Finite Group Databasebeta

← C2×#22619744 #33953513

Subgroups — C2×#22619744 #33953513

Order 18688.

2358 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556590C211yesmaximal, characteristic, complemented
2C2×#17989055C211yesmaximal, characteristic
3#22619744C2116yesmaximal, direct product factor
4C2×#17988881C212yesmaximal, complemented
5C2×#17988871C212yesmaximal, complemented
6C2×#17988876C212yesmaximal, complemented
7C2×#17988866C212yesmaximal, complemented
8C2×#17988630C211yesmaximal, characteristic, complemented
9C2×#17988610C211yesmaximal, characteristic, complemented
10C2×#17988589C211yesmaximal, characteristic
11C2×#17988586C211yesmaximal, characteristic
12C2×#428×C73C211yesmaximal, characteristic, complemented
13C22×#11924028C2211yescharacteristic
14C23×Dic146—21no—
15C22×#11924012C2211yescharacteristic
16C2×#14556590—21no—
17C2×#14556590—21no—
18C2×#14556590—24no—
19C2×#14556590C2212yes—
20C2×#14556590C2212yes—
21C2×#14556590C2218yescomplemented
22#17989055—24no—
23#17989055C2218yes—
24C2×#14556556C2211yescharacteristic
25C2×#14556556C2211yescharacteristic
26C2×#14556553C2211yescharacteristic
27C2×#14556553C2211yescharacteristic
28C2×#14556551—21no—
29C2×#14556551—21no—
30C2×#14556548—21no—
31C22×C4⋊Dic73C2211yescharacteristic
32C22×#11924007—21no—
33C22×C4×Dic73C2211yescharacteristic
34C2×C42×D73—21no—
35#17988881C22116yescomplemented
36#17988876C22116yescomplemented
37#17988871C22116yescomplemented
38#17988866C22116yescomplemented
39#17988630C2218yescomplemented
40C2×#14556524C2212yes—
41C2×#14556522C2212yes—
42#17988610C2218yescomplemented
43C2×#14556520C2212yes—
44C2×#14556520C2212yes—
45#17988589C2218yes—
46#17988586C2218yes—
47C2×#14556516C2212yes—
48C2×#14556514C2212yes—
49C2×#14556513C2212yes—
50C2×#14556511C2212yes—
51C22×M16×C73C2211yescharacteristic
52C2×#117×C73C2211yescharacteristic
53C22×C4⋊C4×C73C2211yescharacteristic
54#428×C73C2218yescomplemented
55C2×#107×C73C2212yes—
56C2×#106×C73C2212yes—
57C2×#11924028—22no—
58C2×#11924028—24no—
59C2×#11924028Q818yescomplemented
60C2×#11924028C2314yes—
61C22×Dic146—22no—
62C22×Dic146—22no—
63C22×Dic146—28no—
64C22×Dic146—48no—
65C22×Dic146C2311yescharacteristic
66C22×D292C2311yescharacteristic
67C22×C22⋊D73—21no—
68C23×Dic73—21no—
69C23×Dic73D411yescharacteristic
70C23×Dic73D411yescharacteristic
71C22×C4×D73—21no—
72C22×C4×D73—21no—
73C22×C4×D73—21no—
74C2×#11924012—24no—
75C2×#11924012C2314yes—
76C22×#10596831C2312yes—
77#14556590—24no—
78#14556590—24no—
79#14556590—28no—
80#14556590—28no—
81#14556590C2318yes—
82#14556590C2318yes—
83#14556556—22no—
84#14556556—22no—
85#14556556C2314yes—
86#14556556C2314yes—
87#14556553—22no—
88#14556553—22no—
89#14556553C2314yes—
90#14556553C2314yes—
91#14556551—28no—
92#14556551—28no—
93#14556548—28no—
94C2×C4⋊Dic73—21no—
95C2×C4⋊Dic73—22no—
96C2×C4⋊Dic73C2312yes—
97C2×C4⋊Dic73C2314yes—
98C2×#11924007—22no—
