Finite Group Databasebeta

← C67×#11043 #30690294

Subgroups — C67×#11043 #30690294

Order 17152.

1454 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#99×C67C212yesmaximal
2C2×#386×C67C211yesmaximal, characteristic
3C2×#380×C67C211yesmaximal, characteristic, complemented
4C2×#342×C67C212yesmaximal, complemented
5C67×C22⋊#95C211yesmaximal, characteristic, complemented
6C23×D4×C67C2211yescharacteristic
7C24×C4×C67C2211yescharacteristic
8C22×C4⋊C4×C67—21no—
9C22×C22⋊C4×C67C412yescomplemented
10C22×C22⋊C4×C67C2211yescharacteristic
11C22×C22⋊C4×C67C2212yes—
12C22×C22⋊C4×C67C2212yes—
13C2×#99×C67—24no—
14C2×#99×C67—24no—
15C2×#99×C67—24no—
16#386×C67—21no—
17#386×C67—22no—
18#386×C67—21no—
19#380×C67—24no—
20C2×#95×C67—21no—
21C2×#95×C67—22no—
22C2×#95×C67C412yescomplemented
23#342×C67—24no—
24C25×C67C2×C411yescharacteristic
25C25×C67C2311yescharacteristic
26C22×D4×C67—21no—
27C22×D4×C67—21no—
28C22×D4×C67—22no—
29C22×D4×C67—22no—
30C22×D4×C67—21no—
31C22×D4×C67—21no—
32C22×D4×C67—22no—
33C22×D4×C67—22no—
34C22×D4×C67D412yes—
35C22×D4×C67Q812yes—
36C23×C4×C67—22no—
37C23×C4×C67—22no—
38C23×C4×C67—22no—
39C23×C4×C67—22no—
40C23×C4×C67—44no—
41C23×C4×C67C2×C411yescharacteristic
42C23×C4×C67C2×C412yes—
43C23×C4×C67C2×C412yes—
44C23×C4×C67D411yescharacteristic
45C23×C4×C67D412yes—
46C23×C4×C67D411yescharacteristic
47C2×C4⋊C4×C67—24no—
48C2×C4⋊C4×C67—44no—
49C2×C22⋊C4×C67—22no—
50C2×C22⋊C4×C67—22no—
51C2×C22⋊C4×C67—22no—
52C2×C22⋊C4×C67—22no—
53C2×C22⋊C4×C67—22no—
54C2×C22⋊C4×C67—22no—
55C2×C22⋊C4×C67—22no—
56C2×C22⋊C4×C67—22no—
57C2×C22⋊C4×C67—22no—
58C2×C22⋊C4×C67—22no—
59C2×C22⋊C4×C67—21no—
60C2×C22⋊C4×C67—21no—
61C2×C22⋊C4×C67—22no—
62C2×C22⋊C4×C67—22no—
63C2×C22⋊C4×C67—22no—
64C2×C22⋊C4×C67—22no—
65C2×C22⋊C4×C67—44no—
66C2×C22⋊C4×C67D414yes—
67C2×C22⋊C4×C67D414yes—
68#99×C67—216no—
69#95×C67—24no—
70#95×C67—22no—
71#95×C67—22no—
72#95×C67—44no—
73C24×C67—21no—
74C24×C67—21no—
75C24×C67—22no—
76C24×C67—22no—
77C24×C67—42no—
78C24×C67—42no—
79C24×C67—42no—
80C24×C67—42no—
81C24×C67C4⋊C412yescomplemented
82C24×C67C22×C411yescharacteristic
83C24×C67C2×D411yescharacteristic
84C24×C67C2×D411yescharacteristic
85C24×C67C2×D411yescharacteristic
86C24×C67C2×D412yes—
87C24×C67C2×D412yes—
88C24×C67C2×Q811yescharacteristic
89C24×C67#4111yescharacteristic
90C24×C67#4111yescharacteristic
91C24×C67#4112yes—
92C24×C67#4112yes—
93C2×D4×C67—24no—
94C2×D4×C67—22no—
95C2×D4×C67—22no—
96C2×D4×C67—22no—
97C2×D4×C67—22no—
98C2×D4×C67—24no—
99C2×D4×C67—24no—
100C2×D4×C67—24no—
101C2×D4×C67—41no—
102C2×D4×C67—42no—
103C2×D4×C67—42no—
104C2×D4×C67—42no—
105C2×D4×C67—41no—
106C2×D4×C67—42no—
107C2×D4×C67—42no—
108C2×D4×C67—41no—
109C2×D4×C67—42no—
110C2×D4×C67—41no—
111C22×C4×C67—21no—
112C22×C4×C67—21no—
113C22×C4×C67—24no—
114C22×C4×C67—21no—
115C22×C4×C67—21no—
116C22×C4×C67—24no—
117C22×C4×C67—22no—
118C22×C4×C67—22no—
119C22×C4×C67—21no—
120C22×C4×C67—24no—
121C22×C4×C67—21no—
