Finite Group Databasebeta

← C2×#22624011 #33994186

Subgroups — C2×#22624011 #33994186

Order 18688.

7758 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624011C2132yesmaximal, direct product factor
2C2×#17989323C213yesmaximal, complemented
3C2×Q8×Dic146C211yesmaximal, characteristic, complemented
4C2×#17989294C213yesmaximal, complemented
5C2×#17989244C212yesmaximal, complemented
6C2×#17989237C216yesmaximal, complemented
7C2×#17989204C216yesmaximal, complemented
8C2×#17989135C213yesmaximal, complemented
9C2×#17989119C213yesmaximal, complemented
10C2×#17989070C212yesmaximal, complemented
11C2×C4×Q8×D73C211yesmaximal, characteristic, complemented
12C2×#550×C73C211yesmaximal, characteristic, complemented
13C22×#11924027C2216yes—
14C22×Q8×D73C2211yescharacteristic
15#17989323C22148yescomplemented
16Q8×Dic146C22116yescomplemented
17#17989294C22148yescomplemented
18#17989244C22132yescomplemented
19#17989237C22196yescomplemented
20#17989204C22196yescomplemented
21#17989135C22148yescomplemented
22C2×#14556591C2213yes—
23C2×#14556590C2216yes—
24#17989119C22148yescomplemented
25C2×#14556584C2216yes—
26C2×#14556582C2213yes—
27C2×#14556578C2213yes—
28C2×#14556578C2216yes—
29C2×#14556578C2216yes—
30C2×#14556573C2216yes—
31C2×#14556572C22112yes—
32C2×#14556571C2211yescharacteristic
33C2×#14556570C2216yes—
34C2×#14556568C22112yes—
35C2×#14556562C2216yes—
36C2×#14556562C2213yes—
37#17989070C22132yescomplemented
38C2×Q8×Dic73C2212yes—
39C2×Q8×Dic73C2211yescharacteristic
40C2×C4×Dic146C2213yes—
41C2×C4×Dic146C2213yes—
42C2×#14556556C2213yes—
43C4×Q8×D73C22116yescomplemented
44C2×D4×Dic73C2216yes—
45C2×D4×Dic73C2216yes—
46C2×D4×Dic73C2216yes—
47C2×C4×C22⋊D73C2216yes—
48C2×#14556550C2216yes—
49C22×C4⋊Dic73C2216yes—
50C2×C4⋊C4×D73C2213yes—
51C2×C42×D73C2213yes—
52C22×#41×C73C2212yes—
53#550×C73C22116yescomplemented
54C2×#127×C73C2213yes—
55C2×#121×C73C2216yes—
56C2×C4×Q8×C73C2211yescharacteristic
57C2×C4×D4×C73C2213yes—
58C2×#11924027—424no—
59C2×#11924027C23148yes—
60C22×Dic146C2313yes—
61C22×Dic146C2311yescharacteristic
62C22×Dic146C2316yes—
63C2×Q8×D73—28no—
64C2×Q8×D73C2318yes—
65C22×C22⋊D73C2316yes—
66C23×Dic73C23112yes—
67C22×C4×D73C2313yes—
68C22×C4×D73C2313yes—
69C22×C4×D73C2311yescharacteristic
70#14556591C23124yes—
71#14556590C23148yes—
72#14556584C23148yes—
73#14556582C23124yes—
74#14556578C23148yes—
75#14556578C23148yes—
76#14556578C23124yes—
77#14556573C23148yes—
78#14556572C23196yes—
79#14556571C2318yes—
80#14556570C23148yes—
81#14556568C23196yes—
82#14556562C23124yes—
83#14556562C23148yes—