99C2×#11924007—22no—
100C2×#11924007—28no—
101C2×#11924007—21no—
102C2×#11924007—21no—
103C2×#11924005—21no—
104C2×#11924005—21no—
105C2×C4×Dic73—21no—
106C2×C4×Dic73—21no—
107C2×C4×Dic73D411yescharacteristic
108C2×C4×Dic73D411yescharacteristic
109C2×C4×Dic73D414yes—
110C2×C4×Dic73C2314yes—
111C2×C4×Dic73C2311yescharacteristic
112C2×C4×Dic73C2311yescharacteristic
113C42×D73—28no—
114#14556524C2318yes—
115#14556522C2318yes—
116#14556520C2318yes—
117#14556520C2318yes—
118#14556516C2318yes—
119#14556514C2318yes—
120#14556513C2318yes—
121#14556511C2318yes—
122C23×C4×C73—21no—
123C23×C4×C73C2311yescharacteristic
124C2×M16×C73—24no—
125C2×M16×C73C2314yes—
126C22×C8×C73C2312yes—
127#117×C73—22no—
128#117×C73C2314yes—
129C2×C4⋊C4×C73—21no—
130C2×C4⋊C4×C73—22no—
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C4⋊C4×C73C2312yes—
133C2×C4⋊C4×C73C2314yes—
134C2×C4⋊C4×C73C2311yescharacteristic
135C2×C22⋊C4×C73—21no—
136C2×C42×C73—21no—
137#107×C73C2318yes—
138#106×C73C2318yes—
139C23×D73—21no—
140#11924028—216no—
141#11924028—28no—
142#11924028C2×Q8116yescomplemented
143C2×Dic146—21no—
144C2×Dic146—22no—
145C2×Dic146—28no—
146C2×Dic146—28no—
147C2×Dic146—48no—
148C2×Dic146—432no—
149C2×Dic146—48no—
150C2×Dic146C2×Q814yescomplemented
151C2×Dic146C2412yes—
152C2×D292—21no—
153C2×D292—22no—
154C2×D292C2×Q814yescomplemented
155C2×D292C2412yes—
156C2×C22⋊D73—28no—
157C2×C22⋊D73—24no—
158C22×Dic73—21no—
159C22×Dic73—24no—
160C22×Dic73—24no—
161C22×Dic73—24no—
162C22×Dic73—22no—
163C22×Dic73—44no—
164C22×Dic73C2×D411yescharacteristic
165C22×Dic73C2×D411yescharacteristic
166C22×Dic73C2×D414yes—
167C22×Dic73C2×D414yes—
168C22×Dic73C2×D411yescharacteristic
169C22×Dic73C2×D411yescharacteristic
170C2×C4×D73—28no—
171C2×C4×D73—24no—
172C2×C4×D73—24no—
173C2×C4×D73—24no—
174C2×C4×D73—44no—
175C2×C4×D73—44no—
176#11924012—28no—
177C2×#10596831—24no—
178C2×#10596831C2414yes—
179C4⋊Dic73—24no—
180C4⋊Dic73—22no—
181C4⋊Dic73C2414yes—
182#11924007—28no—
183#11924007—28no—
184#11924007—24no—
185#11924007—24no—
186#11924005—24no—
187#11924005—24no—
188C4×Dic73—24no—
189C4×Dic73—24no—
190C4×Dic73C2×D414yes—
191C4×Dic73C2×D414yes—
192C4×Dic73C2×D412yes—
193C4×Dic73C2×D412yes—
194C4×Dic73C2412yes—
195C4×Dic73C2412yes—
196C24×C73C2×D411yescharacteristic
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—24no—
200C22×C4×C73—22no—
201C22×C4×C73—44no—
202C22×C4×C73C2×D411yescharacteristic
203C22×C4×C73C2×D412yes—
204C22×C4×C73C2×Q814yes—
205C22×C4×C73#4114yes—
206C22×C4×C73C2411yescharacteristic
207C22×C4×C73C2412yes—
208M16×C73—28no—
209C2×C8×C73—24no—
210C2×C8×C73C2414yes—
211C4⋊C4×C73—24no—
212C4⋊C4×C73—22no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73C2412yes—
216C4⋊C4×C73C2414yes—
217C4⋊C4×C73C2412yes—
218C22⋊C4×C73—24no—
219C42×C73—24no—
220C23×C73—22no—
221C23×C73—22no—
222C23×C73C22⋊Q814yes—
223C23×C73C22×D412yes—