122C22×C4×C67—24no—
123C22×C4×C67—24no—
124C22×C4×C67—22no—
125C22×C4×C67—24no—
126C22×C4×C67—24no—
127C22×C4×C67—24no—
128C22×C4×C67—42no—
129C22×C4×C67—42no—
130C22×C4×C67—42no—
131C22×C4×C67—48no—
132C22×C4×C67—48no—
133C22×C4×C67—44no—
134C22×C4×C67—44no—
135C22×C4×C67—44no—
136C22×C4×C67—44no—
137C22×C4×C67—42no—
138C22×C4×C67—42no—
139C22×C4×C67—42no—
140C22×C4×C67—42no—
141C22×C4×C67—42no—
142C22×C4×C67—42no—
143C22×C4×C67C22⋊C412yes—
144C22×C4×C67C22⋊C412yes—
145C22×C4×C67C4⋊C412yescomplemented
146C4⋊C4×C67—48no—
147C22⋊C4×C67—24no—
148C22⋊C4×C67—24no—
149C22⋊C4×C67—24no—
150C22⋊C4×C67—24no—
151C22⋊C4×C67—44no—
152C22⋊C4×C67—42no—
153C22⋊C4×C67—42no—
154C22⋊C4×C67—42no—
155C22⋊C4×C67—42no—
156C22⋊C4×C67—41no—
157C22⋊C4×C67—42no—
158C22⋊C4×C67—41no—
159C22⋊C4×C67—44no—
160C23×C67—22no—
161C23×C67—22no—
162C23×C67—22no—
163C23×C67—22no—
164C23×C67—22no—
165C23×C67—22no—
166C23×C67—22no—
167C23×C67—24no—
168C23×C67—42no—
169C23×C67—42no—
170C23×C67—42no—
171C23×C67—42no—
172C23×C67—44no—
173C23×C67—42no—
174C23×C67—42no—
175C23×C67—42no—
176C23×C67—42no—
177C23×C67—44no—
178C23×C67—44no—
179C23×C67—44no—
180C23×C67—41no—
181C23×C67—42no—
182C23×C67—41no—
183C23×C67—42no—
184C23×C67—42no—
185C23×C67—82no—
186C23×C67—82no—
187C23×C67—84no—
188C23×C67C2×C22⋊C411yescharacteristic
189C23×C67C2×C4⋊C411yescharacteristic
190C23×C67#11711yescharacteristic
191C23×C67C4×D412yes—
192C23×C67C4×D412yes—
193C23×C67#12011yescharacteristic
194C23×C67#12012yes—
195C23×C67#12111yescharacteristic
196C23×C67#12111yescharacteristic
197C23×C67#12112yes—
198C23×C67#12112yes—
199C23×C67#12112yes—
200C23×C67C22⋊Q811yescharacteristic
201C23×C67C22⋊Q811yescharacteristic
202C23×C67C22⋊Q812yes—
203C23×C67C22⋊Q812yes—
204C23×C67#12311yescharacteristic
205C23×C67#12312yes—
206C23×C67#12312yes—
207C23×C67#12411yescharacteristic
208C23×C67#12412yes—
209C23×C67#12612yes—
210C23×C67#12711yescharacteristic
211D4×C67—42no—
212D4×C67—42no—
213D4×C67—44no—
214D4×C67—44no—
215D4×C67—42no—
216D4×C67—42no—
217C2×C4×C67—24no—
218C2×C4×C67—24no—
219C2×C4×C67—22no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—24no—
223C2×C4×C67—22no—
224C2×C4×C67—22no—
225C2×C4×C67—22no—
226C2×C4×C67—48no—
227C2×C4×C67—48no—
228C2×C4×C67—42no—
229C2×C4×C67—42no—
230C2×C4×C67—41no—
231C2×C4×C67—42no—
232C2×C4×C67—41no—
233C2×C4×C67—44no—
234C2×C4×C67—44no—
235C2×C4×C67—48no—
236C2×C4×C67—42no—
237C2×C4×C67—44no—
238C2×C4×C67—44no—
239C2×C4×C67—42no—
240C2×C4×C67—42no—
241C2×C4×C67—84no—
242C2×C4×C67—84no—
243C2×C4×C67—82no—
244C2×C4×C67—82no—
245C2×C4×C67—84no—
246C2×C4×C67—84no—
247C2×C4×C67#9914yescomplemented
248C22×C67—22no—
249C22×C67—22no—
250C22×C67—22no—
251C22×C67—22no—
252C22×C67—22no—
253C22×C67—22no—
254C22×C67—21no—
255C22×C67—21no—
256C22×C67—22no—
257C22×C67—22no—
258C22×C67—21no—
259C22×C67—22no—
260C22×C67—22no—
261C22×C67—21no—
262C22×C67—24no—
263C22×C67—22no—
264C22×C67—24no—
