Finite Group Databasebeta

← #13320184

Subgroups — #13320184

Order 1792.

1206 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6551C211yesmaximal, characteristic
2#11736736C211yesmaximal, characteristic, complemented
3#11736721C211yesmaximal, characteristic, complemented
4C7×#1479C211yesmaximal, characteristic, complemented
5C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#11728509C211yesmaximal, characteristic, complemented
7#11728508C211yesmaximal, characteristic, complemented
8#11725985C212yesmaximal, complemented
9#11725984C212yesmaximal, complemented
10#11725983C212yesmaximal, complemented
11D7×Q8⋊C8C212yesmaximal, complemented
12C2×#6551C411yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14#90503C411yescharacteristic
15#90265—21no—
16#90265C412yes—
17C4×#1161—21no—
18C4×#1161C2212yes—
19#90263—21no—
20#90263C412yes—
21C4×Q8×D7—21no—
22C4×Q8×D7C2212yes—
23#90195—22no—
24#90195C411yescharacteristic
25#90195C411yescharacteristic
26C4×#1157—22no—
27C4×#1157C2211yescharacteristic
28C4×#1157C2211yescharacteristic
29C7×#431C2211yescharacteristic
30C7×#431C2211yescharacteristic
31C2×C4⋊(C7⋊C8)C2211yescharacteristic
32C2×C4×C7⋊C8C2211yescharacteristic
33#89606—21no—
34#89605C2212yes—
35D7×C4⋊C8C2212yes—
36#89602—21no—
37(C22⋊D7)⋊C8—22no—
38(C22⋊D7)⋊C8—22no—
39C8×D28—21no—
40#89457C2212yes—
41C4×C8×D7C2212yes—
42C8×Dic14—21no—
43C7×Q8⋊C8C2212yes—
44C7×Q8⋊C8C2212yes—
45Q8⋊(C7⋊C8)C2212yes—
46Q8⋊(C7⋊C8)C2212yes—
47#89279C2211yescharacteristic
48#89279C2211yescharacteristic
49#89278C2211yescharacteristic
50#89278C2211yescharacteristic
51#89254C2212yes—
52#89251C2212yes—
53#17439—71nomaximal, Sylow
54C22×C7×Q8C2×C411yescharacteristic
55#6551—24no—
56#6551C818yescomplemented
57C2×#1161—21no—
58C2×#1161C2×C412yes—
59C2×Q8×D7—21no—
60C2×Q8×D7C2×C412yes—
61C2×#1157—22no—
62C2×#1157C2×C411yescharacteristic
63C2×#1157C2×C411yescharacteristic
64C4×C7×Q8—21no—
65C4×C7×Q8—21no—
66C4×C7×Q8C2×C412yes—
67C4×C7×Q8C2312yes—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2×C411yescharacteristic
70C2×C42×C7—21no—
71C2×C42×C7C2311yescharacteristic
72Q8×Dic7—21no—
73Q8×Dic7—21no—
74Q8×Dic7C2×C412yes—
75Q8×Dic7C2×C412yes—
76C4×C22⋊D7—21no—
77C4×C22⋊D7—21no—
78C4×C22⋊D7—22no—
79C4×C22⋊D7—22no—
80C4×C22⋊D7D412yes—
81C4×C22⋊D7D412yes—
82#6489—22no—
83#6489C2×C411yescharacteristic
84#6489C2×C411yescharacteristic
85C22×C7⋊C8—21no—
86C22×C7⋊C8—21no—
87C2×C8×D7—22no—
88C2×C8×D7—22no—
89#6456—21no—
90#6456—22no—
91#6456—22no—
92#6456C2×C412yes—
93#6455—22no—
94#6455—21no—
95#6455—22no—
96#6455C2×C412yes—
97D7×C4⋊C4—22no—
98D7×C4⋊C4—21no—
99D7×C4⋊C4—22no—
100D7×C4⋊C4C2×C412yes—
101#6450—21no—
102#6450—22no—
103#6450—22no—
104#6450C2×C412yes—
105C4×D28—22no—
106C4×D28—22no—
107C4×D28C2×C411yescharacteristic
