Finite Group Databasebeta

← #33011466

Subgroups — #33011466

Order 18688.

1786 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556556C211yesmaximal, characteristic
2C2×#17988508C211yesmaximal, characteristic
3C2×#17988483C212yesmaximal
4C2×#17988469C211yesmaximal, characteristic
5#22608201C211yesmaximal, characteristic
6#22608181C211yesmaximal, characteristic
7#22608165C212yesmaximal
8#22608163C212yesmaximal
9#22608162C212yesmaximal
10#95×Dic73C211yesmaximal, characteristic
11C73×#1444C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C2×#14556556—28no—
14C2×#14556556—24no—
15C22×C4⋊Dic73C2212yes—
16C22×#11924007C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2212yes—
19C22×C4×Dic73C2212yes—
20C22×C4×Dic73C2212yes—
21C22×C4×Dic73C2211yescharacteristic
22#17988508C418yescomplemented
23#17988483—28no—
24#17988469—24no—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2211yescharacteristic
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2212yes—
37C2×#95×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2211yescharacteristic
40C22×Dic146—24no—
41C22×Dic146—24no—
42C22×Dic146—22no—
43C22×Dic146—22no—
44C22×Dic146C2×C414yes—
45C23×Dic73C2311yescharacteristic
46C23×Dic73C2311yescharacteristic
47C23×Dic73C2312yes—
48C23×Dic73C2312yes—
49C23×Dic73C2311yescharacteristic
50C23×Dic73C2311yescharacteristic
51#14556556—232no—
52C2×C4⋊Dic73—24no—
53C2×C4⋊Dic73—24no—
54C2×C4⋊Dic73C2×C418yes—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—22no—
58C2×#11924007—24no—
59C2×#11924007—22no—
60C2×#11924007—22no—
61C2×#11924007C2×C414yes—
62C2×#11924007D418yes—
63C2×C4×Dic73—24no—
64C2×C4×Dic73—24no—
65C2×C4×Dic73—24no—
66C2×C4×Dic73—24no—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—42no—
70C2×C4×Dic73D414yes—
71#14556498—22no—
72#14556498—22no—
73#14556498—22no—
74#14556498—24no—
75#14556498—24no—
76#14556498—24no—
77#14556498—22no—
78#14556498—42no—
79#14556498C2×C418yes—
80#14556498Q818yescomplemented
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2312yes—
86C2×C4⋊C4×C73—24no—
87C2×C4⋊C4×C73—22no—
88C2×C4⋊C4×C73—24no—
89C2×C4⋊C4×C73—24no—
90C2×C4⋊C4×C73—42no—
91#95×C73—22no—
92#95×C73—24no—
93#95×C73C2×C414yes—
94C2×Dic146—28no—
95C2×Dic146—216no—
96C2×Dic146—28no—
97C2×Dic146—28no—
98C2×Dic146—216no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—21no—
117C22×Dic73—21no—
118C22×Dic73—22no—
119C22×Dic73—21no—
120C22×Dic73—22no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—22no—
125C22×Dic73C22×C412yes—
126C22×Dic73C22×C412yes—
127C22×Dic73C22×C414yes—
128C22×Dic73C2×D414yes—
129C22×Dic73C2×D412yes—
130C22×Dic73C2×D414yes—
131C22×Dic73C2×D412yes—
132C22×Dic73C2×Q814yes—
133C22×Dic73C2×Q814yes—
134C22×Dic73#4114yes—
135C4⋊Dic73—216no—
136#11924007—28no—
137#11924007—216no—
138#11924007—44no—
139C4×Dic73—216no—
140C4×Dic73—28no—
141C4×Dic73—48no—
142C24×C73C2411yescharacteristic
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73C22×C412yes—
166C22×C4×C73C22×C414yes—
167C22×C4×C73C2×D414yes—
168C22×C4×C73C2×D412yes—
169C22×C4×C73C2×Q814yes—
170C22×C4×C73#4114yes—
171C22×C4×C73#4112yes—
172C4⋊C4×C73—28no—
173C4⋊C4×C73—48no—
174C23×C73C23×C411yescharacteristic
175C23×C73C22×D411yescharacteristic
176C23×C73C22×D412yes—
177C23×C73C22×Q812yes—
178C23×C73C2×#4112yes—
179C23×C73C2×#4112yes—
180C23×C73C2×#4111yescharacteristic
181C23×C73H(5,𝔽2)11yescharacteristic
182C23×C732−1+411yescharacteristic
183C23×C732−1+411yescharacteristic
184C23×C732−1+411yescharacteristic
185C2×C4×C73—24no—
186C2×C4×C73—24no—
187C2×C4×C73—24no—
188C2×C4×C73—22no—
189C2×C4×C73—22no—
190C2×C4×C73—24no—
191C2×C4×C73—24no—
192C2×C4×C73—24no—
193C2×C4×C73—24no—
