Finite Group Databasebeta

← C2×#22623943 #33994255

Subgroups — C2×#22623943 #33994255

Order 18688.

7118 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623943C2132yesmaximal, direct product factor
2C2×Q8×Dic146C211yesmaximal, characteristic, complemented
3C2×#17989311C213yesmaximal, complemented
4C2×#17989308C216yesmaximal, complemented
5C2×#17989302C213yesmaximal, complemented
6C2×#17989248C212yesmaximal, complemented
7C2×#17989210C216yesmaximal, complemented
8C2×#17989142C213yesmaximal, complemented
9C2×#17989119C213yesmaximal, complemented
10C2×#17989065C212yesmaximal, complemented
11C2×C4×#11924029C211yesmaximal, characteristic, complemented
12C2×D4×Q8×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2211yescharacteristic
14Q8×Dic146C22116yescomplemented
15#17989311C22148yescomplemented
16#17989308C22196yescomplemented
17#17989302C22148yescomplemented
18#17989248C22132yescomplemented
19#17989210C22196yescomplemented
20#17989142C22148yescomplemented
21C2×#14556591C2213yes—
22C2×#14556590C2216yes—
23#17989119C22148yescomplemented
24C2×#14556586C22112yes—
25C2×#14556584C2216yes—
26C2×#14556582C2213yes—
27C2×#14556580C2216yes—
28C2×#14556578C2211yescharacteristic
29C2×#14556577C22112yes—
30C2×#14556573C2216yes—
31C2×#14556571C2216yes—
32C2×#14556571C2213yes—
33C2×#14556567C2216yes—
34C2×#14556565C2216yes—
35C2×#14556562C2213yes—
36#17989065C22132yescomplemented
37C4×#11924029C22116yescomplemented
38C2×Q8×Dic73C2212yes—
39C2×Q8×Dic73C2216yes—
40C2×Q8×Dic73C2211yescharacteristic
41C2×Q8×Dic73C2216yes—
42C2×C4×Dic146C2213yes—
43C2×#14556556C2216yes—
44C2×C4×D292C2213yes—
45C2×#14556553C2213yes—
46C2×C4×C22⋊D73C2216yes—
47C2×#14556550C2216yes—
48C2×#14556550C2216yes—
49C2×#14556549C2213yes—
50C2×#14556549C2216yes—
51C2×C42×D73C2213yes—
52C23×Q8×C73C2212yes—
53D4×Q8×C73C22116yescomplemented
54C2×C4⋊Q8×C73C2213yes—
55C2×C22⋊Q8×C73C2216yes—
56C2×C4×Q8×C73C2211yescharacteristic
57C2×C4×D4×C73C2213yes—
58C2×#11924029—28no—
59C2×#11924029C2318yes—
60C22×Dic146C2311yescharacteristic
61C22×Dic146C2316yes—
62C22×D292C2313yes—
63C22×C22⋊D73C2316yes—
64C22×C4×D73C2313yes—
65C22×C4×D73C2313yes—
66C22×C4×D73C2313yes—
67#14556591C23124yes—
68#14556590C23148yes—
69#14556586C23196yes—
70#14556584C23148yes—
71#14556582C23124yes—
72#14556580C23148yes—
73#14556578C2318yes—
74#14556577C23196yes—
75#14556573C23148yes—
76#14556571C23148yes—
77#14556571C23124yes—
78#14556567C23148yes—
79#14556565C23148yes—
80#14556562C23124yes—
