Finite Group Databasebeta

← #34026764

Subgroups — #34026764

Order 18688.

838 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626440C211yesmaximal, characteristic
2#22626439C211yesmaximal, characteristic
3#22626037C212yesmaximal, complemented
4#22625994C212yesmaximal, complemented
5#22625993C212yesmaximal, complemented
6#22625835C211yesmaximal, characteristic
7#22625834C211yesmaximal, characteristic
8#22625808C211yesmaximal, characteristic
9#22625747C212yesmaximal, complemented
10#22625667C211yesmaximal, characteristic
11C2×#106×D73C211yesmaximal, characteristic
12C2×#14556670—21no—
13C2×#14556660C2212yes—
14#17989555C2211yescharacteristic
15C2×#14556645C2212yes—
16#17989538C2212yes—
17#17989538C2212yes—
18#17989536—22no—
19#17989536—22no—
20#17989530—21no—
21#17989529—21no—
22#17989529C2211yescharacteristic
23#17989529C2211yescharacteristic
24C4⋊C4×#64408C2212yes—
25C4⋊C4×#64408C2212yes—
26C4×C22⋊#64408—21no—
27C4×C22⋊#64408C2211yescharacteristic
28#17989461C2212yes—
29#17989460C2212yes—
30#17989460C2212yes—
31#17989460C2212yes—
32#17989460C2212yes—
33#17989452C2212yes—
34C22×C8×D73C2211yescharacteristic
35C2×C4⋊C4×D73C2211yescharacteristic
36C2×C4⋊C4×D73C2211yescharacteristic
37C2×#14556523C411yescharacteristic, complemented
38C2×#14556522C411yescharacteristic, complemented
39C2×#14556522C411yescharacteristic, complemented
40#106×D73C411yescharacteristic, complemented
41#106×D73C411yescharacteristic, complemented
42#106×D73C411yescharacteristic, complemented
43#106×D73C411yescharacteristic, complemented
44#106×D73C2211yescharacteristic
45#106×D73C2211yescharacteristic
46#106×D73C2211yescharacteristic
47#106×D73C2211yescharacteristic
48C2×#106×C73C411yescharacteristic, complemented
49C2×#11924036C2312yes—
50#14556670—44no—
51C2×C4⋊#64408—22no—
52C2×C4⋊#64408—22no—
53C2×C4⋊#64408C2312yes—
54C2×C4⋊#64408C2312yes—
55C2×C22⋊#64408—21no—
56C2×C22⋊#64408—21no—
57C2×C22⋊#64408D411yescharacteristic
58C2×C22⋊#64408D411yescharacteristic
59C2×C4×#64408—22no—
60C2×C4×#64408—22no—
61C2×C4×#64408C2312yes—
62#14556660—24no—
63#14556648—21no—
64C22⋊#10596833—21no—
65#14556645—24no—
66(C2×#64408)⋊C4—22no—
67(C2×#64408)⋊C4—22no—
68(C2×#64408)⋊C4—22no—
69(C2×#64408)⋊C4—22no—
70(C2×#64408)⋊C4—21no—
71(C2×#64408)⋊C4—21no—
72(C2×#64408)⋊C4D412yes—
73C22×C4×D73—21no—
74C22×C4×D73—21no—
75C22×C4×D73C2311yescharacteristic
76C22×#10596831C2×C411yescharacteristic
77C2×C8×D73—24no—
78C2×C8×D73C2×C411yescharacteristic
79C2×C8×D73C2×C411yescharacteristic
80C2×C8×D73C2311yescharacteristic
81C2×C8×D73C2311yescharacteristic
82C2×C4⋊Dic73C2×C411yescharacteristic
83C2×C4⋊Dic73C2×C411yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C4⋊C4×D73—21no—
87C4⋊C4×D73—21no—
88C4⋊C4×D73—21no—
89C4⋊C4×D73—21no—
90C4⋊C4×D73C2×C411yescharacteristic
91C4⋊C4×D73C2×C411yescharacteristic
92C4⋊C4×D73C2×C411yescharacteristic
93C4⋊C4×D73C2×C411yescharacteristic
94C4⋊C4×D73C2311yescharacteristic
95C4⋊C4×D73C2311yescharacteristic
96C4⋊C4×D73C2311yescharacteristic
97C4⋊C4×D73C2311yescharacteristic
98#14556523C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100#14556523C2×C411yescharacteristic
101#14556523C2×C411yescharacteristic
102#14556522C2×C411yescharacteristic
103#14556522C2×C411yescharacteristic
104#14556522C2×C411yescharacteristic
105#14556522C2×C411yescharacteristic
106#14556522C2×C411yescharacteristic
107#14556522C2×C411yescharacteristic
