Finite Group Databasebeta

← #33008656

Subgroups — #33008656

Order 18688.

1630 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988484C212yesmaximal
2C2×#17988482C212yesmaximal
3C2×#17988451C211yesmaximal, characteristic
4C2×C4⋊C4×Dic73C211yesmaximal, characteristic
5C2×#17988438C211yesmaximal, characteristic
6#22608196C211yesmaximal, characteristic
7#22608195C211yesmaximal, characteristic
8#22608194C211yesmaximal, characteristic
9#22608160C212yesmaximal
10#22608159C212yesmaximal
11C73×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2212yes—
14C22×#11924007C2211yescharacteristic
15C22×#11924007C2211yescharacteristic
16C22×#11924007C2212yes—
17C22×#11924007C2212yes—
18C22×#11924007C2212yes—
19C22×C4×Dic73C2212yes—
20C22×C4×Dic73C2212yes—
21C22×C4×Dic73C2211yescharacteristic
22#17988484—28no—
23#17988482—28no—
24#17988451—24no—
25C4⋊C4×Dic73—24no—
26#17988438C418yescomplemented
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2211yescharacteristic
34C2×#14556498C2211yescharacteristic
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2212yes—
37C2×#95×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2211yescharacteristic
40C23×Dic73C2311yescharacteristic
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2312yes—
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2312yes—
45C23×Dic73C2311yescharacteristic
46C2×C4⋊Dic73—24no—
47C2×C4⋊Dic73—24no—
48C2×C4⋊Dic73—22no—
49C2×C4⋊Dic73—24no—
50C2×C4⋊Dic73—42no—
51C2×#11924007—22no—
52C2×#11924007—22no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007—42no—
60C2×#11924007—42no—
61C2×#11924007C2×C418yes—
62C2×#11924007D418yes—
63C2×#11924007Q818yes—
64C2×C4×Dic73—24no—
65C2×C4×Dic73—24no—
66C2×C4×Dic73—24no—
67C2×C4×Dic73—42no—
68C2×C4×Dic73—42no—
69C2×C4×Dic73C2×C418yes—
70C2×C4×Dic73C2×C414yes—
71C2×C4×Dic73D414yes—
72C2×C4×Dic73Q814yes—
73#14556498—24no—
74#14556498—22no—
75#14556498—22no—
76#14556498—24no—
77#14556498—24no—
78#14556498—22no—
79#14556498—24no—
80#14556498C2×C414yes—
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2312yes—
86C2×C4⋊C4×C73—22no—
87C2×C4⋊C4×C73—24no—
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—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—21no—
97C22×Dic73—21no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—21no—
109C22×Dic73—21no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—21no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—21no—
118C22×Dic73C22×C414yes—
119C22×Dic73C22×C412yes—
120C22×Dic73C22×C412yes—
121C22×Dic73C2×D412yes—
122C22×Dic73C2×D414yes—
123C22×Dic73C2×D412yes—
124C22×Dic73C2×D414yes—
125C22×Dic73C2×Q812yes—
126C22×Dic73C2×Q814yes—
127C22×Dic73C2×Q812yes—
128C22×Dic73C2×Q814yes—
129C22×Dic73#4114yes—
130C4⋊Dic73—28no—
131C4⋊Dic73—48no—
132#11924007—48no—
133#11924007—48no—
134#11924007—44no—
135#11924007—44no—
136#11924007C4⋊C4132yescomplemented
137C4×Dic73—48no—
138C4×Dic73—48no—
139C4×Dic73C4⋊C4116yescomplemented
140C24×C73C2411yescharacteristic
141C22×C4×C73—22no—
142C22×C4×C73—22no—
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—21no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73C22×C412yes—
164C22×C4×C73C22×C414yes—
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D414yes—
167C22×C4×C73C2×Q812yes—
168C22×C4×C73C2×Q814yes—
169C22×C4×C73#4114yes—
170C4⋊C4×C73—28no—
171C4⋊C4×C73—48no—
172C23×C73C23×C411yescharacteristic
173C23×C73C22×D411yescharacteristic
174C23×C73C22×D412yes—
175C23×C73C22×Q811yescharacteristic
176C23×C73C22×Q812yes—
177C23×C73C2×#4112yes—
178C23×C73C2×#4112yes—
179C23×C73H(5,𝔽2)11yescharacteristic
180C23×C73H(5,𝔽2)11yescharacteristic
181C23×C732−1+411yescharacteristic
182C23×C732−1+411yescharacteristic
183C2×C4×C73—24no—
184C2×C4×C73—24no—
185C2×C4×C73—24no—