224C23×C73C22×D411yescharacteristic
225C22×D73—24no—
226C22×D73—22no—
227C22×D73—42no—
228C22×D73—42no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—24no—
232C2×C4×C73—48no—
233C2×C4×C73—44no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73C22⋊Q814yes—
237C2×C4×C73C22⋊Q818yes—
238C2×C4×C73C22×D412yes—
239C2×C4×C73C22×D411yescharacteristic
240C2×C4×C73C22×Q812yes—
241C2×C4×C73C22×Q814yes—
242C2×C4×C73C2×#4112yes—
243C2×C4×C73C2×#4114yes—
244C2×C4×C73C2511yescharacteristic
245C22⋊D73—216no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—28no—
251C2×Dic73—28no—
252C2×Dic73—24no—
253C2×Dic73—48no—
254C2×Dic73—44no—
255C2×Dic73AGL(1,ℤ/8)14yes—
256C2×Dic73#13714yes—
257C2×Dic73C22×D412yes—
258C2×Dic73C22×D412yes—
259C2×Dic73C22×D412yes—
260C2×Dic73C22×D412yes—
261D292—24no—
262D292—22no—
263D292C22×Q814yescomplemented
264C4×D73—216no—
265C4×D73—48no—
266C4×D73—48no—
267Dic146—24no—
268Dic146—22no—
269Dic146—416no—
270Dic146—416no—
271Dic146C22×Q814yescomplemented
272C8×C73—24no—
273#10596831—24no—
274C22×C73—28no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73C2×C22⋊Q814yes—
278C22×C73C2×C22⋊Q812yes—
279C22×C73C2×AGL(1,ℤ/8)12yes—
280C22×C73C2×#13712yes—
281C22×C73C23×D411yescharacteristic
282C2×D73—24no—
283C2×D73—48no—
284C2×D73—42no—
285C2×D73—42no—
286C4×C73—44no—
287C4×C73—42no—
288C4×C73—42no—
289C4×C73C2×C22⋊Q812yes—
290C4×C73C2×C22⋊Q814yes—
291C4×C73C23×Q811yescharacteristic
292C4×C73C22×#4111yescharacteristic
293Dic73—24no—
294Dic73—22no—
295Dic73—22no—
296Dic73—44no—
297Dic73C2×AGL(1,ℤ/8)14yes—
298Dic73C2×#13714yes—
299C2×#3079—731nomaximal, Sylow
300C2×C73—24no—
301C2×C73#307914yescomplemented
302C2×C73C22×C22⋊Q811yescharacteristic
303C2×C73C22×AGL(1,ℤ/8)11yescharacteristic
304C2×C73C22×#13711yescharacteristic
305D73—44no—
306C22×C4⋊Q8—731no—
307C2×C4×#41—731no—
308#3079—7316no—
309C2×#477—732no—
310C2×#475—732no—
311C2×#476—732no—
312C2×#474—732no—
313C2×#428—731no—
314C2×#428D7311yescharacteristic, complemented
315C2×#423—731no—
316C2×#417—731no—
317C2×#417—731no—
318C73C2×#307911yescharacteristic, complemented, Sylow
319C22×#41—731no—
320C23×Q8—1461no—
321C22×M16—731no—
322C22×M16C2×D7311yescharacteristic
323C2×C4⋊Q8—732no—
324C2×C4⋊Q8—732no—
325C2×C4⋊Q8—738no—
326C2×C4⋊Q8—1461no—
327C2×C4⋊Q8—1464no—
328C2×C4⋊Q8—1461no—
329C4×#41—738no—
330C4×#41—1464no—
331C2×C4×Q8—731no—
332C2×C4×Q8—731no—
333C2×C4×D4—731no—
334C2×C4×D4—731no—
335C2×C4×D4—1461no—
336C2×C4×D4—1461no—
337C2×#117—1461no—
338C2×#117C2×D7311yescharacteristic
339C22×C4⋊C4—731no—
340C22×C4⋊C4—1461no—
341C22×C4⋊C4C2×D7311yescharacteristic
342C22×C42—731no—
343C22×C42—1461no—
344#477—7316no—
345#476—7316no—
346#475—7316no—
347#474—7316no—
348#428—738no—
349#428C2×D7318yescomplemented
350C2×#107—732no—
351C2×#107C2×D7312yes—
352C2×#106—732no—
353C2×#106C2×D7312yes—
354#423—738no—
355C2×C4⋊C8—732no—
356C2×C4⋊C8—732no—
357#417—738no—
