Finite Group Databasebeta

← #13372224

Subgroups — #13372224

Order 1792.

926 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic7⋊C4C211yesmaximal, characteristic
2C2×C4×#1144C211yesmaximal, characteristic
3C2×#89562C211yesmaximal, characteristic
4C2×#89562C211yesmaximal, characteristic
5#11733426C211yesmaximal, characteristic
6C7×#1863C211yesmaximal, characteristic
7#11728396C211yesmaximal, characteristic
8#11728360C211yesmaximal, characteristic
9#11727990C211yesmaximal, characteristic
10#11727975C211yesmaximal, characteristic
11#11727373C211yesmaximal, characteristic
12#11726658C211yesmaximal, characteristic
13#11726613C211yesmaximal, characteristic
14#11726437C211yesmaximal, characteristic
15#11726405C211yesmaximal, characteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C2×#6450C2211yescharacteristic
21C2×#6450C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6390C2211yescharacteristic
30C2×#6390C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33C2×C8×Dic7C2211yescharacteristic
34C4×#1144—24no—
35#89757C2211yescharacteristic
36#89748—21no—
37#89748—21no—
38#89748C2211yescharacteristic
39#89748C2211yescharacteristic
40#89745—21no—
41#89745C2211yescharacteristic
42#89744—21no—
43C2×#6384C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47C4×Dic7⋊C4—21no—
48C4×Dic7⋊C4C2211yescharacteristic
49#89700—21no—
50#89700C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52#89562—24no—
53#89562C418yescomplemented
54#89421—21no—
55#89419—21no—
56C7×#340C2211yescharacteristic
57C7×#336C2211yescharacteristic
58#89346C2211yescharacteristic
59#89345C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62C4⋊C4×Q16—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C4×C7×Q8—22no—
68C4×C7×Q8—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76Q8×Dic7—22no—
77Q8×Dic7—22no—
78Q8×Dic7—41no—
79Q8×Dic7C2×C414yes—
80C2×#1144C2×C414yes—
81C2×#1144D414yes—
82C2×#1144Q814yes—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4D411yescharacteristic
99C2×Dic7⋊C4D411yescharacteristic
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7—21no—
105C2×C4×Dic7C2311yescharacteristic
106C2×C4×Dic7C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109#6450—22no—
110#6450—22no—
111#6450—41no—
112#6450C2×C414yes—
113C4×Dic14—22no—
114C4×Dic14—41no—
115C7×#107—41no—
116C7×#103—22no—
117C7×#103C2×C414yes—
118C7×#95—21no—
119#6409—22no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95D411yescharacteristic
124C7⋊#95D411yescharacteristic
125#6392—41no—
126#6390—22no—
127#6390C2×C414yes—
128Dic7⋊C8—22no—
129C8×Dic7—22no—
130C8×Dic7C2×C414yes—
131#6384—22no—
132#6381—22no—
133#6381—22no—
134#6381C2×C414yes—
135C4×C7⋊C8—22no—
136C2×C4×Q16—71no—
137C2×C4×Q16—71no—
138C2×C4×Q16—71no—
139Q8×C4⋊C4—71no—
140Q8×C4⋊C4—71no—
141#1942—71no—
142#1942—71no—
143#1863—71no—
144#1863—71no—
145#1863—71no—
146#1863D711yescharacteristic
147#1845—71no—
148#1803—71no—
149#1803—71no—
150C8×C4⋊C4—71no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8C22×C412yes—
155C2×C7×Q8C2×D412yes—
156C2×C7×Q8C2×Q812yes—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14C22×C412yes—
176C2×Dic14C2×D412yes—
177C2×Dic14C2×Q812yes—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8C22×C412yes—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—42no—
190C7×C4⋊C4—81no—
191C7×C4⋊C4C22×C412yes—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—42no—
195#1144C4⋊C4116yescomplemented
196C4⋊Dic7—21no—
197C4⋊Dic7—24no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—42no—