186C2×C4×C73—22no—
187C2×C4×C73—22no—
188C2×C4×C73—24no—
189C2×C4×C73—24no—
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—42no—
197C2×C4×C73—42no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—41no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73C2×C4⋊C414yes—
211C2×C4×C73C2×C4⋊C418yes—
212C2×Dic73—24no—
213C2×Dic73—22no—
214C2×Dic73—24no—
215C2×Dic73—22no—
216C2×Dic73—22no—
217C2×Dic73—22no—
218C2×Dic73—22no—
219C2×Dic73—22no—
220C2×Dic73—24no—
221C2×Dic73—24no—
222C2×Dic73—24no—
223C2×Dic73—24no—
224C2×Dic73—24no—
225C2×Dic73—24no—
226C2×Dic73—24no—
227C2×Dic73—42no—
228C2×Dic73—42no—
229C2×Dic73—42no—
230C2×Dic73—42no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73C2×C4⋊C414yes—
242C2×Dic73C2×C4⋊C418yes—
243C2×Dic73C2×C4⋊C414yes—
244C2×Dic73C2×C4⋊C418yes—
245C2×Dic73#12714yes—
246C2×Dic73C4⋊Q814yes—
247C2×Dic73C4⋊Q818yes—
248C22×C73C22×C4⋊C411yescharacteristic
249C22×C73C22×C4⋊C412yes—
250C22×C73C4×#4112yes—
251C22×C73#51811yescharacteristic
252C22×C73#51911yescharacteristic
253C22×C73#52511yescharacteristic
254C22×C73#52511yescharacteristic
255C22×C73C2×#12711yescharacteristic
256C22×C73C2×C4⋊Q811yescharacteristic
257C22×C73C2×C4⋊Q812yes—
258C22×C73#53312yes—
259C22×C73#53711yescharacteristic
260C22×C73#53711yescharacteristic
261C22×C73#54411yescharacteristic
262C22×C73#54411yescharacteristic
263C22×C73#54612yes—
264C22×C73#54612yes—
265C22×C73#54812yes—
266C22×C73#54812yes—
267C22×C73#55412yes—
268C22×C73#55412yes—
269C22×C73#55712yes—
270C22×C73#55712yes—
271C4×C73—28no—
272C4×C73—24no—
273C4×C73—44no—
274C4×C73—81no—
275C4×C73—81no—
276Dic73—24no—
277Dic73—28no—
278Dic73—44no—
279Dic73—81no—
280Dic73—81no—
281Dic73#38418yes—
282#13954—731nomaximal, Sylow
283C2×C73C2×#38411yescharacteristic
284C2×C73#229611yescharacteristic
285C2×C73#231611yescharacteristic
286C2×C73#234912yes—
287C2×C73#235112yes—
288C2×C73#254011yescharacteristic
289C2×C73#254011yescharacteristic
290C2×C73#255012yes—
291C2×C73#255012yes—
292C2×C73#258411yescharacteristic
293C2×C73#258411yescharacteristic
294C2×#389—732no—
295C2×#389—732no—
296C2×#382—731no—
297C2×C4×C4⋊C4—731no—
298C2×C4×C4⋊C4—731no—
299#1444—731no—
300#1444—731no—
301#1444—731no—
302#1444D7311yescharacteristic
303C4⋊#95—732no—
304C4⋊#95—732no—
305C73#1395411yescharacteristic, complemented, Sylow
306C22×C4⋊C4—732no—
307C22×C4⋊C4—732no—
308C22×C4⋊C4—732no—
309C22×C4⋊C4—732no—
310C22×C4⋊C4—731no—
311C22×C4⋊C4—731no—
312C22×C4⋊C4—731no—
313C22×C4⋊C4C2×D7311yescharacteristic
314C22×C4⋊C4C2×D7312yes—
315C22×C42—732no—
316C22×C42—731no—
317C22×C42—732no—
318#389—1468no—
319#389—1468no—
320#382—1464no—
321C4×C4⋊C4—738no—
322C4×C4⋊C4—1464no—
323C2×#95—731no—
324C2×#95—732no—
325C2×#95—731no—
326C2×#95—731no—
327C2×#95—732no—
328C2×#95—732no—
329C2×#95—732no—
330C2×#95—731no—
331C2×#95C2×D7312yes—
332C2×#95C2×D7311yescharacteristic
333C2×#95C2×D7311yescharacteristic
334C23×C4—731no—
335C23×C4—732no—
336C23×C4—731no—
337C23×C4—732no—
338C23×C4—731no—
339C23×C4—731no—
340C23×C4C22×D7311yescharacteristic
341C23×C4C22×D7311yescharacteristic
342C23×C4C22×D7312yes—
343C23×C4C22×D7312yes—
344C23×C4C22×D7311yescharacteristic
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—2922no—
351C2×C4⋊C4—2922no—
352C2×C4⋊C4—2922no—
353C2×C4⋊C4—738no—
354C2×C4⋊C4—738no—
355C2×C4⋊C4—738no—
356C2×C4⋊C4—1462no—
357C2×C4⋊C4—1462no—
358C2×C4⋊C4—1464no—
359C2×C4⋊C4—1464no—
360C2×C4⋊C4—1464no—
361C2×C4⋊C4—1464no—
362C2×C4⋊C4—1464no—
363C2×C4⋊C4—1464no—
364C2×C4⋊C4—1464no—
365C2×C4⋊C4—1464no—
366C2×C4⋊C4—1464no—
367C2×C4⋊C4—1462no—
368C2×C42—2922no—
369C2×C42—2922no—
370C2×C42—738no—
371C2×C42—734no—
372C2×C42—734no—
373C2×C42—734no—
374C2×C42—1464no—
375C2×C42—1464no—
376C2×C42—1464no—
377#95—22no—
378#95—24no—
379#95—734no—
380#95—1464no—