84Q8×Dic73C23116yes—
85Q8×Dic73C2318yes—
86C4×Dic146C23124yes—
87C4×Dic146C23124yes—
88#14556556C23124yes—
89D4×Dic73C23148yes—
90D4×Dic73C23148yes—
91D4×Dic73C23148yes—
92C4×C22⋊D73C23148yes—
93#14556550C23148yes—
94C2×C4⋊Dic73C2316yes—
95C2×C4⋊Dic73C2313yes—
96C2×C4⋊Dic73C23148yes—
97C2×C4⋊Dic73C2313yes—
98C2×C4⋊Dic73C2316yes—
99C2×C4⋊Dic73C2312yes—
100C2×#11924007C2316yes—
101C2×#11924007C2316yes—
102C2×#11924007C2316yes—
103C4⋊C4×D73C23124yes—
104C2×C22⋊Dic73C2316yes—
105C2×C22⋊Dic73C2316yes—
106C2×C22⋊Dic73C2316yes—
107C2×C22⋊Dic73C23112yes—
108C2×#11924005C2312yes—
109C2×C4×Dic73C2316yes—
110C2×C4×Dic73C2313yes—
111C2×C4×Dic73C2313yes—
112C2×C4×Dic73C2313yes—
113C42×D73C23124yes—
114C2×#41×C73—216no—
115C2×#41×C73C23116yes—
116C22×Q8×C73C2311yescharacteristic
117C22×D4×C73C2313yes—
118C22×D4×C73C2316yes—
119C22×D4×C73C2316yes—
120C23×C4×C73C2316yes—
121#127×C73C23124yes—
122#121×C73C23148yes—
123C4×Q8×C73C2318yes—
124C4×D4×C73C23124yes—
125C2×C4⋊C4×C73C2313yes—
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C22⋊C4×C73C2316yes—
128C2×C42×C73C2313yes—
129C23×D73C2411yescharacteristic
130#11924027—496no—
131C2×Dic146—212no—
132C2×Dic146—24no—
133C2×Dic146—412no—
134C2×Dic146C24112yes—
135C2×Dic146C2414yes—
136C2×Dic146C24124yes—
137Q8×D73—232no—
138C2×C22⋊D73—224no—
139C2×C22⋊D73C24124yes—
140C22×Dic73—424no—
141C22×Dic73C2411yescharacteristic
142C22×Dic73C2412yes—
143C22×Dic73C2413yes—
144C22×Dic73C2413yes—
145C22×Dic73C2416yes—
146C22×Dic73C24148yes—
147C2×C4×D73—212no—
148C2×C4×D73—212no—
149C2×C4×D73—24no—
150C2×C4×D73C2414yes—
151C2×C4×D73C24112yes—
152C2×C4×D73C24112yes—
153C4⋊Dic73C24112yes—
154C4⋊Dic73C24124yes—
155C4⋊Dic73C24124yes—
156C4⋊Dic73C2418yes—
157C4⋊Dic73C24112yes—
158#11924007C24124yes—
159#11924007C24124yes—
160#11924007C24124yes—
161C22⋊Dic73C24124yes—
162C22⋊Dic73C24124yes—
163C22⋊Dic73C24124yes—
164C22⋊Dic73C24148yes—
165#11924005C2418yes—
166C4×Dic73C24112yes—
167C4×Dic73C24112yes—
168C4×Dic73C24112yes—
169C4×Dic73C24124yes—
170C24×C73C2416yes—
171#41×C73—264no—
172C2×Q8×C73#4118yescomplemented
173C2×Q8×C73C2414yes—
174C2×D4×C73—224no—
175C2×D4×C73—212no—
176C2×D4×C73—412no—
177C2×D4×C73C24112yes—
178C2×D4×C73C24124yes—
179C2×D4×C73C24124yes—
180C22×C4×C73—224no—
181C22×C4×C73C2413yes—
182C22×C4×C73C2412yes—
183C22×C4×C73C24124yes—
184C22×C4×C73C2413yes—
185C22×C4×C73C2411yescharacteristic
186C4⋊C4×C73C2414yes—
187C4⋊C4×C73C24112yes—
188C22⋊C4×C73C24124yes—
189C42×C73C24112yes—
190C23×C73—212no—
191C23×C73—212no—
192C23×C73—212no—
193C23×C73C25112yes—