108C4×D28D411yescharacteristic
109C4×D28D411yescharacteristic
110C4×D28C2311yescharacteristic
111#6435—21no—
112#6435—22no—
113#6435—22no—
114#6435C2×C412yes—
115C42×D7—22no—
116C42×D7—21no—
117C42×D7—22no—
118C42×D7C2312yes—
119C4×Dic14—22no—
120C4×Dic14—22no—
121C4×Dic14C2×C411yescharacteristic
122C4×Dic14D411yescharacteristic
123C4×Dic14D411yescharacteristic
124C4×Dic14C2311yescharacteristic
125C7×C4⋊C8—21no—
126C7×C4⋊C8C2311yescharacteristic
127C7×C4⋊C8C2311yescharacteristic
128C7×C22⋊C8—21no—
129C7×C22⋊C8—21no—
130C4×C7×C8—21no—
131C4×C7×C8C2312yes—
132(C2×D7)⋊C8—22no—
133(C2×D7)⋊C8—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136C8×Dic7—22no—
137C8×Dic7—22no—
138C4⋊(C7⋊C8)—21no—
139C4⋊(C7⋊C8)C2312yes—
140C4×C7⋊C8—21no—
141C4×C7⋊C8C2311yescharacteristic
142C4×C7⋊C8C2311yescharacteristic
143C4×2−1+4—71no—
144#2988—71no—
145C8×#41—71no—
146#1479—72no—
147#1479D711yescharacteristic, complemented
148#1477—71no—
149#1477—72no—
150#1476—71no—
151#1476—72no—
152C2×Q8⋊C8—72no—
153C2×Q8⋊C8—71no—
154C2×C7×Q8—21no—
155C2×C7×Q8—22no—
156C2×C7×Q8C2×C814yescomplemented
157C2×C7×Q8C22×C412yes—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7C22×C411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C2×C22⋊D7—21no—
164C2×C22⋊D7C22⋊C412yes—
165#1161—28no—
166#1161—24no—
167#1161C2×C818yescomplemented
168Q8×D7—24no—
169Q8×D7—28no—
170Q8×D7C2×C818yescomplemented
171#1157—22no—
172#1157—216no—
173#1157C2×C814yes—
174#1157C2×C814yescomplemented
175#1157M1614yes—
176C2×D28—22no—
177C2×D28C22⋊C411yescharacteristic
178C2×D28C22×C411yescharacteristic
179C2×C4×D7—22no—
180C2×C4×D7—21no—
181C2×C4×D7—22no—
182C2×C4×D7—21no—
183C2×C4×D7—22no—
184C2×C4×D7—21no—
185C2×C4×D7—22no—
186C2×C4×D7C22×C412yes—
187C2×C4×D7C2×D412yes—
188C2×C4×D7C2×D412yes—
189C2×Dic14—22no—
190C2×Dic14C22⋊C411yescharacteristic
191C2×Dic14C22×C411yescharacteristic
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4C22×C412yes—
199C42×C7—22no—
200C42×C7—22no—
201C42×C7C22×C411yescharacteristic
202C42×C7C2×D411yescharacteristic
203C42×C7C2×D411yescharacteristic
204C42×C7C2411yescharacteristic
205C22⋊Dic7—21no—
206C22⋊Dic7C22⋊C412yes—
207(C2×D7)⋊C4—22no—
208(C2×D7)⋊C4—21no—
209(C2×D7)⋊C4—21no—
210(C2×D7)⋊C4—22no—
211(C2×D7)⋊C4C22⋊C412yes—
212(C2×D7)⋊C4C22⋊C412yes—
213C4⋊Dic7—22no—
214C4⋊Dic7—22no—
215C4⋊Dic7C22⋊C411yescharacteristic
216C4⋊Dic7C22⋊C411yescharacteristic
217C4⋊Dic7C22×C411yescharacteristic
218C4⋊Dic7C22×C411yescharacteristic
219Dic7⋊C4—21no—
220Dic7⋊C4—21no—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223Dic7⋊C4C22⋊C412yes—
224Dic7⋊C4C22⋊C412yes—
225C4×Dic7—21no—
226C4×Dic7—21no—
227C4×Dic7—21no—
228C4×Dic7—22no—
229C4×Dic7—22no—
230C4×Dic7—22no—
231C4×Dic7—22no—
232C4×Dic7C22×C412yes—
233C4×Dic7C2×D412yes—