108#14556522C2×C411yescharacteristic
109#14556522C2×C411yescharacteristic
110C22×C8×C73C2×C411yescharacteristic
111C2×C4⋊C4×C73C2×C411yescharacteristic
112C2×C4⋊C4×C73C2×C411yescharacteristic
113#106×C73C2×C411yescharacteristic
114#106×C73C2×C411yescharacteristic
115#106×C73C2×C411yescharacteristic
116#106×C73C2×C411yescharacteristic
117C22×#64408—22no—
118C22×#64408—22no—
119C22×#64408C2×D412yes—
120C22×#64408C2×D412yes—
121#11924036—22no—
122#11924036—22no—
123C2×#10596833—22no—
124C4⋊#64408—22no—
125C4⋊#64408—22no—
126C4⋊#64408—22no—
127C4⋊#64408—22no—
128C4⋊#64408—44no—
129C4⋊#64408—82no—
130C22⋊#64408—22no—
131C22⋊#64408—22no—
132C22⋊#64408—42no—
133C22⋊#64408—42no—
134C4×#64408—22no—
135C4×#64408—22no—
136C4×#64408—44no—
137C4×#64408—82no—
138C23×D73C2×D411yescharacteristic
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73#4111yescharacteristic
142C2×C4×D73—22no—
143C2×C4×D73—22no—
144C2×C4×D73—21no—
145C2×C4×D73—21no—
146C2×C4×D73—21no—
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—42no—
153C2×C4×D73—42no—
154C2×C4×D73—82no—
155C2×C4×D73C22×C411yescharacteristic
156C2×C4×D73C2×D411yescharacteristic
157C2×C4×D73#4111yescharacteristic
158C2×C4×D73C2411yescharacteristic
159C2×#10596831—22no—
160C2×#10596831C22×C411yescharacteristic
161C2×#10596831C22×C411yescharacteristic
162C8×D73—22no—
163C8×D73—22no—
164C8×D73—22no—
165C8×D73#4114yes—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73C22×C411yescharacteristic
169C4⋊Dic73C22×C411yescharacteristic
170C4⋊Dic73C22×C411yescharacteristic
171C4⋊Dic73C22×C411yescharacteristic
172#11924007—21no—
173#11924007—21no—
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178#11924007—21no—
179#11924007—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73C22×C411yescharacteristic
183C2×C8×C73—22no—
184C2×C8×C73C22×C411yescharacteristic
185C2×C8×C73C22×C411yescharacteristic
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73C22×C411yescharacteristic
189C4⋊C4×C73C22×C411yescharacteristic
190C4⋊C4×C73C22×C411yescharacteristic
191C4⋊C4×C73C22×C411yescharacteristic
192C23×C73C4×D411yescharacteristic
193C22×D73—22no—
194C22×D73—22no—
195C22×D73C4×D411yescharacteristic
196C22×D73AGL(1,ℤ/8)12yes—
197C22×D73#13712yes—
198C22×D73C22×D411yescharacteristic
199C2×#64408—22no—
200C2×#64408—22no—
201C2×#64408—22no—
202C2×#64408—22no—
203C2×#64408—22no—
204C2×#64408—22no—
205C2×#64408—42no—
206C2×#64408—42no—
207C2×#64408—42no—
208C2×#64408—42no—
209C2×#64408—42no—
210C2×#64408—42no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—42no—
218C2×C4×C73—81no—
219C2×C4×C73C4×D411yescharacteristic
220C2×C4×C73C23×C411yescharacteristic
221C2×Dic73—22no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—42no—
227C2×Dic73—81no—
228C2×Dic73C4×D411yescharacteristic
229C2×Dic73C2×#4111yescharacteristic
230C4×D73—22no—
231C4×D73—21no—
232C4×D73—21no—
233C4×D73—22no—
234C4×D73—41no—
235C4×D73—41no—
236C4×D73—41no—
237C4×D73—41no—
238C4×D73—42no—
239C4×D73—42no—
240C4×D73—42no—
241C4×D73—42no—
242C4×D73—81no—
243C4×D73—81no—
244C4×D73C2×#4112yes—
245C4×D732−1+412yes—
246#10596833—42no—
247C8×C73—21no—
248C8×C73#11712yescomplemented
249#10596831—21no—
250#10596831#11712yescomplemented
251C22×C73—21no—
252C22×C73—21no—
253C22×C73#44012yes—
254C22×C73C2×C4×D411yescharacteristic
255C2×D73—21no—
256C2×D73—21no—
257C2×D73—24no—
258C2×D73—24no—
