Finite Group Databasebeta

← C2×#22608277 #33051926

Subgroups — C2×#22608277 #33051926

Order 18688.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556574C211yesmaximal, characteristic
2C2×#17988551C211yesmaximal, characteristic, complemented
3#22608277C218yesmaximal, direct product factor
4C2×#17988312C212yesmaximal, complemented
5C2×#17988310C212yesmaximal, complemented
6C22×C22⋊C8×C73C211yesmaximal, characteristic, complemented
7C23×C22⋊D73C411yescharacteristic
8C22×#11924012—21no—
9C2×#14556574—21no—
10C2×#14556574—21no—
11C2×#14556574—22no—
12C2×#14556574—21no—
13C2×#14556574—21no—
14C2×#14556574—22no—
15C2×#14556574C414yes—
16C2×#14556574C2214yes—
17C22×#11924007C411yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×C22⋊Dic73C411yescharacteristic
20C22×#11924005C411yescharacteristic
21C22×#11924005C2211yescharacteristic
22#17988551—22no—
23#17988551C2214yescomplemented
24C2×C22⋊#10596831—21no—
25C2×C22⋊#10596831C2212yes—
26#17988312C2218yescomplemented
27#17988310C2218yescomplemented
28C24×C4×C73C2211yescharacteristic
29C23×C8×C73—21no—
30C2×C22⋊C8×C73—22no—
31C2×C22⋊C8×C73—21no—
32C2×C22⋊C8×C73C2212yes—
33C2×C22⋊C8×C73C2214yescomplemented
34C24×D73C2×C411yescharacteristic
35C22×C22⋊D73—21no—
36C22×C22⋊D73—22no—
37C22×C22⋊D73—21no—
38C22×C22⋊D73—44no—
39C22×C22⋊D73C2×C414yes—
40C23×Dic73C2×C411yescharacteristic
41C23×Dic73C2×C411yescharacteristic
42C23×Dic73C2×C411yescharacteristic
43C2×#11924012—24no—
44C2×#11924012—44no—
45C22×#10596831—22no—
46#14556574—24no—
47#14556574—24no—
48#14556574—24no—
49#14556574—24no—
50#14556574—24no—
51#14556574—24no—
52#14556574—24no—
53#14556574—24no—
54C2×#11924007—21no—
55C2×#11924007—21no—
56C2×#11924007—22no—
57C2×#11924007—21no—
58C2×#11924007—21no—
59C2×#11924007—21no—
60C2×#11924007—21no—
61C2×#11924007C2×C414yes—
62C2×#11924007C2×C412yes—
63C2×#11924007C2312yes—
64C2×C22⋊Dic73—21no—
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73—21no—
67C2×C22⋊Dic73C2×C414yes—
68C2×#11924005—21no—
69C2×#11924005—21no—
70C2×#11924005—22no—
71C2×#11924005—21no—
72C2×#11924005—21no—
73C2×#11924005—21no—
74C2×#11924005—21no—
75C2×#11924005C2×C414yes—
76C2×#11924005C2×C412yes—
77C2×#11924005C2312yes—
78C22⋊#10596831—24no—
79C22⋊#10596831—22no—
80C22⋊#10596831C2314yes—
81C25×C73C2×C411yescharacteristic
82C23×C4×C73—44no—
83C23×C4×C73C2×C411yescharacteristic
84C23×C4×C73C2×C412yes—
85C23×C4×C73D411yescharacteristic
86C23×C4×C73D411yescharacteristic
87C23×C4×C73D411yescharacteristic
88C23×C4×C73D412yes—
89C23×C4×C73D411yescharacteristic
90C23×C4×C73D412yes—
91C23×C4×C73C2311yescharacteristic
92C23×C4×C73C2312yes—
93C22×C8×C73—22no—
94C22×C8×C73—24no—
95C22×C8×C73—44no—
96C22⋊C8×C73—22no—
97C22⋊C8×C73—24no—
98C22⋊C8×C73C2314yes—
99C23×D73—21no—
100C23×D73—21no—
101C23×D73—24no—
102C23×D73—42no—
103C23×D73—42no—
104C23×D73C22×C412yes—
105C2×C22⋊D73—24no—
106C2×C22⋊D73—24no—
107C2×C22⋊D73—48no—
108C2×C22⋊D73—44no—
109C2×C22⋊D73—44no—
110C2×C22⋊D73—48no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—24no—
118C22×Dic73—42no—
119C22×Dic73—42no—
120C22×Dic73C22×C412yes—
