Finite Group Databasebeta

← #13395376

Subgroups — #13395376

Order 1792.

1086 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×#1142C211yesmaximal, characteristic, complemented
3C2×#89788C212yesmaximal, complemented
4C2×#89787C211yesmaximal, characteristic
5#11731157C211yesmaximal, characteristic, complemented
6#11729367C212yesmaximal, complemented
7#11727383C212yesmaximal, complemented
8#11727382C212yesmaximal, complemented
9#11727342C211yesmaximal, characteristic, complemented
10#11727334C211yesmaximal, characteristic, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C4×C7×D4C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C22×C7×C4⋊C4—21no—
15C22×#1142C2211yescharacteristic
16C22×#1141C2212yes—
17C2×#6438C2212yes—
18C2×#6437C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C7×#389C2212yes—
21C7×#385—22no—
22C7×#384C2211yescharacteristic
23C7×#380—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C4×#1142—24no—
27#89788—28no—
28#89787—24no—
29#89760—21no—
30#89760—21no—
31C2×#6384C2211yescharacteristic
32C2×#6384C2212yes—
33C2×#6383C2212yes—
34C2×#6383C2212yes—
35C2×#6383C2212yes—
36C2×#6382C2211yescharacteristic
37#89711C2212yes—
38C2×C4⋊(C7⋊C8)C2212yes—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89444—22no—
42#89324C2212yes—
43#89319C2212yes—
44#24731—71nomaximal, Sylow
45C22×C7×D4C2311yescharacteristic
46C23×C4×C7—21no—
47C23×C4×C7—22no—
48C22×D28—21no—
49C22×D28—21no—
50C22×D28C2312yes—
51C22×Dic14C2311yescharacteristic
52C4×C7×D4—24no—
53C4×C7×D4—22no—
54C4×C7×D4—42no—
55C4×C7×D4D414yes—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—44no—
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C4⋊C4C2312yes—
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—22no—
66C2×C42×C7C2312yes—
67C2×C42×C7C2311yescharacteristic
68C2×#1142—22no—
69C2×#1142—22no—
70C2×#1142D414yescomplemented
71C2×#1141—24no—
72C2×#1141—24no—
73C2×#1141—24no—
74C2×(C2×D7)⋊C4—22no—
75C2×(C2×D7)⋊C4—22no—
76C2×(C2×D7)⋊C4—22no—
77C2×(C2×D7)⋊C4—22no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—21no—
80C2×C4×Dic7—21no—
81C22×C7⋊C8C2311yescharacteristic
82C22×C7⋊C8C2312yes—
83C22×C7⋊C8C2311yescharacteristic
84#6438—24no—
85#6438—42no—
86#6437—22no—
87#6437—22no—
88C4×Dic14—22no—
89C4×Dic14—22no—
90C7×#95—22no—
91C7×#95D412yes—
92#6406—22no—
93#6384—24no—
94#6384—22no—
95#6383—24no—
96#6383—24no—
97#6383—42no—
98#6382—22no—
99C4⋊(C7⋊C8)—22no—
100C4⋊(C7⋊C8)—24no—
101C4×C7⋊C8—22no—
102C2×#463—72no—
103C2×#459—71no—
104C2×C4×SD16—71no—
105#2548—71no—
106D4×C4⋊C4D711yescharacteristic, complemented
107#2065—72no—
108#2042—72no—
109#1957—71no—
110#1939—71no—
111#1879—72no—
112#1803—71no—
113C24×C7—21no—
114C23×D7—22no—
115C2×C7×D4—21no—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4C2×D412yes—
119C2×C7×D4C2×D412yes—
120C2×C7×D4#4112yes—
121C22×C4×C7—21no—
122C22×C4×C7—24no—
123C22×C4×C7—22no—
124C22×C4×C7—44no—
125C22×C4×C7—44no—
126C22×C4×C7—42no—
127C22×C4×C7—42no—
128C22×C4×C7—82no—
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—21no—
135C22×Dic7—21no—
136C2×D28—22no—
137C2×D28—22no—
138C2×D28—22no—
139C2×D28—22no—
140C2×D28—22no—
141C2×D28—22no—
142C2×D28—42no—
143C2×D28—42no—
144C2×D28—42no—
145C2×D28—42no—
146C2×D28—81no—
147C2×D28—81no—
148C2×Dic14—21no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14C2×D412yes—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—44no—
160C7×C4⋊C4—42no—
161C7×C4⋊C4—81no—
162C7×C4⋊C4C2×D412yes—
163C7×C22⋊C4—24no—
164C7×C22⋊C4—44no—
165C42×C7—22no—
166C42×C7—22no—
167C42×C7C2×D412yes—
168C42×C7#4112yes—
169#1142—28no—
170#1141—48no—
171(C2×D7)⋊C4—44no—
172(C2×D7)⋊C4—44no—
173(C2×D7)⋊C4—82no—