81Q8×Dic73C23148yes—
82Q8×Dic73C23116yes—
83Q8×Dic73C23148yes—
84Q8×Dic73C2318yes—
85C4×Dic146C23124yes—
86#14556556C23148yes—
87C4×D292C23124yes—
88#14556553C23124yes—
89C4×C22⋊D73C23148yes—
90#14556550C23148yes—
91#14556550C23148yes—
92#14556549C23148yes—
93#14556549C23124yes—
94C2×C4⋊Dic73C2316yes—
95C2×C4⋊Dic73C2313yes—
96C2×C4⋊Dic73C2313yes—
97C2×C4⋊Dic73C2316yes—
98C2×C4⋊Dic73C2316yes—
99C2×#11924007C2316yes—
100C2×#11924007C2316yes—
101C2×C22⋊Dic73C2316yes—
102C2×C22⋊Dic73C2312yes—
103C2×C22⋊Dic73C2312yes—
104C2×#11924005C2316yes—
105C2×#11924005C23112yes—
106C2×#11924005C2316yes—
107C2×C4×Dic73C2316yes—
108C2×C4×Dic73C2311yescharacteristic
109C2×C4×Dic73C2313yes—
110C2×C4×Dic73C2313yes—
111C2×C4×Dic73C23112yes—
112C42×D73C23124yes—
113C22×Q8×C73—216no—
114C22×Q8×C73C23116yes—
115C22×Q8×C73C2316yes—
116C22×Q8×C73C2311yescharacteristic
117C22×D4×C73C2311yescharacteristic
118C23×C4×C73C2316yes—
119C4⋊Q8×C73C23124yes—
120C22⋊Q8×C73C23148yes—
121C4×Q8×C73C2318yes—
122C4×D4×C73C23124yes—
123C2×C4⋊C4×C73C2313yes—
124C2×C4⋊C4×C73C2313yes—
125C2×C4⋊C4×C73C2316yes—
126C2×C22⋊C4×C73C2316yes—
127C2×C42×C73C2313yes—
128C23×D73C2413yes—
129#11924029—232no—
130C2×Dic146—24no—
131C2×Dic146—412no—
132C2×Dic146C2414yes—
133C2×Dic146C24124yes—
134C2×D292—212no—
135C2×D292C24112yes—
136C2×C22⋊D73—412no—
137C2×C22⋊D73C24124yes—
138C22×Dic73C2413yes—
139C22×Dic73C2411yescharacteristic
140C22×Dic73C2411yescharacteristic
141C22×Dic73C2412yes—
142C22×Dic73C2416yes—
143C2×C4×D73—212no—
144C2×C4×D73—212no—
145C2×C4×D73—46no—
146C2×C4×D73C24112yes—
147C2×C4×D73C24112yes—
148C2×C4×D73C24112yes—
149C4⋊Dic73C24112yes—
150C4⋊Dic73C24112yes—
151C4⋊Dic73C24124yes—
152C4⋊Dic73C24124yes—
153C4⋊Dic73C24124yes—
154#11924007C24124yes—
155#11924007C24124yes—
156C22⋊Dic73C24124yes—
157C22⋊Dic73C2418yes—
158C22⋊Dic73C2418yes—
159#11924005C24124yes—
160#11924005C24148yes—
161#11924005C24124yes—
162C4×Dic73C2414yes—
163C4×Dic73C24112yes—
164C4×Dic73C24112yes—
165C4×Dic73C24148yes—
166C4×Dic73C24124yes—
167C24×C73C2412yes—
168C2×Q8×C73—264no—
169C2×Q8×C73—224no—
170C2×Q8×C73#4118yes—
171C2×Q8×C73C24124yes—
172C2×Q8×C73C2414yes—
173C2×D4×C73—24no—
174C2×D4×C73C2414yes—
175C22×C4×C73—224no—
176C22×C4×C73C2413yes—
177C22×C4×C73C2416yes—
178C22×C4×C73C24124yes—
179C22×C4×C73C2413yes—
180C22×C4×C73C2411yescharacteristic
181C4⋊C4×C73C24112yes—
182C4⋊C4×C73C24112yes—
183C4⋊C4×C73C24124yes—
184C22⋊C4×C73C24124yes—
185C42×C73C24112yes—