194C23×C73C2511yescharacteristic
195C22×D73—22no—
196C22×D73—22no—
197C22×D73H(5,𝔽2)14yes—
198C22×D73C2512yes—
199Q8×C73C2×#41116yescomplemented
200D4×C73—248no—
201D4×C73—224no—
202D4×C73—424no—
203C2×C4×C73—24no—
204C2×C4×C73—26no—
205C2×C4×C73—248no—
206C2×C4×C73C2×#41112yes—
207C2×C4×C732−1+414yes—
208C2×C4×C73C2516yes—
209C2×C4×C73C2514yes—
210C2×C4×C73C2516yes—
211C2×C4×C73C2512yes—
212C22⋊D73—248no—
213C2×Dic73—26no—
214C2×Dic73—26no—
215C2×Dic73—212no—
216C2×Dic73—22no—
217C2×Dic73—24no—
218C2×Dic73—448no—
219C2×Dic73C2512yes—
220C2×Dic73C2514yes—
221C2×Dic73C2516yes—
222C2×Dic73C2516yes—
223C2×Dic73C25112yes—
224C4×D73—224no—
225C4×D73—224no—
226C4×D73—28no—
227Dic146—224no—
228Dic146—28no—
229Dic146—424no—
230C22×C73—212no—
231C22×C73—212no—
232C22×C73—212no—
233C22×C73—424no—
234C22×C73C22×#4112yes—
235C22×C73C2×H(5,𝔽2)12yes—
236C22×C73C2×2−1+412yes—
237C22×C73C2611yescharacteristic
238C2×D73—22no—
239C2×D73—22no—
240C2×D73—28no—
241C2×D73C2×H(5,𝔽2)14yes—
242C4×C73—24no—
243C4×C73—26no—
244C4×C73C22×#41112yes—
245C4×C73C2×2−1+414yes—
246Dic73—26no—
247Dic73—212no—
248Dic73—26no—
249Dic73—24no—
250Dic73—22no—
251C2×#3522—731nomaximal, Sylow
252C2×C73—412no—
253C2×C73#352214yescomplemented
254C2×C73C23×#4111yescharacteristic
255C2×C73C22×H(5,𝔽2)11yescharacteristic
256C2×C73C22×2−1+411yescharacteristic
257D73—24no—
258#3522—7332no—
259C2×#559—733no—
260C2×Q82—731no—
261C2×#554—733no—
262C2×#550D7311yescharacteristic, complemented
263C2×#548—732no—
264C2×#547—736no—
265C2×#542—736no—
266C2×#532—733no—
267C2×#529—733no—
268C2×#520—732no—
269C22×C4×Q8—731no—
270C73C2×#352211yescharacteristic, complemented, Sylow
271C22×#41—736no—
272C22×#41C2×D7312yes—
273C23×Q8—731no—
274#559—7348no—
275Q82—7316no—
276#554—7348no—
277#550C2×D73116yescomplemented
278#548—7332no—
279#547—7396no—
280#542—7396no—
281#532—7348no—
282C2×C4⋊Q8—736no—
283C2×C4⋊Q8—733no—
284C2×#127C2×D7313yes—
285#529—7348no—
286C2×#126—736no—
287C2×#125—733no—
288C2×#124—733no—
289C2×#124—736no—
290C2×#124—736no—
291C2×#123—7312no—
292C2×#123—736no—
293C2×C22⋊Q8—736no—
294C2×C22⋊Q8—731no—
295C2×C22⋊Q8—7312no—
296C2×#121—733no—
297C2×#121—736no—
298C2×#121C2×D7316yes—
299#520—7332no—
300C2×C4×Q8—731no—
301C2×C4×Q8—732no—
302C2×C4×Q8—7316no—
303C2×C4×Q8—733no—
304C2×C4×Q8—733no—
305C2×C4×Q8—733no—
306C2×C4×Q8C2×D7311yescharacteristic
307C2×C4×D4—736no—
308C2×C4×D4—736no—
309C2×C4×D4—736no—
310C2×C4×D4—736no—
311C2×C4×D4C2×D7313yes—
312C2×#117—736no—
313C22×C4⋊C4—736no—
314C22×C4⋊C4—733no—
315C22×C42—733no—
316C2×#41—216no—