265C22×C67—22no—
266C22×C67—42no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67—42no—
270C22×C67—42no—
271C22×C67—42no—
272C22×C67—42no—
273C22×C67—42no—
274C22×C67—44no—
275C22×C67—42no—
276C22×C67—84no—
277C22×C67—84no—
278C22×C67—81no—
279C22×C67—81no—
280C22×C67—82no—
281C22×C67#38011yescharacteristic
282C22×C67#38512yes—
283C22×C67#38611yescharacteristic
284C22×C67#38812yes—
285C22×C67#39011yescharacteristic
286C22×C67#39211yescharacteristic
287C22×C67#39311yescharacteristic
288C22×C67#39412yes—
289C22×C67#39612yes—
290C22×C67#39711yescharacteristic
291C22×C67#39911yescharacteristic
292C22×C67C2×#9912yes—
293C22×C67#41012yes—
294C4×C67—24no—
295C4×C67—42no—
296C4×C67—42no—
297C4×C67—84no—
298C4×C67—84no—
299C4×C67—82no—
300#11043C6711yesmaximal, characteristic, direct product factor, Sylow
301C2×C67—21no—
302C2×C67—22no—
303C2×C67—22no—
304C2×C67—22no—
305C2×C67—21no—
306C2×C67—22no—
307C2×C67—21no—
308C2×C67—21no—
309C2×C67—82no—
310C2×C67#143911yescharacteristic
311C2×C67#179912yes—
312C2×C67#189811yescharacteristic
313C2×C67#189911yescharacteristic
314C2×C67#196412yes—
315C22×#99C2×C6712yes—
316C2×#386C2×C6711yescharacteristic
317C2×#380C2×C6711yescharacteristic, complemented
318C2×#342C2×C6712yescomplemented
319C22⋊#95C2×C6711yescharacteristic, complemented
320C67#1104311yescharacteristic, direct product factor, Sylow
321C23×D4C22×C6711yescharacteristic
322C24×C4C22×C6711yescharacteristic
323C22×C4⋊C4—21no—
324C22×C22⋊C4C4×C6712yescomplemented
325C22×C22⋊C4C22×C6711yescharacteristic
326C22×C22⋊C4C22×C6712yes—
327C22×C22⋊C4C22×C6712yes—
328C2×#99—24no—
329C2×#99—24no—
330C2×#99—24no—
331#386—21no—
332#386—22no—
333#386—21no—
334#380—24no—
335C2×#95—21no—
336C2×#95—22no—
337C2×#95C4×C6712yescomplemented
338#342—24no—
339C25C2×C4×C6711yescharacteristic
340C25C23×C6711yescharacteristic
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4D4×C6712yes—
350C22×D4Q8×C6712yes—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—44no—
356C23×C4C2×C4×C6711yescharacteristic
357C23×C4C2×C4×C6712yes—
358C23×C4C2×C4×C6712yes—
359C23×C4D4×C6711yescharacteristic
360C23×C4D4×C6712yes—
361C23×C4D4×C6711yescharacteristic
362C2×C4⋊C4—24no—
363C2×C4⋊C4—44no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—44no—
381C2×C22⋊C4D4×C6714yes—
382C2×C22⋊C4D4×C6714yes—
383#99—216no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—44no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C4⋊C4×C6712yescomplemented
397C24C22×C4×C6711yescharacteristic
398C24C2×D4×C6711yescharacteristic
399C24C2×D4×C6711yescharacteristic
400C24C2×D4×C6711yescharacteristic
401C24C2×D4×C6712yes—
402C24C2×D4×C6712yes—
403C24C2×Q8×C6711yescharacteristic
404C24#41×C6711yescharacteristic
405C24#41×C6711yescharacteristic
406C24#41×C6712yes—
407C24#41×C6712yes—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—41no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4C22⋊C4×C6712yes—