121C22×Dic73C22×C412yes—
122C22×Dic73C22×C412yes—
123#11924012—48no—
124C2×#10596831—24no—
125C2×#10596831—44no—
126#11924007—24no—
127#11924007—24no—
128#11924007—22no—
129#11924007—22no—
130#11924007—22no—
131#11924007—22no—
132C22⋊Dic73—24no—
133C22⋊Dic73—24no—
134#11924005—24no—
135#11924005—24no—
136#11924005—22no—
137#11924005—22no—
138#11924005—22no—
139#11924005—22no—
140C24×C73—24no—
141C24×C73—24no—
142C24×C73—42no—
143C24×C73C22⋊C411yescharacteristic
144C24×C73C22⋊C412yes—
145C24×C73C22⋊C411yescharacteristic
146C24×C73C22×C412yes—
147C24×C73C22×C411yescharacteristic
148C22×C4×C73—24no—
149C22×C4×C73—24no—
150C22×C4×C73—42no—
151C22×C4×C73—44no—
152C22×C4×C73—42no—
153C22×C4×C73—44no—
154C22×C4×C73—44no—
155C22×C4×C73—42no—
156C22×C4×C73—42no—
157C22×C4×C73—44no—
158C22×C4×C73C22⋊C411yescharacteristic
159C22×C4×C73C22⋊C411yescharacteristic
160C22×C4×C73C22⋊C412yes—
161C22×C4×C73C22⋊C412yes—
162C22×C4×C73C22⋊C412yes—
163C22×C4×C73C22⋊C412yes—
164C22×C4×C73C22⋊C412yes—
165C22×C4×C73C22×C412yes—
166C22×C4×C73C22×C411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D412yes—
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D412yes—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2411yescharacteristic
176C2×C8×C73—24no—
177C2×C8×C73—44no—
178C2×C8×C73—48no—
179C23×C73—22no—
180C23×C73—22no—
181C23×C73—22no—
182C23×C73—22no—
183C23×C73—24no—
184C23×C73—24no—
185C23×C73—22no—
186C23×C73—24no—
187C23×C73—22no—
188C23×C73—24no—
189C23×C73—22no—
190C23×C73—42no—
191C23×C73—42no—
192C23×C73—416no—
193C23×C73#9914yes—
194C23×C73C2×C22⋊C412yes—
195C23×C73C2×C22⋊C411yescharacteristic
196C23×C73C2×C22⋊C412yes—
197C23×C73C2×C22⋊C411yescharacteristic
198C23×C73#13114yes—
199C23×C73C23×C411yescharacteristic
200C22×D73—22no—
201C22×D73—24no—
202C22×D73—42no—
203C22×D73—42no—
204C22×D73—42no—
205C22×D73—42no—
206C22×D73—42no—
207C22×D73—42no—
208C22×D73—42no—
209C22×D73—42no—
210C22×D73—88no—
211C2×C4×C73—24no—
212C2×C4×C73—22no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—24no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—41no—
223C2×C4×C73—42no—
224C2×C4×C73—41no—
225C2×C4×C73—42no—
226C2×C4×C73—41no—
227C2×C4×C73—42no—
228C2×C4×C73—41no—
229C2×C4×C73C2×C22⋊C412yes—
230C2×C4×C73C2×C22⋊C412yes—
231C2×C4×C73C2×C22⋊C411yescharacteristic
232C2×C4×C73C2×C22⋊C411yescharacteristic
233C2×C4×C73C22×D411yescharacteristic
234C2×C4×C73C22×D411yescharacteristic
235C22⋊D73—48no—
236C22⋊D73—48no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—24no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C8×C73—44no—
250#10596831—44no—
251C22×C73—22no—
252C22×C73—21no—
253C22×C73—21no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73—22no—
257C22×C73—28no—
258C22×C73—28no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73—22no—
262C22×C73—48no—
263C22×C73—48no—
264C22×C73—44no—
265C22×C73#40812yes—
266C22×C73#40812yes—
267C22×C73C2×#9914yes—
268C22×C73C2×#9912yes—
269C22×C73#41312yes—
270C22×C73C22×C22⋊C411yescharacteristic
271C22×C73C2×#13112yes—
272C2×D73—42no—
273C2×D73—42no—
274C2×D73—42no—
