Finite Group Databasebeta

← C2×#22623526 #33993047

Subgroups — C2×#22623526 #33993047

Order 18688.

7802 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#22623526C2132yesmaximal, direct product factor
2C2×#17989322C213yesmaximal, complemented
3C2×#17989316C211yesmaximal, characteristic, complemented
4C2×#17989310C213yesmaximal, complemented
5C2×#17989285C216yesmaximal, complemented
6C2×#17989259C211yesmaximal, characteristic, complemented
7C2×#17989248C212yesmaximal, complemented
8C2×#17989137C213yesmaximal, complemented
9C2×#17989124C216yesmaximal, complemented
10C2×C4×#11924029C212yesmaximal, complemented
11C2×C4×#11924028C213yesmaximal, complemented
12C22×C4×Q8×C73C211yesmaximal, characteristic, complemented
13C22×#11924029C2212yes—
14C22×#11924028C2213yes—
15#17989322C22148yescomplemented
16#17989316C22116yescomplemented
17#17989310C22148yescomplemented
18#17989285C22196yescomplemented
19#17989259C22116yescomplemented
20#17989248C22132yescomplemented
21#17989137C22148yescomplemented
22C2×#14556591C2213yes—
23C2×#14556588C2213yes—
24#17989124C22196yescomplemented
25C2×#14556586C22112yes—
26C2×#14556582C2216yes—
27C2×#14556580C2216yes—
28C2×#14556579C2216yes—
29C2×#14556576C22112yes—
30C2×#14556567C2216yes—
31C2×#14556566C2216yes—
32C2×#14556565C2216yes—
33C2×#14556565C2216yes—
34C4×#11924029C22132yescomplemented
35C4×#11924028C22148yescomplemented
36C2×Q8×Dic73C2212yes—
37C2×C4×Dic146C2213yes—
38C2×C4×Dic146C2213yes—
39C2×C4×D292C2213yes—
40C2×C4×D292C2213yes—
41C2×C4×D292C2216yes—
42C2×C4×D292C2216yes—
43C2×#14556553C2216yes—
44C2×C4×C22⋊D73C2216yes—
45C2×C4×C22⋊D73C2216yes—
46C2×#14556550C2211yescharacteristic
47C2×#14556549C2216yes—
48C2×#14556548C2216yes—
49C2×C42×D73C2216yes—
50C23×Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2213yes—
53C2×C4×Q8×C73C2211yescharacteristic
54C2×C4×Q8×C73C2213yes—
55C2×C4×Q8×C73C22116yescomplemented
56C22×C4⋊C4×C73C2213yes—
57C22×C42×C73C2213yes—
58C2×#11924029—216no—
59C2×#11924029C23116yes—
60C2×#11924028—224no—
61C2×#11924028C23124yes—
62C22×Dic146C2313yes—
63C22×D292C2313yes—
64C22×D292C2316yes—
65C22×D292C2316yes—
66C22×C22⋊D73C2316yes—
67C22×C4×D73C2316yes—
68C22×C4×D73C2316yes—
69C22×C4×D73C2316yes—
70#14556591C23124yes—
71#14556588C23124yes—
72#14556586C23196yes—
73#14556582C23148yes—
74#14556580C23148yes—
75#14556579C23148yes—
76#14556576C23196yes—
77#14556567C23148yes—
78#14556566C23148yes—
79#14556565C23148yes—
80#14556565C23148yes—
81Q8×Dic73C23116yes—
82C4×Dic146C23124yes—
83C4×Dic146C23124yes—
84C4×D292C23148yes—
85C4×D292C23148yes—
86C4×D292C23124yes—
87C4×D292C23124yes—
88#14556553C23148yes—
89C4×C22⋊D73C23148yes—
90C4×C22⋊D73C23148yes—
91#14556550C2318yes—
92#14556549C23148yes—
93#14556548C23148yes—
94C2×C4⋊Dic73C2316yes—
95C2×C4⋊Dic73C2316yes—
96C2×C4⋊Dic73C2311yescharacteristic
97C2×C4⋊Dic73C2311yescharacteristic
98C2×#11924007C2316yes—
99C2×#11924007C2316yes—
100C2×C22⋊Dic73C2312yes—
101C2×#11924005C2316yes—
102C2×#11924005C2316yes—
103C2×#11924005C23112yes—
104C2×#11924005C23112yes—
105C2×C4×Dic73C2312yes—
