×C4⋊C4C22×D7111yescharacteristic
363C2×C4⋊C4C22×D7111yescharacteristic
364C2×C4⋊C4C22×D7111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—2844no—
368C2×C22⋊C4—2844no—
369C2×C22⋊C4—1422no—
370C2×C22⋊C4—1422no—
371C2×C22⋊C4—1422no—
372C2×C22⋊C4—1422no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—42no—
376C2×C42C22⋊D7111yescharacteristic
377C2×C42C22⋊D7111yescharacteristic
378C2×C42C22×D7111yescharacteristic
379C2×C42C22×D7111yescharacteristic
380#107—2842no—
381#103—2842no—
382#102—2842no—
383#102—2842no—
384#102—718no—
385C22⋊C8—714no—
386C22⋊C8—1422no—
387#97—2842no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393C24—2842no—
394C24—2842no—
395C24—2842no—
396C24—2842no—
397C24—2842no—
398C24—2842no—
399C24—1422no—
400C24—1424no—
401C24C2×C22⋊D7111yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C2×C4×D7112yes—
406C2×D4—2842no—
407C2×D4—2842no—
408C2×D4—2848no—
409C2×D4—2848no—
410C2×D4—714no—
411C2×D4—1422no—
412C2×D4—1422no—
413C2×D4—1422no—
414C2×D4—1422no—
415C2×D4—1422no—
416C2×D4—1424no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—2842no—
430C22×C4—2842no—
431C22×C4—2842no—
432C22×C4—1422no—
433C22×C4C2×C4×D7112yes—
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4#1176586512yes—
440C22×C4C23×D7111yescharacteristic
441SD16—2848no—
442C2×C8—714no—
443C2×C8—1422no—
444C2×C8—1422no—
445C2×C8—1422no—
446C2×C8—1422no—
447C2×C8—1422no—
448C4⋊C4—21no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—2842no—
455C4⋊C4—1422no—
456C4⋊C4—1422no—
457C4⋊C4C2×C4×D7112yes—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—2844no—
461C22⋊C4—2844no—
462C22⋊C4—2844no—
463C22⋊C4—2844no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—42no—
469C42—42no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—2848no—
475C23—2848no—
476C23—2842no—
477C23—2848no—
478C23—2842no—
479C23—2842no—
480C23—2842no—
481C23—2842no—
482C23—2842no—
483C23—5684no—
484C23—5684no—
485C23—5684no—
486C23—5684no—
487C23—1424no—
488C23C4×C22⋊D7112yes—
489C23#1455615312yes—
490C23#1455615912yes—
491C23C22×C22⋊D7111yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—2848no—
495D4—2842no—
496D4—2842no—
497D4—1424no—
498D4—1424no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—2842no—
529C2×C4—2842no—
530C2×C4—2842no—
531C2×C4—5684no—
532C2×C4C4×C22⋊D7112yes—
533C2×C4#1455615312yes—
534C2×C4C22×C4×D7111yescharacteristic
535C2×C4C2×#1176586511yescharacteristic
536C2×C4#1455617312yes—
537C2×C4#1455617412yes—
538C8—2842no—
539C8—1424no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—2844no—
547C22—2844no—
548C22—2844no—
549C22—56816no—
550C22—5682no—
551C22—5682no—
552C22—5682no—
553C22—5682no—
554C22#1798609714yes—
555C22C2×C4×C22⋊D7111yescharacteristic
556C22#1798655811yescharacteristic
557C22#1798656311yescharacteristic
558C22C2×#1455615311yescharacteristic
559C22C2×#1455615311yescharacteristic
560C22C2×#1455615911yescharacteristic
561C22#1798679711yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—5682no—
572C4#1798647112yes—
573C4#1798658212yes—
574C2—24no—
575C2—5684no—
576C2#2259099411yescharacteristic
577C2C2×#1798609711yescharacteristic
578C2C4×#1455615311yescharacteristic
579C2#2259914211yescharacteristic
580C2#2259915011yescharacteristic
581C2#2259933911yescharacteristic
582C2#2259935511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×C22⋊Q8×C71C2
socleC23×C71C22×D4
prime radicalC23×D284C22
cosocleC22×C4×C71C24
hypercenterC4×C22⋊Q8D71
hypocenterC71#22049
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22589224C2semidirect
4. #22589036C2semidirect
5. #22588960C2semidirect
6. #22588695C2semidirect
7. #22588648C2semidirect
8. #22588527C2semidirect
10. #22588102C2semidirect
11. C4×C22⋊Q8×C71C2semidirect
17. #17986149C4semidirect
18. C2×#14556061C22semidirect
19. C2×#14556052C22semidirect
25. C2×#14556039C22semidirect
30. #17985761C22semidirect
31. #17985761C22semidirect
32. #17985754C22semidirect
34. C2×C4×Q8×C71C22semidirect
36. #391×C71C22semidirect
37. #389×C71C22semidirect
42. C4×C4⋊C4×C71C22semidirect
54. C2×#11765852C2×C4semidirect
70. #14556048C2×C4semidirect
293. C4×C22⋊Q8D71semidirect
303. C71#22049semidirect
307. C2×C4×Q8C2×D71semidirect
321. #391C2×D71semidirect
326. #389C2×D71semidirect
331. C4×C4⋊C4C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179861492
C2×#179859012
#225892242
#225890362
#225889602
#225886952
#225886482
#225885272
#225882702
#225881022
C4×C22⋊Q8×C712
#2204971