203C4⋊Dic7—81no—
204C4⋊Dic7C22×C412yes—
205Dic7⋊C4—24no—
206Dic7⋊C4—22no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—42no—
211Dic7⋊C4—81no—
212Dic7⋊C4—81no—
213Dic7⋊C4—81no—
214Dic7⋊C4—81no—
215Dic7⋊C4Q1614yescomplemented
216C4×Dic7—21no—
217C4×Dic7—21no—
218C4×Dic7—24no—
219C4×Dic7—24no—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C4×Dic7C22×C412yes—
224C4×Dic7#4112yes—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8C22×C412yes—
229C2×C7⋊C8C2×D412yes—
230C2×C7⋊C8C2×Q812yes—
231C22×Q16—71no—
232C2×C4×Q8—71no—
233C2×C4×Q8—71no—
234C2×C4×Q8—71no—
235C2×C4×Q8—71no—
236C2×C4×Q8—71no—
237C2×C4×Q8C2×D711yescharacteristic
238C4×Q16—78no—
239C4×Q16—144no—
240C4×Q16—144no—
241C2×#107—71no—
242C2×#107—71no—
243C2×#107—71no—
244C2×#107—71no—
245C2×#107C2×D711yescharacteristic
246C2×C4⋊C8—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#103—71no—
250C2×#103—71no—
251C2×#103C2×D711yescharacteristic
252C2×#103C2×D711yescharacteristic
253C2×C4×C8—71no—
254C2×C4×C8—71no—
255C2×C4×C8—71no—
256#389—71no—
257#389—71no—
258#389—71no—
259#389—141no—
260#389—141no—
261#389—141no—
262#389—141no—
263#389C2×D711yescharacteristic
264#384—71no—
265#384—71no—
266#384—141no—
267#384—141no—
268C4×C4⋊C4—71no—
269C4×C4⋊C4—141no—
270C4×C4⋊C4—141no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340C2×D711yescharacteristic
275#336—71no—
276#336C2×D711yescharacteristic
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8C2×C4⋊C414yes—
282C2×C4×C7—22no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7C2×Q1612yes—
300C2×C4×C7C23×C411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×Q811yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7H(5,𝔽2)11yescharacteristic
305C2×C4×C72−1+411yescharacteristic
306C2×Dic7—22no—
307C2×Dic7—21no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—22no—
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7C4×D412yes—
326C2×Dic7C4×D412yes—
327C2×Dic7C2×Q1612yes—
328C2×Dic7C2×Q1612yes—
329C2×Dic7#13512yes—
330Dic14—22no—
331Dic14—22no—
332Dic14—22no—
333Dic14C2×C4⋊C414yes—
334C7×C8—22no—
335C7×C8—41no—
336C7⋊C8—41no—
337C7⋊C8C2×C4⋊C414yes—
338C22×Q8—71no—
339C22×Q8C22×D711yescharacteristic
340C2×Q16—74no—
341C2×Q16—74no—
342C2×Q16—74no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8C22×D711yescharacteristic
346C22×C8C22×D711yescharacteristic
347C4×Q8—22no—
348C4×Q8—41no—
349C4×Q8—74no—
350C4×Q8—74no—
351C4×Q8—142no—
352C4×Q8—142no—
353C4×Q8—142no—
354C4×Q8—142no—
355C4×Q8—142no—
356C4×Q8—281no—
357C4×Q8—281no—
358C4×Q8—281no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
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×C42—21no—
382C2×C42—71no—
383C2×C42—71no—
384C2×C42—141no—
385C2×C42—141no—
386C2×C42—141no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42C22×D711yescharacteristic
390#107—41no—
391#107—74no—
392#107—142no—
393#107—142no—
394#107—281no—
395C4⋊C8—142no—
396#103—22no—
397#103—74no—
398#103—142no—
399#103—142no—
400#103—142no—
401#103C4×D714yes—
402C4×C8—74no—
403C4×C8—142no—
404C4×C8—142no—
405#95—21no—
406#95—71no—
407#95—71no—
408#95—141no—
409#95—141no—
410#95—141no—
411C22×C7C2×C4×D411yescharacteristic
412C22×C7D4211yescharacteristic
413C22×C7D4×Q811yescharacteristic
414C22×C7C22×Q1611yescharacteristic
415C22×C7C2×#13511yescharacteristic
416C22×C7#57611yescharacteristic
417C22×C7#57811yescharacteristic
418C4×C7—24no—
419C4×C7—22no—
420C4×C7—42no—
421C4×C7—41no—
422C4×C7—81no—
423C4×C7—81no—