194C2×C4×C73—24no—
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—24no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—42no—
201C2×C4×C73—42no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73C4×D414yes—
213C2×C4×C73C4×Q818yescomplemented
214C2×Dic73—24no—
215C2×Dic73—24no—
216C2×Dic73—22no—
217C2×Dic73—24no—
218C2×Dic73—24no—
219C2×Dic73—24no—
220C2×Dic73—24no—
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—22no—
224C2×Dic73—24no—
225C2×Dic73—24no—
226C2×Dic73—24no—
227C2×Dic73—24no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—24no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73C4×D418yes—
248C2×Dic73#12014yes—
249C2×Dic73C4⋊Q814yes—
250Dic146—232no—
251C22×C73C2×C4×D412yes—
252C22×C73C2×C4×D411yescharacteristic
253C22×C73C2×C4×Q812yes—
254C22×C73#51911yescharacteristic
255C22×C73#52011yescharacteristic
256C22×C73C2×#12011yescharacteristic
257C22×C73#52511yescharacteristic
258C22×C73#52512yes—
259C22×C73C2×C4⋊Q811yescharacteristic
260C22×C73#53211yescharacteristic
261C22×C73#53312yes—
262C22×C73#53611yescharacteristic
263C22×C73#53711yescharacteristic
264C22×C73#54111yescharacteristic
265C22×C73#54211yescharacteristic
266C22×C73#54612yes—
267C22×C73#54712yes—
268C22×C73D4×Q812yes—
269C22×C73D4×Q812yes—
270C22×C73#55212yes—
271C22×C73#55412yes—
272C22×C73#55612yes—
273C22×C73#55712yes—
274C4×C73—28no—
275C4×C73—24no—
276C4×C73—44no—
277C4×C73—81no—
278C4×C73—81no—
279Dic73—24no—
280Dic73—24no—
281Dic73—28no—
282Dic73—42no—
283Dic73—44no—
284Dic73—42no—
285#14186—731nomaximal, Sylow
286C2×C73C4×#12011yescharacteristic
287C2×C73C4×C4⋊Q811yescharacteristic
288C2×C73#232311yescharacteristic
289C2×C73#236012yes—
290C2×C73#236512yes—
291C2×C73#243011yescharacteristic
292C2×C73#243411yescharacteristic
293C2×C73#246212yes—
294C2×C73#246512yes—
295C2×C73#258411yescharacteristic
296C2×C73#260411yescharacteristic
297C22×C4×Q8—731no—
298C2×#395—731no—
299C2×#389—732no—
300C2×#387—731no—
301#1446—731no—
302#1444D7311yescharacteristic
303#1441—731no—
304#1436—732no—
305#1436—732no—
306#1436—732no—
307C4×#95—731no—
308C73#1418611yescharacteristic, complemented, Sylow
309C23×Q8—731no—
310C2×C4×Q8—1464no—
311C2×C4×Q8—1468no—
312C22×C4⋊C4—732no—
313C22×C4⋊C4—731no—
314C22×C4⋊C4—732no—
315C22×C4⋊C4—731no—
316C22×C4⋊C4C2×D7311yescharacteristic
317C22×C4⋊C4C2×D7312yes—
318C22×C42—732no—
319C22×C42—731no—
320C22×C42—732no—
321#395—738no—
322#389—1468no—
323#387—1464no—
324C2×#95—732no—
325C2×#95—732no—
326C2×#95—732no—
327C2×#95—731no—
328C2×#95—732no—
329C2×#95—731no—
330C2×#95—731no—
331C2×#95—731no—
332C2×#95—732no—
333C2×#95—732no—
334C2×#95C2×D7312yes—
335C2×#95C2×D7311yescharacteristic
336C2×#95C2×D7311yescharacteristic
337C22×Q8—734no—
338C22×Q8—1464no—
339C22×Q8—1464no—
340C22×Q8—1462no—
341C22×Q8—1462no—
342C23×C4—731no—
343C23×C4—732no—
344C23×C4—731no—
345C23×C4—731no—
346C23×C4—731no—
347C23×C4—732no—
348C23×C4C22×D7311yescharacteristic
349C23×C4C22×D7311yescharacteristic
350C23×C4C22×D7312yes—
351C23×C4C22×D7311yescharacteristic
352C23×C4C22×D7312yes—
353C4×Q8—14632no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4—734no—
360C2×C4⋊C4—738no—
361C2×C4⋊C4—738no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1464no—
364C2×C4⋊C4—1464no—
365C2×C4⋊C4—1464no—
366C2×C4⋊C4—1464no—
367C2×C4⋊C4—1464no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1462no—
370C2×C42—2922no—
371C2×C42—734no—
372C2×C42—1464no—
373C2×C42—1464no—
374C2×C42—1464no—
375C2×C42—1464no—
376C2×C42—1464no—
377C2×C42—1464no—
378#95—22no—
379#95—24no—
380#95—2922no—
381#95—738no—
382#95—738no—
383#95—1464no—
384#95—1462no—
385#95—1462no—
386#95—1462no—