186C23×C73—24no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73C2514yes—
190C23×C73C2511yescharacteristic
191C22×D73—26no—
192C22×D73—26no—
193C22×D73—26no—
194C22×D73C2516yes—
195Q8×C73—248no—
196Q8×C73C2×#41116yes—
197D4×C73—28no—
198C2×C4×C73—212no—
199C2×C4×C73—26no—
200C2×C4×C73—248no—
201C2×C4×C73C2×#41112yes—
202C2×C4×C732−1+414yes—
203C2×C4×C73C2516yes—
204C2×C4×C73C25112yes—
205C2×C4×C73C2516yes—
206C2×C4×C73C2512yes—
207C22⋊D73—424no—
208C2×Dic73—22no—
209C2×Dic73—26no—
210C2×Dic73—24no—
211C2×Dic73—22no—
212C2×Dic73—212no—
213C2×Dic73C2516yes—
214C2×Dic73C2512yes—
215C2×Dic73C2512yes—
216C2×Dic73C2514yes—
217C2×Dic73C25112yes—
218D292—224no—
219C4×D73—224no—
220C4×D73—224no—
221C4×D73—412no—
222Dic146—28no—
223Dic146—424no—
224C22×C73—24no—
225C22×C73—24no—
226C22×C73—24no—
227C22×C73—48no—
228C22×C73C22×#4112yes—
229C22×C73C2×2−1+412yes—
230C22×C73#58512yes—
231C22×C73C2611yescharacteristic
232C2×D73—26no—
233C2×D73—26no—
234C2×D73—26no—
235C2×D73—412no—
236C4×C73—212no—
237C4×C73—26no—
238C4×C73C22×#41112yes—
239C4×C73C2×2−1+414yes—
240Dic73—22no—
241Dic73—24no—
242Dic73—26no—
243Dic73—212no—
244Dic73—22no—
245C2×#3518—731nomaximal, Sylow
246C2×C73—44no—
247C2×C73#351814yescomplemented
248C2×C73C23×#4111yescharacteristic
249C2×C73C22×2−1+411yescharacteristic
250C2×C73C2×#58511yescharacteristic
251D73—46no—
252#3518—7332no—
253C2×Q82—731no—
254C2×#556—733no—
255C2×#556—736no—
256C2×#555—733no—
257C2×D4×Q8D7311yescharacteristic, complemented
258C2×#548—732no—
259C2×#542—736no—
260C2×#533—733no—
261C2×#529—733no—
262C2×#519—732no—
263C2×C4×#41—731no—
264C73C2×#351811yescharacteristic, complemented, Sylow
265C22×#41—731no—
266C23×Q8C2×D7312yes—
267Q82—7316no—
268#556—7348no—
269#556—7396no—
270#555—7348no—
271D4×Q8C2×D73116yescomplemented
272#548—7332no—
273#542—7396no—
274#533—7348no—
275C2×C4⋊Q8—736no—
276C2×C4⋊Q8—733no—
277C2×C4⋊Q8C2×D7313yes—
278#529—7348no—
279C2×#126—7312no—
280C2×#126—736no—
281C2×#125—733no—
282C2×#124—736no—
283C2×#124—731no—
284C2×#124—7312no—
285C2×#123—736no—
286C2×C22⋊Q8—736no—
287C2×C22⋊Q8—733no—
288C2×C22⋊Q8—736no—
289C2×C22⋊Q8C2×D7316yes—
290C2×#121—733no—
291C2×#121—736no—
292#519—7332no—
293C4×#41—7316no—
294C2×C4×Q8—731no—
295C2×C4×Q8—736no—
296C2×C4×Q8—732no—
297C2×C4×Q8—736no—
298C2×C4×Q8—736no—
299C2×C4×Q8—733no—
300C2×C4×Q8C2×D7311yescharacteristic
301C2×C4×D4—736no—
302C2×C4×D4—733no—
303C2×C4×D4—733no—
304C2×C4×D4C2×D7313yes—
305C2×#117—736no—
306C2×#117—736no—