234C4×Dic7C2×D412yes—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—22no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8—42no—
239C2×C7⋊C8—81no—
240C8×D7—44no—
241C8×D7—82no—
242C2×2−1+4—71no—
243#520—71no—
244#520—72no—
245#520—71no—
246#520—142no—
247#520—141no—
248#519—71no—
249#519—72no—
250#519—141no—
251C4×#41—71no—
252C4×#41—72no—
253C4×#41—71no—
254C4×#41—142no—
255C4×#41—141no—
256C2×C4×Q8—72no—
257C2×C4×Q8—141no—
258C2×C4×Q8C2×D711yescharacteristic
259C8×Q8—141no—
260C8×Q8—141no—
261C8×D4—141no—
262C8×D4—142no—
263C8×D4—142no—
264C8×D4—141no—
265#431—72no—
266#431—72no—
267#431C2×D711yescharacteristic
268#431C2×D711yescharacteristic
269C2×C4⋊C8—71no—
270C2×C4⋊C8—72no—
271C2×C4×C8—71no—
272C2×C4×C8—72no—
273Q8⋊C8—72no—
274Q8⋊C8—72no—
275Q8⋊C8—71no—
276Q8⋊C8—71no—
277Q8⋊C8—72no—
278Q8⋊C8C2×D712yes—
279Q8⋊C8C2×D712yes—
280D4⋊C8—71no—
281D4⋊C8—71no—
282D4⋊C8—72no—
283C23×C7C2×C22⋊C411yescharacteristic
284C22×D7—21no—
285C22×D7C2×C22⋊C412yes—
286C7×Q8—22no—
287C7×Q8—24no—
288C7×Q8C22×C814yes—
289C2×C4×C7—22no—
290C2×C4×C7—24no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—41no—
295C2×C4×C7—81no—
296C2×C4×C7C2×C22⋊C411yescharacteristic
297C2×C4×C7C2×C22⋊C411yescharacteristic
298C2×C4×C7C2×C22⋊C411yescharacteristic
299C2×C4×C7C22×C812yes—
300C2×C4×C7C2×M1612yes—
301C2×C4×C7C23×C411yescharacteristic
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C22×D411yescharacteristic
304C22⋊D7—24no—
305C22⋊D7C22⋊C818yes—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7C2×C22⋊C412yes—
310C2×Dic7C2×C22⋊C412yes—
311C2×Dic7C2×C22⋊C412yes—
312D28—28no—
313D28—21no—
314D28C22⋊C814yes—
315D28C22×C812yes—
316C4×D7—28no—
317C4×D7—22no—
318C4×D7—22no—
319C4×D7—24no—
320C4×D7—28no—
321C4×D7—42no—
322C4×D7—41no—
323C4×D7—41no—
324C4×D7—82no—
325C4×D7—81no—
326C4×D7—82no—
327C4×D7C22×C814yes—
328C4×D7C2×M1614yes—
329Dic14—21no—
330Dic14—28no—
331Dic14C22⋊C814yes—
332Dic14C22×C812yes—
333C7×C8—42no—
334C7×C8—41no—
335C7×C8—41no—
336C7⋊C8—22no—
337C7⋊C8—41no—
338C7⋊C8—42no—
3392−1+4—78no—
3402−1+4—144no—
341C2×#41—71no—
342C2×#41—72no—
343C2×#41—71no—
344C2×#41—142no—
345C2×#41—141no—
346C22×Q8—72no—
347C22×Q8—141no—
348C22×Q8C4×D711yescharacteristic
349C22×C8—142no—
350C22×C8—142no—
351C22×C8—141no—
352C22×C8—141no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—71no—
356C4×Q8—71no—
357C4×Q8—72no—
358C4×Q8—72no—
359C4×Q8—72no—
360C4×Q8—71no—
361C4×Q8—71no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×Q8—141no—
367C4×Q8—141no—
368C4×Q8—141no—
369C4×Q8C4×D712yes—
370C4×Q8C22×D712yes—
371C4×D4—72no—
372C4×D4—72no—
373C4×D4—72no—
374C4×D4—71no—
375C4×D4—71no—
376C4×D4—71no—
377C4×D4—71no—
378C4×D4—142no—
379C4×D4—142no—
380C4×D4—142no—
381C4×D4—142no—
382C4×D4—142no—
383C4×D4—141no—
