Finite Group Databasebeta

← #13416819

Subgroups — #13416819

Order 1792.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2452C211yesmaximal, characteristic
2C2×#89866C211yesmaximal, characteristic, complemented
3C2×C4×#1142C211yesmaximal, characteristic
4C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
5#11729537C211yesmaximal, characteristic, complemented
6#11729361C211yesmaximal, characteristic
7#11727414C211yesmaximal, characteristic
8#11727414C211yesmaximal, characteristic
9#11727410C211yesmaximal, characteristic
10#11727406C211yesmaximal, characteristic
11#11727386C211yesmaximal, characteristic, complemented
12#11727386C211yesmaximal, characteristic, complemented
13#11727372C211yesmaximal, characteristic, complemented
14#11727362C211yesmaximal, characteristic, complemented
15#11727236C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C7×#396—21no—
24C7×#393—21no—
25C7×#390C2211yescharacteristic
26C7×#390C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#389C2211yescharacteristic
29C7×#385—21no—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34#89866—24no—
35C4×#1142—24no—
36#89748—21no—
37C4⋊C4×Dic7—21no—
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6382C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46#89707C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89419—21no—
52#89419—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26373—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C22×Dic14C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—41no—
68C7×C22⋊Q8—44no—
69C4×C7×D4—22no—
70C4×C7×D4—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×#1142—22no—
88C2×#1142—22no—
89C2×#1142Q814yescomplemented
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C4×Dic14—22no—
108C4×Dic14—22no—
109C4×Dic14—22no—
110C4×Dic14—22no—
111C4×Dic14—41no—
112C4×Dic14—41no—
113C7×#95—21no—
114C7×#95—21no—
115C7×#95D411yescharacteristic
116C7×#95D411yescharacteristic
117#6406—22no—
118#6406—22no—
119#6406—41no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122#6384—22no—
123#6384—22no—
124#6384D414yes—
125#6382—22no—
126#6382—22no—
127#6382—22no—
128#6382—22no—
129#6382—41no—
130C4⋊(C7⋊C8)—22no—
131C4×C7⋊C8—22no—
132C2×#475—71no—
133C2×C4×SD16—71no—
134#2452D711yescharacteristic
135Q8×C4⋊C4—71no—
136#2094—71no—
137#2084—71no—
138#2065—71no—
139#2058—71no—
140#1941—71no—
141#1916—71no—
142#1880—71no—
143#1880—71no—
144#1865—71no—
145#1865—71no—
146#1864—71no—
147C24×C7—21no—
148C2×C7×Q8—42no—
149C2×C7×Q8—42no—
150C2×C7×Q8—81no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4C2×Q812yes—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—81no—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14C2×D412yes—
178C2×Dic14C2×Q812yes—
179C2×Dic14#4112yes—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—41no—
186C7×C4⋊C4—42no—
187C7×C4⋊C4—42no—
188C7×C4⋊C4—42no—
189C7×C4⋊C4SD1614yescomplemented
190C7×C4⋊C4C2×D412yes—
191C7×C22⋊C4—42no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—81no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202#1142—28no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—42no—
209C4⋊Dic7—81no—
210C4⋊Dic7#4112yes—
211Dic7⋊C4—24no—
212Dic7⋊C4—42no—
213Dic7⋊C4—42no—
214Dic7⋊C4—81no—
215Dic7⋊C4—81no—
216C4×Dic7—24no—
217C4×Dic7—42no—
218C4×Dic7—42no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C2×D412yes—
230C2×C7⋊C8C2×Q812yes—
231C22×SD16—71no—
232C2×C4⋊Q8C2×D711yescharacteristic
233C2×C22⋊Q8—21no—
234C2×C4×Q8—71no—
235C2×C4×Q8—71no—
236C2×C4×Q8—71no—
237C2×C4×D4C2×D711yescharacteristic
238#475—144no—
239C4×SD16—144no—
240C2×#106—71no—
241C2×#106—71no—
242C2×#106—71no—
243C2×#106—71no—
244C2×#106—71no—
245C2×C4⋊C8—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#102—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×C4×C8—71no—
253#396—21no—
254#393—21no—
255#390C2×D711yescharacteristic
256#390C2×D711yescharacteristic
257#389—71no—
258#389—71no—
259#389—141no—
260#389—141no—
261#389C2×D711yescharacteristic
262#389C2×D711yescharacteristic
263#385—21no—
264#384—71no—
265#384—141no—
266C4×C4⋊C4—141no—
267C4×C4⋊C4C2×D711yescharacteristic
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#336—71no—
273#336—71no—
274#336—71no—
275#336—71no—
276C23×C7—22no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7C22×D411yescharacteristic
284C7×Q8—84no—
285C7×D4—22no—
286C7×D4—22no—
287C7×D4—41no—
288C7×D4—41no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—84no—
314C2×C4×C7C2×SD1612yes—
315C2×C4×C7C2×SD1612yes—
316C2×C4×C7C2×SD1612yes—
317C2×C4×C7C22×D411yescharacteristic
318C2×C4×C7C22×Q811yescharacteristic
319C2×C4×C7C2×#4111yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7H(5,𝔽2)11yescharacteristic
322C2×C4×C72−1+411yescharacteristic
323C2×Dic7—22no—
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—22no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335Dic14—22no—
336Dic14—22no—
337Dic14—22no—
338Dic14C22⋊Q814yescomplemented
339C7⋊C8—22no—