424C4×C7C22×C4⋊C411yescharacteristic
425C4×C7#52011yescharacteristic
426C4×C7#54711yescharacteristic
427C4×C7#55411yescharacteristic
428Dic7—24no—
429Dic7—22no—
430Dic7—22no—
431Dic7—42no—
432Dic7—41no—
433Dic7—81no—
434Dic7—81no—
435Dic7C4×Q1614yescomplemented
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—72no—
440C2×Q8—72no—
441C2×Q8—72no—
442C2×Q8—141no—
443C2×Q8—141no—
444C2×Q8—141no—
445C2×Q8C2×C4×D712yes—
446C2×Q8D4×D712yes—
447C2×Q8Q8×D712yes—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—71no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4D4×D711yescharacteristic
461C22×C4D4×D711yescharacteristic
462C22×C4C23×D711yescharacteristic
463Q16—716no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—72no—
470C2×C8—72no—
471C2×C8—72no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8C2×C4×D712yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—74no—
483C4⋊C4—72no—
484C4⋊C4—81no—
485C4⋊C4—142no—
486C4⋊C4—144no—
487C4⋊C4—144no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—282no—
498C4⋊C4—561no—
499C4⋊C4—561no—
500C4⋊C4—561no—
501C4⋊C4—561no—
502C4⋊C4—561no—
503C4⋊C4C2×C4×D712yes—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—72no—
508C42—72no—
509C42—144no—
510C42—144no—
511C42—141no—
512C42—141no—
513C42—282no—
514C42—282no—
515C42—282no—
516C2×C7D4×C4⋊C411yescharacteristic
517C2×C7C2×C4×Q1611yescharacteristic
518C2×C7#295111yescharacteristic
519C2×C7D4×Q1611yescharacteristic
520C2×C7#328711yescharacteristic
521C2×C7Q8×Q1611yescharacteristic
522C2×C7#338311yescharacteristic
523C23C2×D4×D711yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8—74no—
528Q8—142no—
529Q8—142no—
530Q8—142no—
531Q8D7×C4⋊C414yes—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—142no—
558C2×C4—141no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—142no—
564C2×C4—141no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4D7×Q1612yes—
574C2×C4C22×C4×D711yescharacteristic
575C2×C4C2×D4×D711yescharacteristic
576C2×C4#654811yescharacteristic
577C2×C4C2×Q8×D711yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C2×C4#655411yescharacteristic
580C8—22no—
581C8—41no—
582C8—74no—
583C8—281no—
584C7C4⋊C4×Q1611yescharacteristic, complemented, Sylow
585C22C4×D4×D711yescharacteristic
586C22#9028111yescharacteristic
587C22#9032311yescharacteristic
588C22C2×D7×Q1611yescharacteristic
589C22D7×#13511yescharacteristic
590C22#9046511yescharacteristic
591C22#9047011yescharacteristic
592C4—22no—
593C4—24no—
594C4—42no—
595C4—41no—
596C4—74no—
597C4—81no—
598C4—81no—
599C4—142no—
600C4—144no—
601C4—142no—
602C4—282no—
603C4—281no—
604C4—561no—
605C4—561no—
606C4C2×D7×C4⋊C411yescharacteristic
607C4#9023711yescharacteristic
608C4#9032711yescharacteristic
609C4#9034611yescharacteristic
610C2#1173183811yescharacteristic
611C2C4×D7×Q1611yescharacteristic
612C2#1173722311yescharacteristic
613C2#1173757611yescharacteristic
614C2#1173765611yescharacteristic
615C2#1173854711yescharacteristic
616C2#1173866711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1863C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#1863D7
hypocenterC7C4⋊C4×Q16
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
53. #89562C4semidirect
195. #1144C4⋊C4semidirect
215. Dic7⋊C4Q16semidirect
435. Dic7C4×Q16semidirect
584. C7C4⋊C4×Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic7⋊C42
C2×C4×#11442
C2×#895622
#117334262
C7×#18632
#117283962
#117283602
#117279902
#117279752
#117273732
#117266582
#117266132
#117264372
#117264052
C4⋊C4×Q167