384C4×D4—142no—
385C4×D4—141no—
386C4×D4—141no—
387#117—71no—
388#117—72no—
389#117—71no—
390#117—72no—
391#117—142no—
392#117—142no—
393#117—142no—
394#117—142no—
395#117—142no—
396#117—141no—
397#117—141no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—72no—
400C2×C4⋊C4—142no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—142no—
403C2×C4⋊C4C4×D711yescharacteristic
404C2×C42—21no—
405C2×C42—72no—
406C2×C42—142no—
407C2×C42—142no—
408C2×C42—141no—
409C2×C42C22×D711yescharacteristic
410C4⋊C8—21no—
411C4⋊C8—72no—
412C4⋊C8—142no—
413C4⋊C8—142no—
414C4⋊C8—141no—
415C4⋊C8C22×D711yescharacteristic
416C4⋊C8C22×D711yescharacteristic
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8—142no—
420C22⋊C8—142no—
421C4×C8—21no—
422C4×C8—71no—
423C4×C8—71no—
424C4×C8—142no—
425C4×C8—142no—
426C4×C8—141no—
427C4×C8C22×D712yes—
428C22×C7—21no—
429C22×C7—21no—
430C22×C7C2×C22⋊C812yes—
431C22×C7C22×C22⋊C411yescharacteristic
432C2×D7—22no—
433C2×D7—22no—
434C2×D7—22no—
435C2×D7—41no—
436C2×D7—41no—
437C2×D7C2×C22⋊C814yes—
438C4×C7—24no—
439C4×C7—21no—
440C4×C7—21no—
441C4×C7—21no—
442C4×C7—42no—
443C4×C7C2×C22⋊C812yes—
444C4×C7C23×C811yescharacteristic
445C4×C7C22×M1611yescharacteristic
446C4×C7#57711yescharacteristic
447C4×C7#57811yescharacteristic
448Dic7—22no—
449Dic7—22no—
450Dic7—22no—
451Dic7—41no—
452Dic7—41no—
453Dic7C2×C22⋊C814yes—
454#41—74no—
455#41—74no—
456#41—78no—
457#41—74no—
458#41—1416no—
459#41—144no—
460#41—148no—
461#41—142no—
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—71no—
465C2×Q8—78no—
466C2×Q8—71no—
467C2×Q8—142no—
468C2×Q8—144no—
469C2×Q8—148no—
470C2×Q8C8×D714yescomplemented
471C2×Q8C2×C4×D712yes—
472C2×D4—72no—
473C2×D4—71no—
474C2×D4—71no—
475C2×D4—142no—
476C2×D4—141no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—72no—
480C22×C4—72no—
481C22×C4—72no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—141no—
489C22×C4C2×C4×D711yescharacteristic
490C22×C4D4×D711yescharacteristic
491C22×C4D4×D711yescharacteristic
492C2×C8—22no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—142no—
498C2×C8—284no—
499C2×C8—282no—
500C2×C8—562no—
501C2×C8—561no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—72no—
506C4⋊C4—72no—
507C4⋊C4—71no—
508C4⋊C4—71no—
509C4⋊C4—71no—
510C4⋊C4—71no—
511C4⋊C4—142no—
512C4⋊C4—142no—
513C4⋊C4—142no—
514C4⋊C4—142no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4C2×C4×D712yes—
518C22⋊C4—72no—
519C22⋊C4—72no—
520C22⋊C4—72no—
521C22⋊C4—142no—
522C22⋊C4—142no—
523C22⋊C4—141no—
524C22⋊C4—141no—
525C22⋊C4—141no—
526C42—22no—
527C42—22no—
528C42—72no—
529C42—72no—
530C42—72no—
531C42—142no—
532C42—142no—
533C42—142no—
534C42—142no—
535C42—141no—
536C42—141no—
537C42—141no—
538C42C2×C4×D711yescharacteristic
539C42D4×D711yescharacteristic