340C7⋊C8—22no—
341C7⋊C8—41no—
342C7⋊C8—41no—
343C22×Q8—21no—
344C22×Q8—71no—
345C22×D4C22×D711yescharacteristic
346C23×C4—21no—
347C2×SD16—74no—
348C2×SD16—142no—
349C2×SD16—142no—
350C22×C8—71no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354C4⋊Q8—22no—
355C4⋊Q8—41no—
356C22⋊Q8—44no—
357C4×Q8—142no—
358C4×Q8—142no—
359C4×Q8—142no—
360C4×Q8—142no—
361C4×Q8—281no—
362C4×Q8—281no—
363C4×D4—22no—
364C4×D4—41no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4C22⋊D711yescharacteristic
379C2×C4⋊C4C22⋊D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C4⋊C4C22×D711yescharacteristic
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C42—141no—
389C2×C42—141no—
390C2×C42—141no—
391C2×C42C22×D711yescharacteristic
392C2×C42C22×D711yescharacteristic
393C2×C42C22×D711yescharacteristic
394#106—142no—
395#106—142no—
396#106—142no—
397#106—142no—
398#106—281no—
399C4⋊C8—142no—
400#103—74no—
401#103—142no—
402#103—142no—
403#102—142no—
404#102—142no—
405#102—281no—
406C4×C8—142no—
407#95—21no—
408#95—21no—
409#95—141no—
410#95—141no—
411#95C22⋊D711yescharacteristic
412#95C22⋊D711yescharacteristic
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—81no—
417C22×C7—81no—
418C22×C7—81no—
419C22×C7—81no—
420C22×C7#52511yescharacteristic
421C22×C7D4211yescharacteristic
422C22×C7D4×Q811yescharacteristic
423C22×C7C22×SD1611yescharacteristic
424C22×C7#57511yescharacteristic
425C22×C7#57711yescharacteristic
426C22×C7#57711yescharacteristic
427C4×C7—22no—
428C4×C7—22no—
429C4×C7—22no—
430C4×C7—41no—
431C4×C7—41no—
432C4×C7—41no—
433C4×C7—42no—
434C4×C7—81no—
435C4×C7—81no—
436C4×C7C2×C22⋊Q811yescharacteristic
437C4×C7#53911yescharacteristic
438C4×C7#54711yescharacteristic
439C4×C7#55411yescharacteristic
440Dic7—24no—
441Dic7—42no—
442Dic7—81no—
443Dic7—81no—
444C24—21no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—72no—
448C2×Q8—72no—
449C2×Q8—72no—
450C2×Q8—81no—
451C2×Q8—141no—
452C2×Q8—141no—
453C2×Q8—141no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4Q8×D712yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C2×C22⋊D711yescharacteristic
474C22×C4C2×C22⋊D711yescharacteristic
475C22×C4C2×C22⋊D711yescharacteristic
476C22×C4C23×D711yescharacteristic
477SD16—148no—
478C2×C8—72no—
479C2×C8—72no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—72no—
500C4⋊C4—144no—
501C4⋊C4—141no—
502C4⋊C4—141no—
503C4⋊C4—141no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—282no—
507C4⋊C4—282no—
508C4⋊C4—282no—
509C4⋊C4—561no—
510C4⋊C4—561no—
511C4⋊C4—561no—
512C4⋊C4#114214yescomplemented
513C4⋊C4D4×D712yes—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—144no—
526C42—282no—
527C42—282no—
528C2×C7—82no—
529C2×C7#244911yescharacteristic
530C2×C7#310311yescharacteristic
531C2×C7#312511yescharacteristic
532C2×C7#327111yescharacteristic
533C2×C7D4×SD1611yescharacteristic
534C2×C7Q8×SD1611yescharacteristic
535C2×C7#338811yescharacteristic
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23C22×C22⋊D711yescharacteristic
544Q8—74no—
545Q8—84no—
546Q8—142no—
547Q8—142no—
548Q8—142no—
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4C2×#114212yes—
591C2×C4C2×#114212yes—
592C2×C4C2×#114212yes—
593C2×C4C2×D4×D711yescharacteristic
594C2×C4#654811yescharacteristic
595C2×C4C2×Q8×D711yescharacteristic
596C2×C4#655111yescharacteristic
597C2×C4D7×#4111yescharacteristic
598C2×C4D7×#4111yescharacteristic
599C8—142no—
600C8—142no—
601C8—281no—
602C8—281no—
603C7#2637311yescharacteristic, complemented, Sylow
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C22×#114211yescharacteristic
612C22D4×C22⋊D711yescharacteristic
613C22Q8×C22⋊D711yescharacteristic
614C22#9051311yescharacteristic
615C22#9051911yescharacteristic
616C22#9052711yescharacteristic
617C22#9052811yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—144no—
628C4—282no—
629C4—561no—
630C4—561no—
631C4D7×C22⋊Q811yescharacteristic
632C4#9032211yescharacteristic
633C4#9032711yescharacteristic
634C4#9032911yescharacteristic
635C2—82no—
636C2#1174026211yescharacteristic
637C2#1174076011yescharacteristic
638C2#1174090711yescharacteristic
639C2D4×#114211yescharacteristic
640C2#1174099211yescharacteristic
641C2Q8×#114211yescharacteristic
642C2#1174107711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2452C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2452D7
hypocenterC7#26373
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
2. C2×#89866C2semidirect
4. C4⋊C4×Dic14C2semidirect
5. #11729537C2semidirect
11. #11727386C2semidirect
12. #11727386C2semidirect
13. #11727372C2semidirect
14. #11727362C2semidirect
15. #11727236C2semidirect
89. C2×#1142Q8semidirect
189. C7×C4⋊C4SD16semidirect
338. Dic14C22⋊Q8semidirect
512. C4⋊C4#1142semidirect
603. C7#26373semidirect

Maximal subgroups

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

subgroupindex
C7×#24522
C2×#898662
C2×C4×#11422
C4⋊C4×Dic142
#117295372
#117293612
#117274142
#117274102
#117274062
#117273862
#117273722
#117273622
#117272362
#263737