459C22×C4C22⋊C4×C6712yes—
460C22×C4C4⋊C4×C6712yescomplemented
461C4⋊C4—48no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—41no—
474C22⋊C4—44no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—82no—
501C23—82no—
502C23—84no—
503C23C2×C22⋊C4×C6711yescharacteristic
504C23C2×C4⋊C4×C6711yescharacteristic
505C23#117×C6711yescharacteristic
506C23C4×D4×C6712yes—
507C23C4×D4×C6712yes—
508C23#120×C6711yescharacteristic
509C23#120×C6712yes—
510C23#121×C6711yescharacteristic
511C23#121×C6711yescharacteristic
512C23#121×C6712yes—
513C23#121×C6712yes—
514C23#121×C6712yes—
515C23C22⋊Q8×C6711yescharacteristic
516C23C22⋊Q8×C6711yescharacteristic
517C23C22⋊Q8×C6712yes—
518C23C22⋊Q8×C6712yes—
519C23#123×C6711yescharacteristic
520C23#123×C6712yes—
521C23#123×C6712yes—
522C23#124×C6711yescharacteristic
523C23#124×C6712yes—
524C23#126×C6712yes—
525C23#127×C6711yescharacteristic
526D4—42no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4#99×C6714yescomplemented
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—21no—
570C22—21no—
571C22—22no—
572C22—22no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—21no—
577C22—24no—
578C22—22no—
579C22—24no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—84no—
592C22—84no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22#380×C6711yescharacteristic
597C22#385×C6712yes—
598C22#386×C6711yescharacteristic
599C22#388×C6712yes—
600C22#390×C6711yescharacteristic
601C22#392×C6711yescharacteristic
602C22#393×C6711yescharacteristic
603C22#394×C6712yes—
604C22#396×C6712yes—
605C22#397×C6711yescharacteristic
606C22#399×C6711yescharacteristic
607C22C2×#99×C6712yes—
608C22#410×C6712yes—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—84no—
613C4—84no—
614C4—82no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—22no—
619C2—21no—
620C2—22no—
621C2—21no—
622C2—21no—
623C2—82no—
624C2C67×#143911yescharacteristic
625C2C67×#179912yes—
626C2C67×#189811yescharacteristic
627C2C67×#189911yescharacteristic
628C2C67×#196412yes—
subgroupG/H
center, socleC23×C67C2×C22⋊C4
commutatorC24C22×C4×C67
Frattini, cosocleC25C23×C67
prime radicalC23×D4×C67C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C24×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#380×C67C2semidirect
4. C2×#342×C67C2semidirect
5. C67×C22⋊#95C2semidirect
9. C22×C22⋊C4×C67C4semidirect
22. C2×#95×C67C4semidirect
81. C24×C67C4⋊C4semidirect
145. C22×C4×C67C4⋊C4semidirect
247. C2×C4×C67#99semidirect
300. #11043C67direct (N × H)
317. C2×#380C2×C67semidirect
318. C2×#342C2×C67semidirect
319. C22⋊#95C2×C67semidirect
320. C67#11043direct (N × H)
324. C22×C22⋊C4C4×C67semidirect
337. C2×#95C4×C67semidirect
396. C24C4⋊C4×C67semidirect
460. C22×C4C4⋊C4×C67semidirect
562. C2×C4#99×C67semidirect

Maximal subgroups

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

subgroupindex
C22×#99×C672
C2×#386×C672
C2×#380×C672
C2×#342×C672
C67×C22⋊#952
#1104367