307C2×#117—733no—
308C2×#117—736no—
309C22×C42—733no—
310C2×#41—738no—
311C2×#41—1468no—
312C22×Q8—216no—
313C22×Q8—736no—
314C22×Q8—731no—
315C22×Q8C22×D7316yes—
316C22×Q8C22×D73116yes—
317C22×Q8C22×D7311yescharacteristic
318C22×D4—736no—
319C22×D4—733no—
320C22×D4C22×D7311yescharacteristic
321C23×C4—733no—
322C23×C4—733no—
323C23×C4—733no—
324C23×C4C22×D7316yes—
325C4⋊Q8—7348no—
326C4⋊Q8—7324no—
327C4⋊Q8C22×D73124yes—
328#126—7396no—
329#126—7348no—
330#125—7324no—
331#124—7348no—
332#124—738no—
333#124—7396no—
334#123—7348no—
335C22⋊Q8—7348no—
336C22⋊Q8—7324no—
337C22⋊Q8—7348no—
338C22⋊Q8C22×D73148yes—
339#121—7324no—
340#121—7348no—
341C4×Q8—738no—
342C4×Q8—7348no—
343C4×Q8—7316no—
344C4×Q8—7348no—
345C4×Q8—7348no—
346C4×Q8—7324no—
347C4×Q8C22×D7318yes—
348C4×D4—7348no—
349C4×D4—7324no—
350C4×D4—7324no—
351C4×D4C22×D73124yes—
352#117—7348no—
353#117—7348no—
354#117—7324no—
355#117—7348no—
356C2×C4⋊C4—733no—
357C2×C4⋊C4—733no—
358C2×C4⋊C4—736no—
359C2×C4⋊C4—736no—
360C2×C4⋊C4—736no—
361C2×C4⋊C4—736no—
362C2×C4⋊C4—736no—
363C2×C4⋊C4C22×D7313yes—
364C2×C4⋊C4C22×D7313yes—
365C2×C4⋊C4C22×D7316yes—
366C2×C22⋊C4—736no—
367C2×C22⋊C4—732no—
368C2×C22⋊C4—732no—
369C2×C22⋊C4—736no—
370C2×C22⋊C4—7312no—
371C2×C22⋊C4—736no—
372C2×C22⋊C4C22×D7316yes—
373C2×C42—733no—
374C2×C42—7312no—
375C2×C42—733no—
376C2×C42—736no—
377C2×C42—731no—
378C2×C42—7324no—
379C2×C42C22×D7313yes—
380C24—733no—
381C24C23×D7312yes—
382#41—14632no—
383C2×Q8—264no—
384C2×Q8—224no—
385C2×Q8—29212no—
386C2×Q8—7324no—
387C2×Q8—734no—
388C2×Q8—1464no—
389C2×Q8#1192402718yes—
390C2×Q8C23×D73124yes—
391C2×Q8C23×D7314yes—
392C2×D4—24no—
393C2×D4—29212no—
394C2×D4—7324no—
395C2×D4—7312no—
396C2×D4—14612no—
397C2×D4C23×D7314yes—
398C22×C4—224no—
399C22×C4—2926no—
400C22×C4—7312no—
401C22×C4—733no—
402C22×C4—731no—
403C22×C4—731no—
404C22×C4—732no—
405C22×C4—736no—
406C22×C4—7312no—
407C22×C4—7312no—
408C22×C4—14612no—
409C22×C4—14612no—
410C22×C4C23×D73124yes—
411C22×C4C23×D7316yes—
412C22×C4C23×D7313yes—
413C22×C4C23×D7313yes—
414C22×C4C23×D7311yescharacteristic
415C4⋊C4—7324no—
416C4⋊C4—7324no—
417C4⋊C4—7312no—
418C4⋊C4—7312no—
419C4⋊C4—7324no—
420C4⋊C4—7324no—
421C4⋊C4—7324no—
422C4⋊C4C23×D73112yes—
423C4⋊C4C23×D73112yes—
424C4⋊C4C23×D73124yes—
425C22⋊C4—7324no—
426C22⋊C4—738no—
427C22⋊C4—738no—
428C22⋊C4—7324no—
429C22⋊C4—7348no—
430C22⋊C4—7324no—
431C22⋊C4C23×D73124yes—
432C42—734no—
433C42—7312no—
434C42—7312no—
435C42—7348no—
436C42—7324no—
