Finite Group Databasebeta

← #34026386

Subgroups — #34026386

Order 18688.

1354 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊#64408C212yesmaximal, complemented
2#22626122C214yesmaximal, complemented
3#22626121C212yesmaximal, complemented
4#22625793C212yesmaximal
5#22625791C211yesmaximal, characteristic
6#22625678C212yesmaximal
7C4×#14556649C211yesmaximal, characteristic
8C2×#17989250C211yesmaximal, characteristic
9C2×Q8×#64408—22no—
10C2×#14556649C2211yescharacteristic
11C2×#14556649C2212yes—
12C2×#14556649C2211yescharacteristic
13(C2×#64408)⋊Q8—22no—
14(C2×#64408)⋊Q8—24no—
15#17989529C2214yes—
16#17989529C2212yes—
17C4×C22⋊#64408C2212yes—
18C4×C22⋊#64408C2214yes—
19#17989465C2214yes—
20#17989465C2214yes—
21#17989465C2212yes—
22#17989465C2212yes—
23C23×Dic146C411yescharacteristic, complemented
24C22×Q8×D73C2211yescharacteristic
25#17989250—22no—
26#17989250—22no—
27#17989250—24no—
28C2×#14556590C412yescomplemented
29C2×#14556589C411yescharacteristic, complemented
30C2×#14556571C2211yescharacteristic
31C2×#14556569C2212yes—
32C2×#14556568C412yescomplemented
33C2×#14556556C411yescharacteristic, complemented
34C2×C4×C22⋊D73C2211yescharacteristic
35C2×#14556551C2212yes—
36C2×C22⋊Q8×C73C411yescharacteristic, complemented
37Q8×#64408—48no—
38C2×C4⋊#64408—24no—
39C2×C4⋊#64408—22no—
40C2×C22⋊#64408C2312yes—
41C2×C22⋊#64408C2314yes—
42C2×C22⋊#64408C2312yes—
43C2×C4×#64408—24no—
44C2×C4×#64408—22no—
45#14556649—22no—
46#14556649—24no—
47#14556649Q814yescomplemented
48(C2×#64408)⋊C4—24no—
49(C2×#64408)⋊C4—22no—
50(C2×#64408)⋊C4—24no—
51(C2×#64408)⋊C4—22no—
52C22×Dic146—22no—
53C22×Dic146—21no—
54C22×Dic146—21no—
55C22×Dic146—44no—
56C22×Dic146C2×C411yescharacteristic
57C22×Dic146C2×C411yescharacteristic
58C22×Dic146C2×C412yes—
59C22×Dic146C2×C412yes—
60C2×Q8×D73—44no—
61C2×Q8×D73D412yes—
62C2×Q8×D73D412yes—
63C2×Q8×D73D414yes—
64C22×C22⋊D73C2311yescharacteristic
65C23×Dic73C2×C412yes—
66C22×C4×D73C2312yes—
67C22×C4×D73C2311yescharacteristic
68#14556590—24no—
69#14556590—24no—
70#14556589—22no—
71#14556589—22no—
72#14556571—22no—
73#14556571—22no—
74#14556569—24no—
75#14556569—24no—
76#14556568—24no—
77#14556568—24no—
78#14556556—22no—
79#14556556—21no—
80#14556556—21no—
81C4×C22⋊D73—22no—
82C4×C22⋊D73—22no—
83#14556551—24no—
84#14556551—24no—
85C2×C4⋊Dic73C2×C412yes—
86C2×#11924007C2×C412yes—
87C2×#11924007C2×C412yes—
88C2×#11924007C2×C411yescharacteristic
89C2×#11924007C2×C412yes—
90C2×C22⋊Dic73C2×C411yescharacteristic
91C2×#11924005C2312yes—
92C2×#11924005C2311yescharacteristic
93C2×C4×Dic73C2×C411yescharacteristic
94C2×C4×Dic73C2×C412yes—
95C22×Q8×C73C2×C411yescharacteristic
96C23×C4×C73C2×C411yescharacteristic
97C22⋊Q8×C73—22no—
98C22⋊Q8×C73—22no—
99C2×C4⋊C4×C73C2×C412yes—
100C2×C4⋊C4×C73C2×C411yescharacteristic
101C2×C22⋊C4×C73C2×C412yes—
102C22×#64408—22no—
103C22×#64408—24no—
104C22×#64408—22no—
105C4⋊#64408—44no—
106C4⋊#64408—48no—
107C22⋊#64408—22no—
108C22⋊#64408—22no—
109C22⋊#64408—24no—
110C22⋊#64408—22no—
111C22⋊#64408—24no—
112C22⋊#64408C2×Q814yescomplemented
113C4×#64408—44no—
114C4×#64408—48no—
115C23×D73C2411yescharacteristic
116C2×Dic146—24no—
