Finite Group Databasebeta

← #33007794

Subgroups — #33007794

Order 18688.

1942 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556556C211yesmaximal, characteristic
2C2×#17988508C211yesmaximal, characteristic
3C2×C4⋊C4×Dic73C212yesmaximal
4C2×#17988438C211yesmaximal, characteristic
5#22608201C211yesmaximal, characteristic
6#22608194C212yesmaximal
7#22608183C212yesmaximal
8#22608181C211yesmaximal, characteristic
9#95×Dic73C212yesmaximal
10#95×Dic73C211yesmaximal, characteristic
11C73×#1444C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C2×#14556556—24no—
14C2×#14556556—28no—
15C22×C4⋊Dic73C2212yes—
16C22×#11924007C2211yescharacteristic
17C22×#11924007C2212yes—
18C22×#11924007C2211yescharacteristic
19C22×C4×Dic73C2211yescharacteristic
20C22×C4×Dic73C2211yescharacteristic
21C22×C4×Dic73C2212yes—
22C22×C4×Dic73C2211yescharacteristic
23C22×C4×Dic73C2212yes—
24#17988508C418yescomplemented
25C4⋊C4×Dic73—28no—
26#17988438—24no—
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2212yes—
36C22×C4⋊C4×C73C2211yescharacteristic
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—22no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—22no—
59C2×#11924007—22no—
60C2×#11924007—24no—
61C2×#11924007—22no—
62C2×#11924007—22no—
63C2×#11924007C2×C414yes—
64C2×C4×Dic73—22no—
65C2×C4×Dic73—24no—
66C2×C4×Dic73—24no—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—22no—
71C2×C4×Dic73—22no—
72C2×C4×Dic73—24no—
73C2×C4×Dic73D414yes—
74C2×C4×Dic73D418yes—
75C2×C4×Dic73Q814yescomplemented
76C2×C4×Dic73Q818yescomplemented
77#14556498—22no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—24no—
82#14556498—24no—
83#14556498—22no—
84#14556498C2×C418yes—
85C23×C4×C73C2312yes—
86C23×C4×C73C2311yescharacteristic
87C23×C4×C73C2311yescharacteristic
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2312yes—
90C2×C4⋊C4×C73—24no—
91C2×C4⋊C4×C73—24no—
92C2×C4⋊C4×C73—22no—
93C2×C4⋊C4×C73—24no—
94C2×C4⋊C4×C73—24no—
95#95×C73—22no—
96#95×C73—24no—
97#95×C73C2×C414yes—
98C2×Dic146—28no—
99C2×Dic146—216no—
100C2×Dic146—28no—
101C2×Dic146—28no—
102C2×Dic146—216no—
103C22×Dic73—21no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—21no—
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—21no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—22no—
125C22×Dic73C22×C412yes—
126C22×Dic73C22×C414yes—
127C22×Dic73C22×C412yes—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×D414yes—
130C22×Dic73C2×D412yes—
131C22×Dic73C2×D414yes—
132C22×Dic73C2×Q814yes—
133C22×Dic73C2×Q812yes—
134C22×Dic73C2×Q812yes—
135C22×Dic73C2×Q814yes—
136C22×Dic73#4114yes—
137C22×Dic73#4112yes—
138C22×Dic73#4112yes—
139C4⋊Dic73—216no—
140#11924007—28no—
141#11924007—216no—
142#11924007—28no—
143C4×Dic73—216no—
144C4×Dic73—28no—
145C4×Dic73—28no—
146C4×Dic73—216no—
147C4×Dic73—44no—
148C24×C73C2411yescharacteristic
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—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C414yes—
171C22×C4×C73C2×D414yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2×Q812yes—
174C22×C4×C73C2×Q814yes—
175C22×C4×C73#4112yes—
176C22×C4×C73#4114yes—
177C22×C4×C73#4112yes—
178C4⋊C4×C73—216no—
179C4⋊C4×C73—28no—
180C23×C73C23×C411yescharacteristic
181C23×C73C22×D412yes—
182C23×C73C22×D411yescharacteristic
183C23×C73C22×Q811yescharacteristic
184C23×C73C22×Q812yes—
185C23×C73C2×#4112yes—
186C23×C73C2×#4111yescharacteristic
187C23×C73C2×#4111yescharacteristic
188C23×C73C2×#4112yes—
189C23×C73C2×#4111yescharacteristic
190C23×C73C2×#4111yescharacteristic
191C2×C4×C73—24no—
192C2×C4×C73—24no—
193C2×C4×C73—24no—
194C2×C4×C73—22no—
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—22no—
198C2×C4×C73—24no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—22no—
203C2×C4×C73—24no—
204C2×C4×C73—24no—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—24no—
208C2×C4×C73—41no—