106C2×C4×Dic73C2316yes—
107C2×C4×Dic73C2316yes—
108C42×D73C23148yes—
109C22×Q8×C73—28no—
110C22×Q8×C73C2318yes—
111C22×Q8×C73C2313yes—
112C22×Q8×C73C2311yescharacteristic
113C23×C4×C73C2313yes—
114C23×C4×C73C2313yes—
115C23×C4×C73C2311yescharacteristic
116C4×Q8×C73C23124yes—
117C4×Q8×C73C2318yes—
118C4×Q8×C73C23124yes—
119C4×Q8×C73C2318yes—
120C2×C4⋊C4×C73C2313yes—
121C2×C4⋊C4×C73C2313yes—
122C2×C4⋊C4×C73C2316yes—
123C2×C4⋊C4×C73C23124yes—
124C2×C42×C73C2313yes—
125C2×C42×C73C23124yes—
126C2×C42×C73C2313yes—
127C2×C42×C73C2313yes—
128C2×C42×C73C2313yes—
129C23×D73C2416yes—
130#11924029—264no—
131#11924028—296no—
132C2×Dic146—212no—
133C2×Dic146C24112yes—
134C2×D292—212no—
135C2×D292—224no—
136C2×D292—412no—
137C2×D292C24124yes—
138C2×D292C24112yes—
139C2×D292C24124yes—
140C2×C22⋊D73—224no—
141C2×C22⋊D73C24124yes—
142C22×Dic73C2412yes—
143C22×Dic73C2412yes—
144C22×Dic73C2416yes—
145C2×C4×D73—224no—
146C2×C4×D73—224no—
147C2×C4×D73—412no—
148C2×C4×D73C24124yes—
149C2×C4×D73C24124yes—
150C2×C4×D73C24124yes—
151C4⋊Dic73C24124yes—
152C4⋊Dic73C2414yes—
153C4⋊Dic73C2414yes—
154C4⋊Dic73C24124yes—
155#11924007C24124yes—
156#11924007C24124yes—
157C22⋊Dic73C2418yes—
158#11924005C24124yes—
159#11924005C24124yes—
160#11924005C24148yes—
161#11924005C24148yes—
162C4×Dic73C2418yes—
163C4×Dic73C24124yes—
164C4×Dic73C24124yes—
165C24×C73C2411yescharacteristic
166C2×Q8×C73—232no—
167C2×Q8×C73—212no—
168C2×Q8×C73#4118yescomplemented
169C2×Q8×C73C24112yes—
170C2×Q8×C73C2414yes—
171C22×C4×C73—212no—
172C22×C4×C73—46no—
173C22×C4×C73#4118yescomplemented
174C22×C4×C73C2413yes—
175C22×C4×C73C2413yes—
176C22×C4×C73C2413yes—
177C22×C4×C73C24112yes—
178C22×C4×C73C24112yes—
179C22×C4×C73C2411yescharacteristic
180C22×C4×C73C2413yes—
181C22×C4×C73C2414yes—
182C22×C4×C73C2411yescharacteristic
183C4⋊C4×C73C24112yes—
184C4⋊C4×C73C24112yes—
185C4⋊C4×C73C24124yes—
186C42×C73C24112yes—
187C42×C73C24112yes—
188C42×C73C24112yes—
189C42×C73C24112yes—
190C23×C73—22no—
191C23×C73—22no—
192C23×C73C2×#4114yes—
193C23×C73C2512yes—
194C23×C73C2511yescharacteristic
195C22×D73—212no—
196C22×D73—212no—
197C22×D73—212no—
198C22×D73C25112yes—
199Q8×C73—224no—
200Q8×C73C2×#41116yescomplemented
201C2×C4×C73—26no—
202C2×C4×C73—26no—
203C2×C4×C73—26no—
204C2×C4×C73—224no—
205C2×C4×C73—412no—
206C2×C4×C73C2×#41112yes—
207C2×C4×C73C2×#4114yes—
208C2×C4×C73C2×#4114yes—
209C2×C4×C73C2×#41116yescomplemented
210C2×C4×C73C2516yes—
211C2×C4×C73C2516yes—
212C2×C4×C73C2516yes—
213C2×C4×C73C2512yes—
214C2×C4×C73C2512yes—
215C2×C4×C73C2516yes—
216C22⋊D73—248no—
217C2×Dic73—24no—
218C2×Dic73—212no—
219C2×Dic73—24no—
220C2×Dic73C2514yes—
221C2×Dic73C2514yes—
222C2×Dic73C25112yes—
223D292—224no—
224D292—248no—
225D292—424no—
226C4×D73—248no—
227C4×D73—248no—
228C4×D73—424no—
229Dic146—224no—
230C22×C73—22no—
231C22×C73—22no—
232C22×C73—28no—
233C22×C73C22×#4112yes—
234C22×C73C22×#4114yes—
235C22×C73C22×#4112yes—
236C22×C73#58512yes—
237C22×C73C2611yescharacteristic
238C2×D73—212no—
239C2×D73—212no—
240C2×D73—212no—
241C2×D73—424no—
242C4×C73—26no—