275C2×D73—84no—
276C2×D73—84no—
277C2×D73—84no—
278C4×C73—21no—
279C4×C73—21no—
280C4×C73—22no—
281C4×C73—21no—
282C4×C73—21no—
283C4×C73—22no—
284Dic73—42no—
285Dic73—42no—
286Dic73—42no—
287C2×#1468—731nomaximal, Sylow
288C2×C73—22no—
289C2×C73—22no—
290C2×C73—24no—
291C2×C73—42no—
292C2×C73#146814yescomplemented
293C2×C73C2×#40811yescharacteristic
294C2×C73C22×#9911yescharacteristic
295C2×C73C2×#41311yescharacteristic
296D73—82no—
297C22×#123—731no—
298C2×#407—731no—
299C22×C22⋊C8D7311yescharacteristic, complemented
300#1468—738no—
301C2×#324—732no—
302C2×#323—732no—
303C73C2×#146811yescharacteristic, complemented, Sylow
304C23×D4—731no—
305C24×C4C2×D7311yescharacteristic
306C22×M16—1461no—
307C23×C8—21no—
308C2×#123—734no—
309C2×#123—734no—
310C2×#123—1461no—
311C2×#123—1461no—
312C2×#123—1462no—
313C2×#123—1461no—
314C2×#123—1462no—
315C2×#123—1461no—
316C22×C4⋊C4—731no—
317C22×C4⋊C4—731no—
318C22×C22⋊C4—731no—
319C22×C22⋊C4—731no—
320C22×C22⋊C4—731no—
321#407—734no—
322#407—1462no—
323C2×C22⋊C8—22no—
324C2×C22⋊C8—21no—
325C2×C22⋊C8—732no—
326C2×C22⋊C8—1461no—
327C2×C22⋊C8C2×D7312yes—
328C2×C22⋊C8C2×D7314yescomplemented
329#324—738no—
330#323—738no—
331C25—731no—
332C25C4×D7311yescharacteristic
333C22×D4—2924no—
334C22×D4—734no—
335C22×D4—1462no—
336C22×D4—1461no—
337C22×D4—1461no—
338C23×C4—44no—
339C23×C4—731no—
340C23×C4—731no—
341C23×C4—731no—
342C23×C4C4×D7311yescharacteristic
343C23×C4C4×D7312yes—
344C23×C4D29211yescharacteristic
345C23×C4D29211yescharacteristic
346C23×C4D29212yes—
347C23×C4C22⋊D7311yescharacteristic
348C23×C4C22⋊D7312yes—
349C23×C4C22⋊D7311yescharacteristic
350C23×C4C22×D7311yescharacteristic
351C23×C4C22×D7312yes—
352C2×M16—2924no—
353C2×M16—1464no—
354C22×C8—22no—
355C22×C8—24no—
356C22×C8—44no—
357C22×C8—1462no—
358#123—1464no—
359#123—1464no—
360#123—1464no—
361#123—1464no—
362#123—1464no—
363#123—1464no—
364#123—1464no—
365#123—1464no—
366C2×C4⋊C4—734no—
367C2×C4⋊C4—732no—
368C2×C4⋊C4—732no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C22⋊C4—734no—
377C2×C22⋊C4—732no—
378C2×C22⋊C4—734no—
379C2×C22⋊C4—732no—
380C2×C22⋊C4—1462no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1462no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C22⋊C8—22no—
391C22⋊C8—24no—
392C22⋊C8—734no—
393C22⋊C8—1462no—
394C22⋊C8—1464no—
395C22⋊C8C22×D7314yes—
396C24—24no—
397C24—24no—
398C24—42no—
399C24—2922no—
400C24—2922no—
401C24—732no—
402C24—1461no—
403C24—1461no—
404C24—1464no—
405C24#1192400511yescharacteristic
406C24#1192400512yes—
407C24#1192400511yescharacteristic
408C24C2×C4×D7312yes—
409C24C2×C4×D7311yescharacteristic
410C2×D4—2928no—
411C2×D4—2924no—
412C2×D4—2924no—
413C2×D4—2928no—
414C2×D4—1464no—
415C2×D4—1464no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—2922no—
427C22×C4—2922no—
428C22×C4—732no—
429C22×C4—732no—
430C22×C4—732no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4—1464no—
438C22×C4#1192400512yes—
439C22×C4#1192400512yes—
440C22×C4#1192400511yescharacteristic
441C22×C4#1192400511yescharacteristic
442C22×C4#1192400512yes—
443C22×C4#1192400512yes—