209C2×C4×C73—42no—
210C2×C4×C73—41no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—41no—
216C2×C4×C73—42no—
217C2×C4×C73—41no—
218C2×C4×C73C4×D414yes—
219C2×C4×C73C4×Q818yescomplemented
220C2×Dic73—22no—
221C2×Dic73—24no—
222C2×Dic73—24no—
223C2×Dic73—22no—
224C2×Dic73—24no—
225C2×Dic73—22no—
226C2×Dic73—22no—
227C2×Dic73—24no—
228C2×Dic73—24no—
229C2×Dic73—24no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—24no—
234C2×Dic73—22no—
235C2×Dic73—24no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—41no—
241C2×Dic73—42no—
242C2×Dic73—41no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73C4×D418yes—
250C2×Dic73C4×Q814yescomplemented
251C2×Dic73#12014yes—
252C2×Dic73C22⋊Q814yes—
253C2×Dic73C22⋊Q818yes—
254C2×Dic73C4⋊Q814yes—
255C2×Dic73C4⋊Q818yes—
256Dic146—232no—
257C22×C73C2×C4×D412yes—
258C22×C73C2×C4×D411yescharacteristic
259C22×C73C2×C4×Q811yescharacteristic
260C22×C73C2×C4×Q812yes—
261C22×C73C4×#4111yescharacteristic
262C22×C73C2×#12011yescharacteristic
263C22×C73C2×C22⋊Q811yescharacteristic
264C22×C73C2×C22⋊Q812yes—
265C22×C73#52512yes—
266C22×C73#52511yescharacteristic
267C22×C73#52511yescharacteristic
268C22×C73#52512yes—
269C22×C73#52911yescharacteristic
270C22×C73#52912yes—
271C22×C73C2×C4⋊Q811yescharacteristic
272C22×C73C2×C4⋊Q812yes—
273C22×C73#53212yes—
274C22×C73#53211yescharacteristic
275C22×C73#53312yes—
276C22×C73#53312yes—
277C22×C73#53312yes—
278C22×C73#53311yescharacteristic
279C22×C73#53312yes—
280C4×C73—24no—
281C4×C73—28no—
282C4×C73—44no—
283C4×C73—42no—
284C4×C73—42no—
285Dic73—28no—
286Dic73—24no—
287Dic73—42no—
288Dic73—44no—
289Dic73—42no—
290Dic73#39518yescomplemented
291C4×#395—731nomaximal, Sylow
292C2×C73C4×#12011yescharacteristic
293C2×C73C4×C22⋊Q811yescharacteristic
294C2×C73C4×C22⋊Q812yes—
295C2×C73C4×C4⋊Q811yescharacteristic
296C2×C73C4×C4⋊Q812yes—
297C2×C73C2×#39511yescharacteristic
298C2×C73#238811yescharacteristic
299C2×C73#239611yescharacteristic
300C2×C73#239612yes—
301C2×C73#239912yes—
302C2×C73#239911yescharacteristic
303C22×C4×Q8—731no—
304C2×#395—731no—
305C2×C4×C4⋊C4—731no—
306C2×C4×C4⋊C4—732no—
307#1446—731no—
308#1444—732no—
309#1444D7311yescharacteristic
310#1441—732no—
311#1441—731no—
312C4×#95—732no—
313C4×#95—731no—
314C73C4×#39511yescharacteristic, complemented, Sylow
315C23×Q8—731no—
316C2×C4×Q8—1464no—
317C2×C4×Q8—1468no—
318C22×C4⋊C4—731no—
319C22×C4⋊C4—732no—
320C22×C4⋊C4—732no—
321C22×C4⋊C4—731no—
322C22×C4⋊C4C2×D7311yescharacteristic
323C22×C4⋊C4C2×D7312yes—
324C22×C42—732no—
325C22×C42—731no—
326C22×C42—731no—
327C22×C42—732no—
328C22×C42—731no—
329#395—738no—
330C4×C4⋊C4—1464no—
331C4×C4⋊C4—1468no—
332C2×#95—732no—
333C2×#95—732no—
334C2×#95—732no—
335C2×#95—732no—
336C2×#95—731no—
337C2×#95—731no—
338C2×#95—732no—
339C2×#95—732no—
340C2×#95C2×D7312yes—
341C2×#95C2×D7311yescharacteristic
342C2×#95C2×D7311yescharacteristic
343C22×Q8—734no—
344C22×Q8—1464no—
345C22×Q8—1464no—
346C22×Q8—1462no—
347C22×Q8—1462no—
348C23×C4—731no—
349C23×C4—732no—
350C23×C4—731no—
351C23×C4—731no—
352C23×C4—731no—
353C23×C4—732no—
354C23×C4C22×D7312yes—
355C23×C4C22×D7311yescharacteristic
356C23×C4C22×D7311yescharacteristic
357C23×C4C22×D7311yescharacteristic
358C23×C4C22×D7312yes—
359C4×Q8—14632no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—734no—
366C2×C4⋊C4—738no—
367C2×C4⋊C4—1464no—
368C2×C4⋊C4—1464no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1464no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4—1464no—
373C2×C4⋊C4—1464no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1462no—
377C2×C42—738no—
378C2×C42—734no—
379C2×C42—734no—
380C2×C42—738no—
381C2×C42—1462no—
382C2×C42—1464no—
383C2×C42—1464no—
384C2×C42—1464no—
385C2×C42—1464no—
386C2×C42—1464no—
387C2×C42—1462no—
388C2×C42—1464no—
389C2×C42—1462no—
390#95—22no—
391#95—24no—
392#95—738no—
393#95—1464no—
394#95—1464no—
395#95—1462no—