358#417—738no—
359C2×#103—732no—
360C2×#103—732no—
361C2×#102—732no—
362C2×#102—732no—
363C2×#41—738no—
364C2×#41—734no—
365C2×#41—1462no—
366C2×#41—1464no—
367C22×Q8—2928no—
368C22×Q8—731no—
369C22×Q8—1468no—
370C22×Q8—1462no—
371C22×Q8—1462no—
372C22×D4—731no—
373C22×D4—1461no—
374C23×C4—21no—
375C23×C4—731no—
376C23×C4—731no—
377C23×C4—1461no—
378C23×C4—1461no—
379C23×C4—1461no—
380C23×C4—1461no—
381C23×C4C22×D7311yescharacteristic
382C2×M16—24no—
383C2×M16—734no—
384C2×M16—1464no—
385C2×M16C22×D7314yes—
386C22×C8—732no—
387C22×C8C22×D7312yes—
388C4⋊Q8—738no—
389C4⋊Q8—738no—
390C4⋊Q8—1468no—
391C4⋊Q8—1468no—
392C4⋊Q8—1464no—
393C4⋊Q8—1464no—
394C4×Q8—734no—
395C4×Q8—734no—
396C4×Q8—1462no—
397C4×Q8—1462no—
398C4×D4—734no—
399C4×D4—734no—
400C4×D4—1462no—
401C4×D4—1462no—
402C4×D4—1468no—
403C4×D4—1468no—
404#117—22no—
405#117—1468no—
406#117C22×D7314yes—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—732no—
410C2×C4⋊C4—734no—
411C2×C4⋊C4—1468no—
412C2×C4⋊C4—1462no—
413C2×C4⋊C4—1462no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4—1462no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4C22×D7312yes—
419C2×C4⋊C4C22×D7314yes—
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C4⋊C4C22×D7311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C42—21no—
426C2×C42—731no—
427C2×C42—734no—
428C2×C42—731no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—734no—
432C2×C42—1468no—
433C2×C42—1461no—
434C2×C42—1461no—
435#107—738no—
436#107C22×D7318yes—
437#106—738no—
438#106C22×D7318yes—
439C4⋊C8—738no—
440C4⋊C8—738no—
441#103—738no—
442#103—738no—
443#102—738no—
444#102—738no—
445C24—1461no—
446C24D4×D7311yescharacteristic
447#41—7316no—
448#41—14616no—
449#41—1468no—
450C2×Q8—2928no—
451C2×Q8—29232no—
452C2×Q8—2928no—
453C2×Q8—734no—
454C2×Q8—732no—
455C2×Q8—1468no—
456C2×Q8—1468no—
457C2×Q8—1461no—
458C2×Q8—1462no—
459C2×D4—734no—
460C2×D4—732no—
461C2×D4—1468no—
462C2×D4—1464no—
463C2×D4—1461no—
464C2×D4—1462no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—2924no—
471C22×C4—2924no—
472C22×C4—2924no—
473C22×C4—731no—
474C22×C4—731no—
475C22×C4—734no—
476C22×C4—734no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—1464no—
480C22×C4—1464no—
481C22×C4—1464no—
482C22×C4—1462no—
483C22×C4—1464no—
484C22×C4—1464no—
485C22×C4—1461no—
486C22×C4—1468no—
487C22×C4—1464no—
488C22×C4D4×D7312yes—
489C22×C4D4×D7311yescharacteristic
490C22×C4Q8×D7314yes—
491C22×C4#1192402814yes—
492C22×C4C23×D7312yes—
493C22×C4C23×D7311yescharacteristic
494M16—28no—
495M16—1468no—
496C2×C8—24no—
497C2×C8—734no—
498C2×C8—1464no—
499C2×C8C23×D7314yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—734no—
505C4⋊C4—1468no—
506C4⋊C4—1468no—
507C4⋊C4—1464no—
508C4⋊C4—1464no—
509C4⋊C4—1464no—
510C4⋊C4—1462no—
511C4⋊C4C23×D7314yes—
512C4⋊C4C23×D7312yes—
513C4⋊C4C23×D7312yes—
514C22⋊C4—24no—
515C22⋊C4—1464no—
516C22⋊C4—1464no—