317C2×#41—29224no—
318C2×#41—7348no—
319C2×#41C22×D73116yes—
320C22×Q8—738no—
321C22×Q8—736no—
322C22×Q8—731no—
323C22×Q8—733no—
324C22×Q8—1468no—
325C22×Q8C22×D7311yescharacteristic
326C22×D4—736no—
327C22×D4C22×D7313yes—
328C22×D4C22×D7316yes—
329C22×D4C22×D7316yes—
330C23×C4—7312no—
331C23×C4—733no—
332C23×C4—733no—
333C23×C4—731no—
334C23×C4C22×D7316yes—
335C4⋊Q8—7348no—
336C4⋊Q8—7324no—
337#127C22×D73124yes—
338#126—7348no—
339#125—7324no—
340#124—7324no—
341#124—7348no—
342#124—7348no—
343#123—7396no—
344#123—7348no—
345C22⋊Q8—7348no—
346C22⋊Q8—738no—
347C22⋊Q8—7396no—
348#121—7324no—
349#121—7348no—
350#121C22×D73148yes—
351C4×Q8—738no—
352C4×Q8—7316no—
353C4×Q8—7324no—
354C4×Q8—7324no—
355C4×Q8—7324no—
356C4×Q8C22×D7318yes—
357C4×D4—7348no—
358C4×D4—7348no—
359C4×D4—7348no—
360C4×D4—7348no—
361C4×D4C22×D73124yes—
362#117—7348no—
363C2×C4⋊C4—733no—
364C2×C4⋊C4—736no—
365C2×C4⋊C4—736no—
366C2×C4⋊C4—736no—
367C2×C4⋊C4—7348no—
368C2×C4⋊C4—732no—
369C2×C4⋊C4—733no—
370C2×C4⋊C4—736no—
371C2×C4⋊C4—736no—
372C2×C4⋊C4—7324no—
373C2×C4⋊C4C22×D7313yes—
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C22⋊C4—736no—
376C2×C22⋊C4—736no—
377C2×C22⋊C4—7312no—
378C2×C22⋊C4—736no—
379C2×C22⋊C4—732no—
380C2×C22⋊C4C22×D7316yes—
381C2×C42—733no—
382C2×C42—736no—
383C2×C42—733no—
384C2×C42—733no—
385C2×C42—7324no—
386C2×C42C22×D7313yes—
387C24—731no—
388C24C23×D7316yes—
389#41—264no—
390#41—29296no—
391C2×Q8—29212no—
392C2×Q8—7324no—
393C2×Q8—734no—
394C2×Q8—7312no—
395C2×Q8—14632no—
396C2×Q8—14612no—
397C2×Q8—1464no—
398C2×Q8#1192402718yescomplemented
399C2×Q8C23×D7314yes—
400C2×D4—224no—
401C2×D4—212no—
402C2×D4—412no—
403C2×D4—7324no—
404C2×D4—14624no—
405C2×D4C23×D73112yes—
406C2×D4C23×D73124yes—
407C2×D4C23×D73124yes—
408C22×C4—224no—
409C22×C4—29224no—
410C22×C4—7348no—
411C22×C4—731no—
412C22×C4—732no—
413C22×C4—733no—
414C22×C4—733no—
415C22×C4—736no—
416C22×C4—734no—
417C22×C4—7312no—
418C22×C4—7312no—
419C22×C4—14612no—
420C22×C4—14612no—
421C22×C4—1464no—
422C22×C4C23×D73124yes—
423C22×C4C23×D7313yes—
424C22×C4C23×D7312yes—
425C22×C4C23×D7313yes—
426C22×C4C23×D7311yescharacteristic
427C4⋊C4—7324no—
428C4⋊C4—7324no—
429C4⋊C4—7324no—
430C4⋊C4—7312no—
431C4⋊C4—7324no—
432C4⋊C4—7324no—
433C4⋊C4—738no—
434C4⋊C4—7312no—
435C4⋊C4C23×D73112yes—
436C4⋊C4C23×D7314yes—
437C22⋊C4—7324no—
438C22⋊C4—7324no—
439C22⋊C4—7348no—
440C22⋊C4—7324no—
441C22⋊C4—738no—
442C22⋊C4C23×D73124yes—
443C42—7312no—
444C42—7312no—
445C42—7312no—
446C42—7324no—
447C42C23×D73112yes—
448C23—212no—
449C23—212no—
450C23—212no—