387#95—1464no—
388#95—1464no—
389#95—1462no—
390#95C4×D7314yes—
391C24C23×D7311yescharacteristic
392C2×Q8—14616no—
393C2×Q8—1468no—
394C2×Q8—1468no—
395C2×Q8—14616no—
396C2×Q8—1468no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—734no—
420C22×C4—732no—
421C22×C4—732no—
422C22×C4—734no—
423C22×C4—734no—
424C22×C4—734no—
425C22×C4—734no—
426C22×C4—732no—
427C22×C4—732no—
428C22×C4—734no—
429C22×C4—1462no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4—1462no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1461no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4—1462no—
441C22×C4—1461no—
442C22×C4—1461no—
443C22×C4—1461no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4C2×C4×D7312yes—
456C22×C4C2×C4×D7314yes—
457C22×C4D4×D7314yes—
458C22×C4D4×D7312yes—
459C22×C4Q8×D7314yes—
460C22×C4#1192402714yes—
461C22×C4#1192402912yes—
462C4⋊C4—28no—
463C4⋊C4—48no—
464C4⋊C4—2924no—
465C4⋊C4—14616no—
466C4⋊C4—14616no—
467C4⋊C4—1468no—
468C42—2928no—
469C42—1468no—
470C42—14616no—
471C23C22×C4×D7311yescharacteristic
472C23C2×D4×D7311yescharacteristic
473C23C2×D4×D7312yes—
474C23C2×Q8×D7312yes—
475C23#41×D7312yes—
476C23C2×#1192402712yes—
477C23C2×#1192402911yescharacteristic
478C23#1455663611yescharacteristic
479C23#1455663811yescharacteristic
480C23#1455663811yescharacteristic
481C23#1455663911yescharacteristic
482Q8—14632no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4—2922no—
515C2×C4—2922no—
516C2×C4—2922no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—734no—
522C2×C4—738no—
523C2×C4—734no—
524C2×C4—1464no—
525C2×C4—1464no—
526C2×C4—1464no—
527C2×C4—1464no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1462no—
531C2×C4—1462no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1464no—
536C2×C4—1464no—
537C2×C4—1464no—
538C2×C4—1464no—
539C2×C4—1464no—
540C2×C4—1464no—
541C2×C4—1462no—
542C2×C4—1464no—
543C2×C4—1464no—
544C2×C4—1464no—
545C2×C4—1464no—
546C2×C4#1455655314yes—
547C2×C4#1455655618yescomplemented
548C22C4×D4×D7312yes—
549C22C2×#1455655311yescharacteristic
550C22C2×#1455655612yes—
551C22#1798906311yescharacteristic
552C22#1798906611yescharacteristic
553C22#120×D7311yescharacteristic
554C22#1798909511yescharacteristic
555C22#1798910012yes—
556C22C4⋊Q8×D7311yescharacteristic
557C22#1798913511yescharacteristic
558C22#1798914212yes—
559C22#1798916011yescharacteristic
560C22#1798916611yescharacteristic
561C22#1798920011yescharacteristic
562C22#1798921111yescharacteristic
563C22#1798922412yes—
564C22#1798923712yes—
565C22#1798925012yes—
566C22#1798925212yes—
567C22#1798927412yes—
568C22#1798929112yes—
569C22#1798930812yes—
570C22#1798931412yes—
571C4—28no—
572C4—24no—
573C4—44no—
574C4—81no—
575C4—81no—
576C4—2922no—
577C4—2924no—
578C4—2922no—
579C4—1464no—
580C4—1464no—
581C4—1468no—
582C2#2261243211yescharacteristic
583C2#2261249811yescharacteristic
584C2#2261261111yescharacteristic
585C2#2261293112yes—
586C2#2261298412yes—
587C2#2261353411yescharacteristic
588C2#2261357011yescharacteristic
589C2#2261391612yes—
590C2#2261395512yes—
591C2#2261531511yescharacteristic
592C2#2261557011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1444C2
socle, prime radical, cosocleC24×C73C24
hypercenter#1444D73
hypocenterC73#14186
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
22. #17988508C4semidirect
80. #14556498Q8semidirect
213. C2×C4×C73C4×Q8semidirect
308. C73#14186semidirect
547. C2×C4#14556556semidirect

Maximal subgroups

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

subgroupindex
C22×#145565562
C2×#179885082
C2×#179884832
C2×#179884692
#226082012
#226081812
#226081652
#226081632
#226081622
#95×Dic732
C73×#14442
#1418673