174(C2×D7)⋊C4—82no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177Dic7⋊C4—42no—
178C4×Dic7—42no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8C2×D412yes—
188C22×SD16—71no—
189C22×D8—72no—
190C2×#127—71no—
191C2×#124—72no—
192C2×C4×Q8—71no—
193C2×C4×D4C2×D712yes—
194C2×C4×D4C2×D711yescharacteristic
195C22×C4⋊C4—21no—
196#463—148no—
197#459—144no—
198C4×SD16—144no—
199C2×#106—71no—
200C2×C4⋊C8—71no—
201C2×C4⋊C8—72no—
202C2×#103—72no—
203C2×#103—71no—
204C2×#102—71no—
205C2×#102—72no—
206C2×#102—72no—
207C2×#102—72no—
208C2×C4×C8—71no—
209#394—142no—
210#390—72no—
211#390—141no—
212#390—141no—
213#389C2×D712yes—
214#385—22no—
215#384C2×D711yescharacteristic
216#380—21no—
217C4×C4⋊C4C2×D711yescharacteristic
218#340—72no—
219#336—72no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7C22×D411yescharacteristic
228C22×D7—42no—
229C22×D7—42no—
230C22×D7—42no—
231C22×D7—42no—
232C22×D7—42no—
233C22×D7—42no—
234C22×D7—42no—
235C7×D4—22no—
236C7×D4—22no—
237C7×D4—22no—
238C7×D4#12114yes—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—48no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—88no—
255C2×C4×C7#12112yes—
256C2×C4×C7#12112yes—
257C2×C4×C7#13512yes—
258C2×C4×C7AGL(1,ℤ/8)12yes—
259C2×C4×C7C22×D411yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C2×#4111yescharacteristic
262C2×C4×C7C2×#4111yescharacteristic
263C2×C4×C7H(5,𝔽2)11yescharacteristic
264C2×C4×C72−1+411yescharacteristic
265C2×Dic7—22no—
266C2×Dic7—41no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271D28—42no—
272D28—42no—
273D28—42no—
274D28—42no—
275D28—84no—
276D28—84no—
277Dic14—22no—
278Dic14—22no—
279Dic14—41no—
280Dic14—41no—
281C7⋊C8—22no—
282C7⋊C8—42no—
283C7⋊C8—41no—
284C22×Q8—71no—
285C22×D4—72no—
286C22×D4—141no—
287C22×D4—141no—
288C22×D4C22×D711yescharacteristic
289C23×C4—21no—
290C23×C4—22no—
291C2×SD16—74no—
292C2×SD16—142no—
293C2×SD16—142no—
294C2×D8—144no—
295C2×D8—144no—
296C2×D8—144no—
297C22×C8—71no—
298C22×C8—72no—
299C22×C8—71no—
300#127—142no—
301#127—142no—
302#124—144no—
303#124—282no—
304C4×Q8—142no—
305C4×Q8—142no—
306C4×D4—24no—
307C4×D4—22no—
308C4×D4—42no—
309C4×D4D2814yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—71no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4C22⋊D711yescharacteristic
317C2×C4⋊C4C22⋊D711yescharacteristic
318C2×C4⋊C4C22×D712yes—
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—142no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4—142no—
325C2×C22⋊C4—142no—
326C2×C42—141no—
327C2×C42C22×D712yes—
328C2×C42C22×D711yescharacteristic
329#106—142no—
330C4⋊C8—144no—
331C4⋊C8—142no—
332#103—144no—
333#103—142no—
334#102—144no—
335#102—142no—
336#102—144no—
337#102—282no—
338C4×C8—142no—
339#95—22no—
340#95C22⋊D712yes—
341C22×C7—24no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7C2×#12111yescharacteristic
349C22×C7D4211yescharacteristic
350C22×C7#54811yescharacteristic
351C22×C7C2×#13511yescharacteristic
352C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
353C22×C7#57611yescharacteristic
354C22×C7#57711yescharacteristic
355C2×D7—82no—
356C2×D7—82no—
357C2×D7—82no—
358C2×D7—82no—
359C2×D7—82no—
360C2×D7—82no—
361C2×D7—82no—
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—42no—
366C4×C7—82no—
367C4×C7#46314yescomplemented
368C4×C7C2×#12111yescharacteristic
369C4×C7#53711yescharacteristic
370C4×C7#54711yescharacteristic
371C4×C7#55311yescharacteristic
372Dic7—42no—
373Dic7—81no—
374C24—21no—
375C24—142no—
376C2×Q8—72no—
377C2×Q8—141no—
378C2×Q8—141no—
379C2×Q8—141no—
380C2×Q8—141no—
381C2×Q8—141no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—142no—
386C2×D4—142no—
387C2×D4—142no—
388C2×D4—142no—
389C2×D4—142no—
390C2×D4—142no—
391C2×D4—282no—