540C42D4×D711yescharacteristic
541C42C23×D711yescharacteristic
542C2×C7—22no—
543C2×C7C22×C22⋊C811yescharacteristic
544C2×C7#289811yescharacteristic
545C2×C7#290111yescharacteristic
546D7—24no—
547D7—42no—
548C23—72no—
549C23—141no—
550C23D7×C22⋊C411yescharacteristic
551Q8—22no—
552Q8—24no—
553Q8—74no—
554Q8—72no—
555Q8—148no—
556Q8—141no—
557Q8C2×C8×D714yes—
558D4—78no—
559D4—72no—
560D4—74no—
561D4—148no—
562D4—144no—
563D4—141no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—72no—
571C2×C4—74no—
572C2×C4—74no—
573C2×C4—72no—
574C2×C4—72no—
575C2×C4—81no—
576C2×C4—148no—
577C2×C4—148no—
578C2×C4—141no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—144no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—282no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—561no—
588C2×C4—562no—
589C2×C4—562no—
590C2×C4D7×C22⋊C411yescharacteristic
591C2×C4D7×C22⋊C411yescharacteristic
592C2×C4D7×C22⋊C411yescharacteristic
593C2×C4C2×C8×D712yes—
594C2×C4D7×M1612yes—
595C2×C4C22×C4×D711yescharacteristic
596C2×C4C2×D4×D711yescharacteristic
597C2×C4C2×D4×D711yescharacteristic
598C8—42no—
599C8—41no—
600C8—41no—
601C8—142no—
602C8—281no—
603C8—282no—
604C7#1743911yescharacteristic, complemented, Sylow
605C22—21no—
606C22—21no—
607C22—74no—
608C22—142no—
609C22—142no—
610C22—142no—
611C22—281no—
612C22—281no—
613C22D7×C22⋊C812yes—
614C22C2×D7×C22⋊C411yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—24no—
619C4—42no—
620C4—74no—
621C4—142no—
622C4—142no—
623C4—142no—
624C4—281no—
625C4—281no—
626C4D7×C22⋊C812yes—
627C4C22×C8×D711yescharacteristic
628C4C2×D7×M1611yescharacteristic
629C4#9046911yescharacteristic
630C4#9047011yescharacteristic
631C2—22no—
632C2—144no—
633C2—282no—
634C2C2×D7×C22⋊C811yescharacteristic
635C2#1173640211yescharacteristic
636C2#1173643711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC4×C7C23×C8
FrattiniC42C23×D7
FittingC7×#1479C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#6551C4
cosocleC42×C7C24
hypercenter#1479D7
hypocenterC7#17439
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
2. #11736736C2semidirect
3. #11736721C2semidirect
4. C7×#1479C2semidirect
5. C2×Q8⋊(C7⋊C8)C2semidirect
6. #11728509C2semidirect
7. #11728508C2semidirect
8. #11725985C2semidirect
9. #11725984C2semidirect
10. #11725983C2semidirect
11. D7×Q8⋊C8C2semidirect
56. #6551C8semidirect
147. #1479D7semidirect
156. C2×C7×Q8C2×C8semidirect
167. #1161C2×C8semidirect
170. Q8×D7C2×C8semidirect
174. #1157C2×C8semidirect
470. C2×Q8C8×D7semidirect
604. C7#17439semidirect

Maximal subgroups

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

subgroupindex
C4×#65512
#117367362
#117367212
C7×#14792
C2×Q8⋊(C7⋊C8)2
#117285092
#117285082
#117259852
#117259842
#117259832
D7×Q8⋊C82
#174397