517C42—24no—
518C42—734no—
519C42—734no—
520C42—732no—
521C42—732no—
522C42—732no—
523C42—732no—
524C42—1464no—
525C42—1464no—
526C23—22no—
527C23—22no—
528C23—2922no—
529C23—2922no—
530C23—1462no—
531C23—1464no—
532C23#1455656914yes—
533C23C2×D4×D7312yes—
534C23C2×D4×D7311yescharacteristic
535Q8—29216no—
536Q8—29216no—
537Q8—734no—
538Q8—1462no—
539Q8—1464no—
540D4—734no—
541D4—14616no—
542D4—1462no—
543D4—1464no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—2928no—
552C2×C4—2928no—
553C2×C4—2928no—
554C2×C4—2924no—
555C2×C4—734no—
556C2×C4—732no—
557C2×C4—732no—
558C2×C4—734no—
559C2×C4—732no—
560C2×C4—732no—
561C2×C4—1464no—
562C2×C4—1468no—
563C2×C4—1468no—
564C2×C4—1462no—
565C2×C4—1462no—
566C2×C4—1464no—
567C2×C4—1462no—
568C2×C4—14616no—
569C2×C4#1455656914yes—
570C2×C4#1455656918yes—
571C2×C4C2×D4×D7312yes—
572C2×C4C2×D4×D7311yescharacteristic
573C2×C4C2×Q8×D7312yes—
574C2×C4C2×Q8×D7314yes—
575C2×C4C2×#1192402812yes—
576C2×C4C2×#1192402814yes—
577C2×C4C24×D7311yescharacteristic
578C8—24no—
579C8—1464no—
580C22—28no—
581C22—22no—
582C22—22no—
583C22—2928no—
584C22—2922no—
585C22—2922no—
586C22—1464no—
587C22C2×#1455656914yes—
588C22C2×#1455656912yes—
589C22AGL(1,ℤ/8)×D7312yes—
590C22#137×D7312yes—
591C22C22×D4×D7311yescharacteristic
592C4—44no—
593C4—42no—
594C4—42no—
595C4—2924no—
596C4—734no—
597C4—734no—
598C4—1462no—
599C4—1462no—
600C4—1464no—
601C4C2×#1455656914yes—
602C4C2×#1455656912yes—
603C4C22×Q8×D7311yescharacteristic
604C4C22×#1192402811yescharacteristic
605C2—24no—
606C2—2924no—
607C2#2261974414yesdirect product factor
608C2C22×#1455656911yescharacteristic
609C2C2×AGL(1,ℤ/8)×D7311yescharacteristic
610C2C2×#137×D7311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#428×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
hypercenterC2×#428D73
hypocenterC73C2×#3079
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C22×#14556590C2semidirect
3. #22619744C2direct (N × H)
4. C2×#17988881C2semidirect
5. C2×#17988871C2semidirect
6. C2×#17988876C2semidirect
7. C2×#17988866C2semidirect
8. C2×#17988630C2semidirect
9. C2×#17988610C2semidirect
12. C2×#428×C73C2semidirect
21. C2×#14556590C22semidirect
35. #17988881C22semidirect
36. #17988876C22semidirect
37. #17988871C22semidirect
38. #17988866C22semidirect
39. #17988630C22semidirect
42. #17988610C22semidirect
54. #428×C73C22semidirect
59. C2×#11924028Q8semidirect
142. #11924028C2×Q8semidirect
150. C2×Dic146C2×Q8semidirect
154. C2×D292C2×Q8semidirect
263. D292C22×Q8semidirect
271. Dic146C22×Q8semidirect
301. C2×C73#3079semidirect
314. C2×#428D73semidirect
318. C73C2×#3079semidirect
349. #428C2×D73semidirect
607. C2#22619744direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565902
C2×#179890552
#226197442
C2×#179888812
C2×#179888712
C2×#179888762
C2×#179888662
C2×#179886302
C2×#179886102
C2×#179885892
C2×#179885862
C2×#428×C732
C2×#307973