392C2×D4—282no—
393C2×D4—282no—
394C2×D4—282no—
395C2×D4—561no—
396C2×D4—561no—
397C2×D4C2×D2812yes—
398C2×D4D4×D712yes—
399C2×D4#116112yes—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4—141no—
409C22×C4—141no—
410C22×C4C2×C22⋊D712yes—
411C22×C4C2×C22⋊D712yes—
412C22×C4C2×C22⋊D711yescharacteristic
413C22×C4C2×C22⋊D711yescharacteristic
414C22×C4C23×D711yescharacteristic
415SD16—148no—
416D8—288no—
417C2×C8—72no—
418C2×C8—141no—
419C2×C8—141no—
420C2×C8—141no—
421C2×C8—142no—
422C2×C8—142no—
423C2×C8—141no—
424C2×C8—142no—
425C2×C8—141no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—141no—
435C4⋊C4—141no—
436C4⋊C4—282no—
437C4⋊C4C2×D2812yes—
438C22⋊C4—24no—
439C22⋊C4—44no—
440C22⋊C4—284no—
441C22⋊C4—284no—
442C22⋊C4—562no—
443C22⋊C4—562no—
444C42—22no—
445C42—22no—
446C42—282no—
447C42C2×D2812yes—
448C42#115712yes—
449C2×C7—44no—
450C2×C7#241611yescharacteristic
451C2×C7C2×#46311yescharacteristic
452C2×C7#304511yescharacteristic
453C2×C7#327411yescharacteristic
454C2×C7#328411yescharacteristic
455C2×C7#336111yescharacteristic
456C2×C7#337311yescharacteristic
457D7—162no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—282no—
466C23—282no—
467C23—282no—
468C23—282no—
469C23—282no—
470C23—282no—
471C23—282no—
472C23C22×C22⋊D711yescharacteristic
473Q8—142no—
474Q8—142no—
475Q8—281no—
476Q8—281no—
477D4—22no—
478D4—22no—
479D4—22no—
480D4—282no—
481D4—282no—
482D4—282no—
483D4—282no—
484D4—564no—
485D4—564no—
486D4#645814yes—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—88no—
503C2×C4—142no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4#649312yes—
510C2×C4#649312yes—
511C2×C4#649512yes—
512C2×C4#651312yes—
513C2×C4C22×D2811yescharacteristic
514C2×C4C2×#115711yescharacteristic
515C2×C4C2×D4×D711yescharacteristic
516C2×C4#654811yescharacteristic
517C2×C4C2×#116111yescharacteristic
518C2×C4#655111yescharacteristic
519C8—142no—
520C8—282no—
521C8—281no—
522C7#2473111yescharacteristic, complemented, Sylow
523C22—24no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—562no—
531C22—562no—
532C22—562no—
533C22—562no—
534C22—562no—
535C22—562no—
536C22—562no—
537C22C2×#649311yescharacteristic
538C22C2×#649511yescharacteristic
539C22D4×C22⋊D711yescharacteristic
540C22#9050811yescharacteristic
541C22C2×#651311yescharacteristic
542C22#9052611yescharacteristic
543C22#9052811yescharacteristic
544C4—22no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—82no—
549C4—282no—
550C4—561no—
551C4#8978814yescomplemented
552C4C2×#645811yescharacteristic
553C4#9019811yescharacteristic
554C4#9020111yescharacteristic
555C4#9020511yescharacteristic
556C2—44no—
557C2—1122no—
558C2#1173963811yescharacteristic
559C2C2×#8978811yescharacteristic
560C2#1174001011yescharacteristic
561C2#1174093711yescharacteristic
562C2#1174097511yescharacteristic
563C2#1174101411yescharacteristic
564C2#1174108811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×D4×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC2×#89787C2
hypercenterD4×C4⋊C4D7
hypocenterC7#24731
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C7×D4×C4⋊C4C2semidirect
2. C2×C4×#1142C2semidirect
3. C2×#89788C2semidirect
5. #11731157C2semidirect
6. #11729367C2semidirect
7. #11727383C2semidirect
8. #11727382C2semidirect
9. #11727342C2semidirect
10. #11727334C2semidirect
11. #11727238C2semidirect
70. C2×#1142D4semidirect
106. D4×C4⋊C4D7semidirect
367. C4×C7#463semidirect
522. C7#24731semidirect
551. C4#89788semidirect

Maximal subgroups

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

subgroupindex
C7×D4×C4⋊C42
C2×C4×#11422
C2×#897882
C2×#897872
#117311572
#117293672
#117273832
#117273822
#117273422
#117273342
#117272382
#247317