117C2×Dic146—21no—
118C2×Dic146—21no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146—44no—
122C2×Dic146—44no—
123C2×Dic146—48no—
124C2×Dic146—44no—
125C2×Dic146—44no—
126C2×Dic146—42no—
127C2×Dic146C22⋊C411yescharacteristic
128C2×Dic146C22⋊C412yes—
129C2×Dic146C22⋊C414yes—
130C2×Dic146C22⋊C414yes—
131C2×Dic146C22⋊C411yescharacteristic
132Q8×D73—28no—
133Q8×D73—48no—
134Q8×D73—44no—
135C2×C22⋊D73—22no—
136C2×C22⋊D73—22no—
137C2×C22⋊D73C2×Q812yes—
138C2×C22⋊D73#4112yes—
139C22×Dic73—22no—
140C22×Dic73—22no—
141C22×Dic73—44no—
142C22×Dic73C22×C412yes—
143C22×Dic73C22×C411yescharacteristic
144C22×Dic73C22×C411yescharacteristic
145C22×Dic73C22×C412yes—
146C22×Dic73C22×C411yescharacteristic
147C2×C4×D73—44no—
148C2×C4×D73—42no—
149C2×C4×D73C2×D414yes—
150C2×C4×D73C2×D412yes—
151C2×C4×D73C2×D414yes—
152C2×C4×D73C2×D412yes—
153C4⋊Dic73—24no—
154#11924007—24no—
155#11924007—24no—
156#11924007—22no—
157#11924007—22no—
158#11924007—22no—
159C22⋊Dic73—22no—
160#11924005—24no—
161#11924005—22no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—22no—
165C4×Dic73—22no—
166C24×C73C22×C411yescharacteristic
167C2×Q8×C73—42no—
168C2×Q8×C73C22⋊C412yes—
169C2×Q8×C73C22⋊C412yes—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—42no—
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C22×C412yes—
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C22×C412yes—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—24no—
180C22⋊C4×C73—24no—
181C23×C73—21no—
182C23×C73—21no—
183C23×C73—41no—
184C23×C73—41no—
185C23×C73C4×Q812yescomplemented
186C23×C73C23×C411yescharacteristic
187C22×D73—41no—
188C22×D73—41no—
189C22×D73C22×D412yes—
190C22×D73C22×Q811yescharacteristic
191C22×D73C2×#4111yescharacteristic
192C22×D732−1+412yes—
193C2×#64408—24no—
194C2×#64408—42no—
195C2×#64408—42no—
196C2×#64408—42no—
197C2×#64408—44no—
198C2×#64408—44no—
199C2×#64408—42no—
200C2×#64408—42no—
201C2×#64408—44no—
202Q8×C73—24no—
203Q8×C73—42no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—21no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—41no—
211C2×C4×C73—42no—
212C2×C4×C73C2×C22⋊C414yes—
213C2×C4×C73C2×C22⋊C411yescharacteristic
214C2×C4×C73C2×C22⋊C411yescharacteristic
215C22⋊D73—24no—
216C22⋊D73—24no—
217C2×Dic73—21no—
218C2×Dic73—21no—
219C2×Dic73—24no—
220C2×Dic73—21no—
221C2×Dic73—22no—
222C2×Dic73—41no—
223C2×Dic73—44no—
224C2×Dic73—44no—
225C2×Dic73—42no—
226C2×Dic73—41no—
227C2×Dic73C2×C22⋊C412yes—
228C2×Dic73C2×C22⋊C412yes—
229C2×Dic73C2×C22⋊C412yes—
230C2×Dic73C4×Q812yescomplemented
231C4×D73—24no—
232C4×D73—28no—
233C4×D73—44no—
234C4×D73—42no—
235Dic146—22no—
236Dic146—28no—
237Dic146—24no—
238Dic146—44no—
239Dic146—44no—
240Dic146—42no—
241Dic146—42no—
242Dic146—44no—
243Dic146#9914yescomplemented
244C22×C73—21no—
245C22×C73—21no—
246C22×C73—22no—
247C22×C73—42no—
248C22×C73—42no—
249C22×C73—41no—
250C22×C73—41no—
251C22×C73C22×C22⋊C411yescharacteristic
252C22×C73C2×C4×Q811yescharacteristic
253C22×C73#51911yescharacteristic
254C2×D73—22no—
255C2×D73—22no—
256C2×D73—41no—
257C2×D73—41no—
258C2×D73—42no—
259C2×D73#54812yes—
260C2×D73D4×Q812yes—
261#64408—44no—
262#64408—82no—