444C22×C4#1192400512yes—
445C22×C4C2×C4×D7312yes—
446C22×C4C2×C4×D7311yescharacteristic
447C22×C4C2×C22⋊D7312yes—
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C2×C22⋊D7311yescharacteristic
450C22×C4C2×C22⋊D7312yes—
451C22×C4C2×D29212yes—
452C22×C4C2×D29211yescharacteristic
453C22×C4C2×D29211yescharacteristic
454C22×C4C2×D29212yes—
455C22×C4C23×D7311yescharacteristic
456M16—2928no—
457C2×C8—24no—
458C2×C8—44no—
459C2×C8—48no—
460C2×C8—2924no—
461C2×C8—1464no—
462C4⋊C4—1464no—
463C4⋊C4—1464no—
464C4⋊C4—1462no—
465C4⋊C4—1462no—
466C4⋊C4—1462no—
467C4⋊C4—1462no—
468C22⋊C4—1464no—
469C22⋊C4—1464no—
470C22⋊C4—1464no—
471C22⋊C4—1464no—
472C22⋊C4—1462no—
473C22⋊C4—1462no—
474C22⋊C4—1462no—
475C22⋊C4—1462no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—416no—
490C23—2922no—
491C23—2922no—
492C23—2922no—
493C23—2922no—
494C23—2922no—
495C23—2922no—
496C23—2922no—
497C23—2922no—
498C23—5848no—
499C23—1462no—
500C23—1464no—
501C23#1455650514yes—
502C23C2×#1192400512yes—
503C23C2×#1192400511yescharacteristic
504C23C2×#1192400512yes—
505C23C2×#1192400511yescharacteristic
506C23#1455659714yes—
507C23C22×C4×D7311yescharacteristic
508D4—2928no—
509D4—2928no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—1462no—
537C2×C4—1462no—
538C2×C4—1462no—
539C2×C4—1464no—
540C2×C4C2×#1192400511yescharacteristic
541C2×C4C2×#1192400511yescharacteristic
542C2×C4C2×#1192400512yes—
543C2×C4C2×#1192400512yes—
544C2×C4C22×C22⋊D7311yescharacteristic
545C2×C4C22×D29211yescharacteristic
546C8—44no—
547C8—2924no—
548C22—22no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—28no—
555C22—28no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—2922no—
563C22—2922no—
564C22—2922no—
565C22—5844no—
566C22—5844no—
567C22—5844no—
568C22#1798855312yes—
569C22#1798855312yes—
570C22C2×#1455650514yes—
571C22C2×#1455650512yes—
572C22#1798856912yes—
573C22C22×#1192400511yescharacteristic
574C22C2×#1455659712yes—
575C4—21no—
576C4—21no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—2922no—
582C4—2922no—
583C4—2922no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—5842no—
589C2#2260827714yesdirect product factor
590C2C2×#1798855311yescharacteristic
591C2C22×#1455650511yescharacteristic
592C2C2×#1798856911yescharacteristic
subgroupG/H
centerC23C2×#11924005
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC22×C22⋊C8×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC23×C22⋊D73C4
cosocleC22×C4×C73C24
hypercenterC22×C22⋊C8D73
hypocenterC73C2×#1468
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#17988551C2semidirect
3. #22608277C2direct (N × H)
4. C2×#17988312C2semidirect
5. C2×#17988310C2semidirect
6. C22×C22⋊C8×C73C2semidirect
23. #17988551C22semidirect
26. #17988312C22semidirect
27. #17988310C22semidirect
33. C2×C22⋊C8×C73C22semidirect
292. C2×C73#1468semidirect
299. C22×C22⋊C8D73semidirect
303. C73C2×#1468semidirect
328. C2×C22⋊C8C2×D73semidirect
589. C2#22608277direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565742
C2×#179885512
#226082772
C2×#179883122
C2×#179883102
C22×C22⋊C8×C732
C2×#146873