259C2×D73#44012yes—
260C2×D73#54712yes—
261C2×D73C2×AGL(1,ℤ/8)11yescharacteristic
262C2×D73C2×#13711yescharacteristic
263C2×D73#57712yes—
264#64408—42no—
265#64408—42no—
266#64408—82no—
267#64408—82no—
268C4×C73—21no—
269C4×C73—41no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—41no—
273C4×C73C2×#11711yescharacteristic
274C4×C73#51911yescharacteristic
275Dic73—21no—
276Dic73—41no—
277Dic73—41no—
278Dic73—41no—
279Dic73—41no—
280Dic73#55212yes—
281#22615—731nomaximal, Sylow
282C2×C73—22no—
283C2×C73#234411yescharacteristic
284C2×C73C2×#44011yescharacteristic
285C2×C73#294911yescharacteristic
286D73—22no—
287D73#334412yes—
288D73#334512yes—
289C22×#106—731no—
290#2345—731no—
291#2344—731no—
292#1944—732no—
293#1920—732no—
294#1919—732no—
295#1824—731no—
296#1823—731no—
297#1808—731no—
298#1776—732no—
299#1752—731no—
300C73#2261511yescharacteristic, complemented, Sylow
301C23×C8—731no—
302C2×#117—1461no—
303C22×C4⋊C4—731no—
304C22×C4⋊C4—731no—
305C2×#107—732no—
306C2×#106—731no—
307C2×#106—731no—
308C2×#106—731no—
309C2×#106—731no—
310C2×#106—731no—
311C2×#106—731no—
312C2×#106—731no—
313C2×#106—731no—
314C2×#106—731no—
315C2×#106—731no—
316C2×#106—731no—
317C2×#106#6440811yescharacteristic, complemented
318C2×C22⋊C8—731no—
319C2×#97—732no—
320#389—732no—
321#389—732no—
322#387—1462no—
323#387—1462no—
324#381—1461no—
325#380—731no—
326#380—731no—
327#380—1461no—
328C4×C4⋊C4—732no—
329C4×C4⋊C4—732no—
330C4×C22⋊C4—731no—
331C4×C22⋊C4—1461no—
332#340—732no—
333#340—732no—
334#340—732no—
335#340—732no—
336#340—732no—
337#336—732no—
338C23×C4—731no—
339C23×C4—1461no—
340C23×C4—1461no—
341C22×C8—731no—
342C22×C8—731no—
343C22×C8—731no—
344C22×C8—731no—
345C22×C8—731no—
346C22×C8—732no—
347C22×C8—1464no—
348C22×C8C2×#6440811yescharacteristic
349#117—2924no—
350C2×C4⋊C4—731no—
351C2×C4⋊C4—731no—
352C2×C4⋊C4—731no—
353C2×C4⋊C4—731no—
354C2×C4⋊C4—731no—
355C2×C4⋊C4—731no—
356C2×C4⋊C4—731no—
357C2×C4⋊C4—731no—
358C2×C4⋊C4—731no—
359C2×C4⋊C4—731no—
360C2×C4⋊C4—732no—
361C2×C4⋊C4—732no—
362C2×C4⋊C4—1461no—
363C2×C4⋊C4—1461no—
364C2×C4⋊C4—1461no—
365C2×C4⋊C4—1461no—
366C2×C4⋊C4—1462no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4C2×#6440811yescharacteristic
371C2×C4⋊C4C2×#6440811yescharacteristic
372C2×C22⋊C4—731no—
373C2×C22⋊C4—731no—
374C2×C22⋊C4—1461no—
375C2×C22⋊C4—1461no—
376C2×C42—732no—
377C2×C42—1462no—
378C2×C42—1462no—
379#107—1464no—
380#106—731no—
381#106—731no—
382#106—731no—
383#106—731no—
384#106—731no—
385#106—731no—
386#106—731no—
387#106—731no—
388#106—731no—
389#106—731no—
390#106—731no—
391#106—731no—
392#106C2×#6440811yescharacteristic
393#106C2×#6440811yescharacteristic
394#106C2×#6440811yescharacteristic
395#106C2×#6440811yescharacteristic
396C22⋊C8—1461no—
397C22⋊C8—1461no—
398#97—1464no—
399#95—732no—
400#95—1462no—
401#95—1462no—
402#95—1462no—
403#95—1461no—
404#95—1461no—
405#95—1462no—
406C24—731no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—2922no—
410C22×C4—2922no—
411C22×C4—5842no—
412C22×C4—731no—
413C22×C4—731no—
414C22×C4—731no—
415C22×C4—731no—
416C22×C4—731no—
417C22×C4—732no—
418C22×C4—732no—
419C22×C4—1461no—
420C22×C4—1461no—
421C22×C4—1461no—
422C22×C4—1461no—
423C22×C4—1461no—
424C22×C4—1461no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4—1462no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4C22×#6440811yescharacteristic
434C2×C8—22no—