381#95—1462no—
382#95—1462no—
383#95—1464no—
384#95—1464no—
385#95—1462no—
386#95—1464no—
387#95C4×D7314yes—
388C24C23×D7311yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—732no—
412C22×C4—732no—
413C22×C4—734no—
414C22×C4—732no—
415C22×C4—734no—
416C22×C4—734no—
417C22×C4—732no—
418C22×C4—732no—
419C22×C4—734no—
420C22×C4—732no—
421C22×C4—734no—
422C22×C4—734no—
423C22×C4—1461no—
424C22×C4—1461no—
425C22×C4—1462no—
426C22×C4—1462no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1462no—
434C22×C4—1462no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1461no—
442C22×C4—1462no—
443C22×C4—1461no—
444C22×C4—1462no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4C2×C4×D7312yes—
448C22×C4C2×C4×D7314yes—
449C22×C4D4×D7312yes—
450C22×C4D4×D7314yes—
451C22×C4Q8×D7312yes—
452C22×C4Q8×D7314yes—
453C22×C4#1192402714yes—
454C4⋊C4—28no—
455C4⋊C4—48no—
456C4⋊C4—2928no—
457C4⋊C4—2928no—
458C4⋊C4—2924no—
459C4⋊C4—2924no—
460C4⋊C4—2928no—
461C4⋊C4—7332no—
462C4⋊C4—1468no—
463C42—2928no—
464C42—2928no—
465C42—7316no—
466C23C22×C4×D7311yescharacteristic
467C23C2×D4×D7311yescharacteristic
468C23C2×D4×D7312yes—
469C23C2×Q8×D7311yescharacteristic
470C23C2×Q8×D7312yes—
471C23#41×D7312yes—
472C23C2×#1192402712yes—
473C23#1455663611yescharacteristic
474C23#1455663711yescharacteristic
475C23#1455663811yescharacteristic
476C23#1455663911yescharacteristic
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—2922no—
505C2×C4—2922no—
506C2×C4—2922no—
507C2×C4—2922no—
508C2×C4—2921no—
509C2×C4—2921no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2922no—
513C2×C4—2922no—
514C2×C4—2922no—
515C2×C4—2922no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—734no—
519C2×C4—734no—
520C2×C4—738no—
521C2×C4—734no—
522C2×C4—738no—
523C2×C4—734no—
524C2×C4—738no—
525C2×C4—1464no—
526C2×C4—1464no—
527C2×C4—1464no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1462no—
531C2×C4—1462no—
532C2×C4—1464no—
533C2×C4—1464no—
534C2×C4—1464no—
535C2×C4—1464no—
536C2×C4—1464no—
537C2×C4—1462no—
538C2×C4—1464no—
539C2×C4—1462no—
540C2×C4C4⋊C4×D7314yes—
541C2×C4C4⋊C4×D7318yes—
542C22C2×C4⋊C4×D7311yescharacteristic
543C22C2×C4⋊C4×D7312yes—
544C22C4×#1192402712yes—
545C22#1798905911yescharacteristic
546C22#1798906311yescharacteristic
547C22#1798909311yescharacteristic
548C22#1798909511yescharacteristic
549C22#127×D7311yescharacteristic
550C22C4⋊Q8×D7311yescharacteristic
551C22C4⋊Q8×D7312yes—
552C22#1798913912yes—
553C22#1798916611yescharacteristic
554C22#1798916711yescharacteristic
555C22#1798921411yescharacteristic
556C22#1798921711yescharacteristic
557C22#1798922412yes—
558C22#1798922912yes—
559C22#1798924312yes—
560C22#1798924712yes—
561C22#1798929112yes—
562C22#1798929512yes—
563C22#1798931212yes—
564C22#1798931412yes—
565C4—28no—
566C4—24no—
567C4—44no—
568C4—81no—
569C4—81no—
570C4—2924no—
571C4—5841no—
572C4—5841no—
573C4—738no—
574C4—1464no—
575C4—1468no—
576C2#384×D7311yescharacteristic
577C2#2261241011yescharacteristic
578C2#2261254611yescharacteristic
579C2#2261283712yes—
580C2#2261285012yes—
581C2#2261486411yescharacteristic
582C2#2261486611yescharacteristic
583C2#2261494512yes—
584C2#2261494712yes—
585C2#2261531311yescharacteristic
586C2#2261531511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1444C2
socle, prime radical, cosocleC24×C73C24
hypercenter#1444D73
hypocenterC73#13954
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
26. #17988438C4semidirect
136. #11924007C4⋊C4semidirect
139. C4×Dic73C4⋊C4semidirect
305. C73#13954semidirect

Maximal subgroups

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

subgroupindex
C2×#179884842
C2×#179884822
C2×#179884512
C2×C4⋊C4×Dic732
C2×#179884382
#226081962
#226081952
#226081942
#226081602
#226081592
C73×#14442
#1395473