451C23—734no—
452C23—732no—
453C23—1462no—
454C23—1462no—
455C23C24×D73112yes—
456C23C24×D7311yescharacteristic
457Q8—29224no—
458Q8—14624no—
459Q8—1468no—
460Q8C2×#11924027116yescomplemented
461D4—248no—
462D4—224no—
463D4—424no—
464D4—14648no—
465C2×C4—24no—
466C2×C4—26no—
467C2×C4—248no—
468C2×C4—29248no—
469C2×C4—732no—
470C2×C4—734no—
471C2×C4—736no—
472C2×C4—736no—
473C2×C4—7312no—
474C2×C4—14624no—
475C2×C4—1466no—
476C2×C4—1466no—
477C2×C4—14612no—
478C2×C4—1462no—
479C2×C4—1464no—
480C2×C4—14624no—
481C2×C4—1468no—
482C2×C4C2×#11924027112yes—
483C2×C4#1455663814yes—
484C2×C4C24×D7316yes—
485C2×C4C24×D7314yes—
486C2×C4C24×D7316yes—
487C2×C4C24×D7312yes—
488C22—212no—
489C22—212no—
490C22—212no—
491C22—424no—
492C22—734no—
493C22—1468no—
494C22—1462no—
495C22—1462no—
496C22C22×#1192402712yes—
497C22H(5,𝔽2)×D7312yes—
498C22C2×#1455663812yes—
499C22C25×D7311yescharacteristic
500C4—24no—
501C4—26no—
502C4—1466no—
503C4—1466no—
504C4—14612no—
505C4—1462no—
506C4—1464no—
507C4C22×#11924027112yes—
508C4C2×#1455663814yes—
509C2—412no—
510C2—1464no—
511C2#2262401114yesdirect product factor
512C2C23×#1192402711yescharacteristic
513C2C2×H(5,𝔽2)×D7311yescharacteristic
514C2C22×#1455663811yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC2×#550×C73C2
socleC23×C73C25
hypercenterC2×#550D73
hypocenterC73C2×#3522
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #22624011C2direct (N × H)
2. C2×#17989323C2semidirect
3. C2×Q8×Dic146C2semidirect
4. C2×#17989294C2semidirect
5. C2×#17989244C2semidirect
6. C2×#17989237C2semidirect
7. C2×#17989204C2semidirect
8. C2×#17989135C2semidirect
9. C2×#17989119C2semidirect
10. C2×#17989070C2semidirect
11. C2×C4×Q8×D73C2semidirect
12. C2×#550×C73C2semidirect
15. #17989323C22semidirect
16. Q8×Dic146C22semidirect
17. #17989294C22semidirect
18. #17989244C22semidirect
19. #17989237C22semidirect
20. #17989204C22semidirect
21. #17989135C22semidirect
24. #17989119C22semidirect
37. #17989070C22semidirect
43. C4×Q8×D73C22semidirect
53. #550×C73C22semidirect
172. C2×Q8×C73#41semidirect
199. Q8×C73C2×#41semidirect
253. C2×C73#3522semidirect
262. C2×#550D73semidirect
270. C73C2×#3522semidirect
277. #550C2×D73semidirect
398. C2×Q8#11924027semidirect
460. Q8C2×#11924027semidirect
511. C2#22624011direct (N × H)

Maximal subgroups

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

subgroupindex
#226240112
C2×#179893232
C2×Q8×Dic1462
C2×#179892942
C2×#179892442
C2×#179892372
C2×#179892042
C2×#179891352
C2×#179891192
C2×#179890702
C2×C4×Q8×D732
C2×#550×C732
C2×#352273