435C2×C8—731no—
436C2×C8—734no—
437C2×C8—731no—
438C2×C8—1462no—
439C2×C8—1462no—
440C2×C8—1462no—
441C2×C8—1462no—
442C2×C8—1462no—
443C2×C8—1462no—
444C2×C8—1462no—
445C2×C8C22×#6440811yescharacteristic
446C2×C8C22×#6440811yescharacteristic
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—2924no—
450C4⋊C4—5842no—
451C4⋊C4—731no—
452C4⋊C4—731no—
453C4⋊C4—731no—
454C4⋊C4—731no—
455C4⋊C4—1461no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1462no—
464C4⋊C4—1462no—
465C4⋊C4—1462no—
466C4⋊C4—1462no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4C22×#6440811yescharacteristic
470C4⋊C4C22×#6440811yescharacteristic
471C4⋊C4C22×#6440811yescharacteristic
472C4⋊C4C22×#6440811yescharacteristic
473C22⋊C4—2922no—
474C22⋊C4—2922no—
475C22⋊C4—1462no—
476C22⋊C4—1462no—
477C42—2924no—
478C42—5842no—
479C42—1462no—
480C42—1462no—
481C23—731no—
482C23—731no—
483C23—732no—
484C23—732no—
485C23—1462no—
486C23—1462no—
487C23D4×#6440811yescharacteristic
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—42no—
495C2×C4—81no—
496C2×C4—2921no—
497C2×C4—2922no—
498C2×C4—2921no—
499C2×C4—2922no—
500C2×C4—2921no—
501C2×C4—2921no—
502C2×C4—2922no—
503C2×C4—2922no—
504C2×C4—2922no—
505C2×C4—2922no—
506C2×C4—2922no—
507C2×C4—2922no—
508C2×C4—2922no—
509C2×C4—2922no—
510C2×C4—2922no—
511C2×C4—5841no—
512C2×C4—5841no—
513C2×C4—5841no—
514C2×C4—731no—
515C2×C4—732no—
516C2×C4—732no—
517C2×C4—731no—
518C2×C4—1462no—
519C2×C4—1462no—
520C2×C4—1462no—
521C2×C4—1462no—
522C2×C4—1462no—
523C2×C4—1462no—
524C2×C4—1462no—
525C2×C4—1462no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1462no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4D4×#6440811yescharacteristic
534C2×C4C23×#6440811yescharacteristic
535C8—21no—
536C8—2922no—
537C8—732no—
538C8—1461no—
539C8#1455667012yescomplemented
540C22—21no—
541C22—21no—
542C22—731no—
543C22—731no—
544C22—732no—
545C22—732no—
546C22—732no—
547C22—1464no—
548C22—1464no—
549C22—1461no—
550C22—1461no—
551C22#1798962612yes—
552C22C2×D4×#6440811yescharacteristic
553C4—21no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—2922no—
559C4—2922no—
560C4—2921no—
561C4—2921no—
562C4—2921no—
563C4—2921no—
564C4—5842no—
565C4—5842no—
566C4—732no—
567C4—1461no—
568C4C2×#1455667011yescharacteristic
569C4#1798965511yescharacteristic
570C2—22no—
571C2—732no—
572C2—732no—
573C2—1462no—
574C2#2262643911yescharacteristic
575C2C2×#1798962611yescharacteristic
576C2#2262669111yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
FittingC2×#106×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×D73C2×D4
cosocleC2×C4×D73C24
hypercenterC2×#106#64408
hypocenterC73#22615
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
3. #22626037C2semidirect
4. #22625994C2semidirect
5. #22625993C2semidirect
9. #22625747C2semidirect
37. C2×#14556523C4semidirect
38. C2×#14556522C4semidirect
39. C2×#14556522C4semidirect
40. #106×D73C4semidirect
41. #106×D73C4semidirect
42. #106×D73C4semidirect
43. #106×D73C4semidirect
48. C2×#106×C73C4semidirect
248. C8×C73#117semidirect
250. #10596831#117semidirect
300. C73#22615semidirect
317. C2×#106#64408semidirect
539. C8#14556670semidirect

Maximal subgroups

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

subgroupindex
#226264402
#226264392
#226260372
#226259942
#226259932
#226258352
#226258342
#226258082
#226257472
#226256672
C2×#106×D732
#2261573