263#64408—84no—
264C4×C73—21no—
265C4×C73—24no—
266C4×C73—41no—
267C4×C73—42no—
268C4×C73C2×#9912yes—
269Dic73—22no—
270Dic73—22no—
271Dic73—22no—
272Dic73—42no—
273Dic73—41no—
274Dic73C2×#9914yes—
275Q8×#99—731nomaximal, Sylow
276C2×C73—21no—
277C2×C73—21no—
278C2×C73—42no—
279C2×C73Q8×C22⋊C411yescharacteristic
280C2×C73C22×#9911yescharacteristic
281C2×C73#288411yescharacteristic
282D73—22no—
283D73—22no—
284C2×D4×Q8—731no—
285Q8×C22⋊C4—732no—
286#1996—732no—
287#1996—734no—
288#1799—732no—
289#1799—731no—
290C4×#99—731no—
291C4×#99—732no—
292C73Q8×#9911yescharacteristic, complemented, Sylow
293C23×Q8—731no—
294C23×Q8—731no—
295D4×Q8—1462no—
296D4×Q8—1464no—
297D4×Q8—1462no—
298C2×C4⋊Q8—731no—
299C2×C4⋊Q8—732no—
300C2×C22⋊Q8—732no—
301C2×C22⋊Q8—731no—
302C2×C22⋊Q8—732no—
303C2×C22⋊Q8#6440811yescharacteristic, complemented
304C2×C4×Q8—731no—
305C2×C4×Q8—1462no—
306C2×C4×D4—731no—
307C2×C4×D4—732no—
308C2×#99—731no—
309C2×#99—731no—
310C2×#99—732no—
311#391—1462no—
312#391—1464no—
313#380—734no—
314#380—732no—
315C4×C22⋊C4—732no—
316C4×C22⋊C4—734no—
317#342—734no—
318#342—732no—
319#342—734no—
320#342—732no—
321C22×Q8—2924no—
322C22×Q8—2924no—
323C22×Q8—732no—
324C22×Q8—734no—
325C22×Q8—732no—
326C22×Q8—732no—
327C22×Q8—731no—
328C22×Q8—732no—
329C22×Q8—731no—
330C22×Q8—1461no—
331C22×Q8—1462no—
332C22×Q8—1461no—
333C22×Q8C2×#6440811yescharacteristic
334C22×D4—731no—
335C23×C4—731no—
336C23×C4—732no—
337C23×C4—732no—
338C23×C4C2×#6440811yescharacteristic
339C4⋊Q8—1464no—
340C4⋊Q8—1464no—
341C4⋊Q8—1462no—
342C4⋊Q8—1462no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—1464no—
346C22⋊Q8—1464no—
347C22⋊Q8—1464no—
348C22⋊Q8—1464no—
349C22⋊Q8—1462no—
350C22⋊Q8—1462no—
351C4×Q8—2928no—
352C4×Q8—1461no—
353C4×Q8—1462no—
354C4×Q8—1461no—
355C4×D4—1464no—
356C4×D4—1464no—
357C4×D4—1462no—
358C4×D4—1462no—
359C2×C4⋊C4—731no—
360C2×C4⋊C4—732no—
361C2×C4⋊C4—732no—
362C2×C4⋊C4—732no—
363C2×C4⋊C4—732no—
364C2×C4⋊C4—1464no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4C2×#6440811yescharacteristic
367C2×C4⋊C4C2×#6440812yes—
368C2×C22⋊C4—732no—
369C2×C22⋊C4—731no—
370C2×C22⋊C4—731no—
371C2×C22⋊C4—732no—
372C2×C22⋊C4—732no—
373C2×C22⋊C4—734no—
374C2×C22⋊C4C2×#6440812yes—
375C2×C42—732no—
376C2×C42—731no—
377C2×C42—1464no—
378C2×C42—1462no—
379#99—734no—
380#99—1462no—
381#99—1464no—
382#95—1464no—
383#95—1462no—
384#95—1464no—
385#95—1462no—
386C24—731no—
387C24C22×#6440811yescharacteristic
388C2×Q8—42no—
389C2×Q8—2922no—
390C2×Q8—2924no—
391C2×Q8—2928no—
392C2×Q8—2924no—
393C2×Q8—2924no—
394C2×Q8—2924no—
395C2×Q8—2924no—
396C2×Q8—2928no—
397C2×Q8—734no—
398C2×Q8—731no—
399C2×Q8—732no—
400C2×Q8—731no—
401C2×Q8—734no—
402C2×Q8—1462no—
403C2×Q8—1462no—
404C2×Q8—1464no—
405C2×Q8—1468no—
406C2×Q8—1461no—
407C2×Q8—1461no—
408C2×Q8C22⋊#6440812yes—
409C2×Q8C22⋊#6440812yes—
410C2×D4—732no—
411C2×D4—732no—
412C2×D4—1462no—
413C2×D4—1462no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—2922no—
418C22×C4—2924no—
419C22×C4—2924no—
420C22×C4—731no—
421C22×C4—734no—
422C22×C4—732no—
423C22×C4—731no—
424C22×C4—732no—
425C22×C4—731no—
426C22×C4—732no—
427C22×C4—734no—
428C22×C4—732no—