396#95—1464no—
397#95—1464no—
398#95—1464no—
399#95—1462no—
400#95C4×D7314yes—
401C24C23×D7311yescharacteristic
402C2×Q8—14616no—
403C2×Q8—1468no—
404C2×Q8—1468no—
405C2×Q8—14616no—
406C2×Q8—1468no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—734no—
428C22×C4—732no—
429C22×C4—732no—
430C22×C4—734no—
431C22×C4—732no—
432C22×C4—732no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—732no—
436C22×C4—732no—
437C22×C4—734no—
438C22×C4—732no—
439C22×C4—732no—
440C22×C4—734no—
441C22×C4—1462no—
442C22×C4—1461no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1461no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1461no—
453C22×C4—1462no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1461no—
459C22×C4—1462no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1462no—
463C22×C4C2×C4×D7312yes—
464C22×C4C2×C4×D7314yes—
465C22×C4D4×D7314yes—
466C22×C4D4×D7312yes—
467C22×C4Q8×D7312yes—
468C22×C4Q8×D7314yes—
469C22×C4#1192402712yes—
470C22×C4#1192402714yes—
471C22×C4#1192402912yes—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—14616no—
475C4⋊C4—1468no—
476C4⋊C4—14616no—
477C4⋊C4—1468no—
478C42—2924no—
479C42—14616no—
480C42—1468no—
481C42—1468no—
482C42—14616no—
483C23C22×C4×D7311yescharacteristic
484C23C2×D4×D7312yes—
485C23C2×D4×D7311yescharacteristic
486C23C2×Q8×D7311yescharacteristic
487C23C2×Q8×D7312yes—
488C23#41×D7311yescharacteristic
489C23#41×D7311yescharacteristic
490C23#41×D7312yes—
491C23C2×#1192402712yes—
492C23C2×#1192402711yescharacteristic
493C23C2×#1192402911yescharacteristic
494Q8—14632no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—2921no—
526C2×C4—2922no—
527C2×C4—2921no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—734no—
532C2×C4—738no—
533C2×C4—734no—
534C2×C4—734no—
535C2×C4—738no—
536C2×C4—738no—
537C2×C4—734no—
538C2×C4—1462no—
539C2×C4—1464no—
540C2×C4—1464no—
541C2×C4—1464no—
542C2×C4—1462no—
543C2×C4—1462no—
544C2×C4—1464no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1464no—
549C2×C4—1462no—
550C2×C4—1464no—
551C2×C4—1462no—
552C2×C4—1464no—
553C2×C4—1462no—
554C2×C4—1464no—
555C2×C4—1462no—
556C2×C4—1464no—
557C2×C4—1464no—
558C2×C4#1455655314yes—
559C2×C4#1455655618yescomplemented
560C22C4×D4×D7312yes—
561C22C2×#1455655311yescharacteristic
562C22C4×Q8×D7311yescharacteristic
563C22C2×#1455655612yes—
564C22C4×#1192402711yescharacteristic
565C22#120×D7311yescharacteristic
566C22C22⋊Q8×D7311yescharacteristic
567C22C22⋊Q8×D7312yes—
568C22#1798909312yes—
569C22#1798909511yescharacteristic
570C22#1798909711yescharacteristic
571C22#1798910012yes—
572C22#1798911412yes—
573C22#1798912211yescharacteristic
574C22C4⋊Q8×D7311yescharacteristic
575C22C4⋊Q8×D7312yes—
576C22#1798913212yes—
577C22#1798913511yescharacteristic
578C22#1798913912yes—
579C22#1798913912yes—
580C22#1798914212yes—
581C22#1798914211yescharacteristic
582C22#1798914312yes—
583C4—24no—
584C4—28no—
585C4—44no—
586C4—42no—
587C4—42no—
588C4—2922no—
589C4—2922no—
590C4—2924no—
591C4—738no—
592C4—1468no—
593C4—1464no—
594C2#2261243211yescharacteristic
595C2#2261246111yescharacteristic
596C2#2261246312yes—
597C2#2261249811yescharacteristic
598C2#2261250212yes—
599C2#395×D7311yescharacteristic
600C2#2261314311yescharacteristic
601C2#2261320011yescharacteristic
602C2#2261320812yes—
603C2#2261322112yes—
604C2#2261322311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1444C2
socle, prime radical, cosocleC24×C73C24
hypercenter#1444D73
hypocenterC73C4×#395
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
24. #17988508C4semidirect
75. C2×C4×Dic73Q8semidirect
76. C2×C4×Dic73Q8semidirect
219. C2×C4×C73C4×Q8semidirect
250. C2×Dic73C4×Q8semidirect
290. Dic73#395semidirect
314. C73C4×#395semidirect
559. 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×C4⋊C4×Dic732
C2×#179884382
#226082012
#226081942
#226081832
#226081812
#95×Dic732
C73×#14442
C4×#39573