437C42C23×D73112yes—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—736no—
442C23—1466no—
443C23—1466no—
444C23—1466no—
445C23C24×D7314yes—
446C23C24×D7311yescharacteristic
447Q8—248no—
448Q8—29224no—
449Q8—1468no—
450Q8C2×#11924027116yes—
451D4—28no—
452D4—29224no—
453D4—14624no—
454C2×C4—212no—
455C2×C4—26no—
456C2×C4—248no—
457C2×C4—29212no—
458C2×C4—736no—
459C2×C4—732no—
460C2×C4—732no—
461C2×C4—734no—
462C2×C4—7312no—
463C2×C4—1462no—
464C2×C4—1466no—
465C2×C4—1464no—
466C2×C4—1462no—
467C2×C4—14612no—
468C2×C4—14624no—
469C2×C4—14624no—
470C2×C4C2×#11924027112yes—
471C2×C4#1455663914yes—
472C2×C4C24×D73112yes—
473C2×C4C24×D7316yes—
474C2×C4C24×D7316yes—
475C2×C4C24×D7312yes—
476C22—24no—
477C22—24no—
478C22—24no—
479C22—48no—
480C22—29212no—
481C22—1466no—
482C22—1466no—
483C22—1466no—
484C22C22×#1192402712yes—
485C22C2×#1455663912yes—
486C22#1798942412yes—
487C22C25×D7311yescharacteristic
488C4—212no—
489C4—26no—
490C4—1462no—
491C4—1466no—
492C4—1464no—
493C4—1462no—
494C4—14612no—
495C4C22×#11924027112yes—
496C4C2×#1455663914yes—
497C2—44no—
498C2—2926no—
499C2#2262394314yesdirect product factor
500C2C23×#1192402711yescharacteristic
501C2C22×#1455663911yescharacteristic
502C2C2×#1798942411yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×D4×Q8×C73C2
socleC23×C73C25
hypercenterC2×D4×Q8D73
hypocenterC73C2×#3518
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #22623943C2direct (N × H)
2. C2×Q8×Dic146C2semidirect
3. C2×#17989311C2semidirect
4. C2×#17989308C2semidirect
5. C2×#17989302C2semidirect
6. C2×#17989248C2semidirect
7. C2×#17989210C2semidirect
8. C2×#17989142C2semidirect
9. C2×#17989119C2semidirect
10. C2×#17989065C2semidirect
11. C2×C4×#11924029C2semidirect
12. C2×D4×Q8×C73C2semidirect
14. Q8×Dic146C22semidirect
15. #17989311C22semidirect
16. #17989308C22semidirect
17. #17989302C22semidirect
18. #17989248C22semidirect
19. #17989210C22semidirect
20. #17989142C22semidirect
23. #17989119C22semidirect
36. #17989065C22semidirect
37. C4×#11924029C22semidirect
53. D4×Q8×C73C22semidirect
247. C2×C73#3518semidirect
257. C2×D4×Q8D73semidirect
264. C73C2×#3518semidirect
271. D4×Q8C2×D73semidirect
499. C2#22623943direct (N × H)

Maximal subgroups

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

subgroupindex
#226239432
C2×Q8×Dic1462
C2×#179893112
C2×#179893082
C2×#179893022
C2×#179892482
C2×#179892102
C2×#179891422
C2×#179891192
C2×#179890652
C2×C4×#119240292
C2×D4×Q8×C732
C2×#351873