429C22×C4—1464no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4C22×#6440811yescharacteristic
435C22×C4C22×#6440812yes—
436C22×C4C22×#6440811yescharacteristic
437C22×C4C22×#6440812yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—2928no—
442C4⋊C4—2924no—
443C4⋊C4—1462no—
444C4⋊C4—1462no—
445C4⋊C4—1462no—
446C4⋊C4—1464no—
447C4⋊C4—1464no—
448C4⋊C4—1464no—
449C22⋊C4—24no—
450C22⋊C4—734no—
451C22⋊C4—1464no—
452C22⋊C4—1462no—
453C22⋊C4—1462no—
454C22⋊C4—1464no—
455C22⋊C4—1462no—
456C22⋊C4—1464no—
457C22⋊C4—1462no—
458C22⋊C4—1462no—
459C42—2928no—
460C42—2924no—
461C42—1462no—
462C42—1462no—
463C42—1461no—
464C42—1461no—
465C23—21no—
466C23—21no—
467C23—41no—
468C23—41no—
469C23—2921no—
470C23—2921no—
471C23—731no—
472C23—731no—
473C23—732no—
474C23—732no—
475C23Q8×#6440812yescomplemented
476C23C23×#6440811yescharacteristic
477Q8—24no—
478Q8—42no—
479Q8—2922no—
480Q8—2924no—
481Q8—2924no—
482Q8—2924no—
483Q8—2922no—
484Q8—734no—
485Q8—1464no—
486Q8—1468no—
487Q8—1462no—
488D4—1464no—
489D4—1464no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—2922no—
499C2×C4—2924no—
500C2×C4—2922no—
501C2×C4—2922no—
502C2×C4—2924no—
503C2×C4—2924no—
504C2×C4—2922no—
505C2×C4—2921no—
506C2×C4—2924no—
507C2×C4—2922no—
508C2×C4—2924no—
509C2×C4—2922no—
510C2×C4—2924no—
511C2×C4—2922no—
512C2×C4—2921no—
513C2×C4—732no—
514C2×C4—732no—
515C2×C4—732no—
516C2×C4—732no—
517C2×C4—1468no—
518C2×C4—1464no—
519C2×C4—1461no—
520C2×C4—1462no—
521C2×C4—1464no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1464no—
525C2×C4C2×C22⋊#6440811yescharacteristic
526C2×C4C2×C22⋊#6440814yes—
527C2×C4C2×C22⋊#6440811yescharacteristic
528C22—21no—
529C22—21no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—2921no—
536C22—2921no—
537C22—2922no—
538C22—732no—
539C22—732no—
540C22—1462no—
541C22—1462no—
542C22C22×C22⋊#6440811yescharacteristic
543C22C2×Q8×#6440811yescharacteristic
544C22#1798965511yescharacteristic
545C4—21no—
546C4—24no—
547C4—41no—
548C4—42no—
549C4—2924no—
550C4—2922no—
551C4—2921no—
552C4—5844no—
553C4—5842no—
554C4—734no—
555C4—1462no—
556C4—1462no—
557C4—1462no—
558C4D73⋊#9912yes—
559C2—21no—
560C2—21no—
561C2—42no—
562C2—1462no—
563C2—1462no—
564C2Q8×C22⋊#6440811yescharacteristic
565C2C2×D73⋊#9911yescharacteristic
566C2#2262653311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C22⋊Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×C22⋊D73C23
cosocleC23×D73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73Q8×#99
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊#64408C2semidirect
2. #22626122C2semidirect
3. #22626121C2semidirect
23. C23×Dic146C4semidirect
28. C2×#14556590C4semidirect
29. C2×#14556589C4semidirect
32. C2×#14556568C4semidirect
33. C2×#14556556C4semidirect
36. C2×C22⋊Q8×C73C4semidirect
47. #14556649Q8semidirect
112. C22⋊#64408C2×Q8semidirect
185. C23×C73C4×Q8semidirect
230. C2×Dic73C4×Q8semidirect
243. Dic146#99semidirect
292. C73Q8×#99semidirect
303. C2×C22⋊Q8#64408semidirect
475. C23Q8×#64408semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊#644082
#226261222
#226261212
#226257932
#226257912
#226256782
C4×#145566492
C2×#179892502
Q8×#9973