243C4×C73—26no—
244C4×C73—26no—
245C4×C73C22×#41112yes—
246C4×C73C22×#4114yes—
247C4×C73C22×#4114yes—
248Dic73—212no—
249Dic73—24no—
250Dic73—24no—
251C2×#3480—731nomaximal, Sylow
252C2×C73—24no—
253C2×C73#348014yescomplemented
254C2×C73C23×#4111yescharacteristic
255C2×C73C23×#4111yescharacteristic
256C2×C73C2×#58511yescharacteristic
257D73—412no—
258#3480—7332no—
259C2×#559—733no—
260C2×#557—731no—
261C2×#556—733no—
262C2×#553—736no—
263C2×#550—731no—
264C2×#548—732no—
265C2×#532—733no—
266C2×#529—736no—
267C2×C4×#41—733no—
268C2×C4×#41—732no—
269C22×C4×Q8D7311yescharacteristic, complemented
270C73C2×#348011yescharacteristic, complemented, Sylow
271C22×#41—732no—
272C22×#41—733no—
273C23×Q8C2×D7311yescharacteristic
274#559—7348no—
275#557—7316no—
276#556—7348no—
277#553—7396no—
278#550—7316no—
279#548—7332no—
280#532—7348no—
281C2×C4⋊Q8—733no—
282C2×#127—733no—
283#529—7396no—
284C2×#126—7312no—
285C2×#125—736no—
286C2×#124—736no—
287C2×#124—736no—
288C2×#123—7312no—
289C2×C22⋊Q8—736no—
290C2×#121—736no—
291C2×#121—736no—
292C2×#121—736no—
293C4×#41—7332no—
294C4×#41—7348no—
295C2×C4×Q8—732no—
296C2×C4×Q8—733no—
297C2×C4×Q8—733no—
298C2×C4×Q8C2×D7311yescharacteristic
299C2×C4×Q8C2×D7313yes—
300C2×C4×Q8C2×D7311yescharacteristic
301C2×C4×Q8C2×D7313yes—
302C2×C4×Q8C2×D73116yescomplemented
303C2×C4×D4—736no—
304C2×C4×D4—736no—
305C2×C4×D4—736no—
306C2×C4×D4—736no—
307C2×C4×D4—733no—
308C2×C4×D4—733no—
309C2×C4×D4—736no—
310C2×#117—731no—
311C2×#117—736no—
312C2×#117—736no—
313C22×C4⋊C4C2×D7313yes—
314C22×C42—736no—
315C22×C42C2×D7313yes—
316C2×#41—7316no—
317C2×#41—7324no—
318C2×#41—14616no—
319C2×#41—14624no—
320C22×Q8—28no—
321C22×Q8—733no—
322C22×Q8C22×D7313yes—
323C22×Q8C22×D7318yes—
324C22×Q8C22×D7311yescharacteristic
325C22×D4—736no—
326C22×D4—736no—
327C22×D4—736no—
328C22×D4—733no—
329C23×C4—736no—
330C23×C4—736no—
331C23×C4—736no—
332C23×C4C22×D7313yes—
333C23×C4C22×D7313yes—
334C23×C4C22×D7311yescharacteristic
335C4⋊Q8—7324no—
336#127—7324no—
337#126—7396no—
338#125—7348no—
339#124—7348no—
340#124—7348no—
341#123—7396no—
342C22⋊Q8—7348no—
343#121—7348no—
344#121—7348no—
345#121—7348no—
346C4×Q8—7316no—
347C4×Q8—7324no—
348C4×Q8—7324no—
349C4×Q8C22×D7318yes—
350C4×Q8C22×D73124yes—
351C4×Q8C22×D7318yes—
352C4×Q8C22×D73124yes—
353C4×D4—7348no—
354C4×D4—7348no—
355C4×D4—7348no—
356C4×D4—7348no—
357C4×D4—7324no—
358C4×D4—7324no—
359C4×D4—7348no—
360#117—738no—
361#117—7348no—
362#117—7348no—
363C2×C4⋊C4—731no—
364C2×C4⋊C4—731no—
365C2×C4⋊C4—736no—
366C2×C4⋊C4—736no—
367C2×C4⋊C4—736no—
368C2×C4⋊C4—736no—
369C2×C4⋊C4C22×D7313yes—
370C2×C4⋊C4C22×D7313yes—
371C2×C4⋊C4C22×D7316yes—
372C2×C4⋊C4C22×D73124yes—
373C2×C22⋊C4—732no—
374C2×C22⋊C4—736no—
375C2×C22⋊C4—7312no—
376C2×C22⋊C4—736no—
377C2×C22⋊C4—7312no—
378C2×C42—736no—
379C2×C42—736no—
380C2×C42—732no—
381C2×C42—7348no—
382C2×C42C22×D7313yes—
383C2×C42C22×D7313yes—
384C2×C42C22×D73124yes—
385C2×C42C22×D7313yes—
386C2×C42C22×D7313yes—
387C24—736no—
388C24C23×D7311yescharacteristic
389#41—14664no—
390#41—14696no—
391C2×Q8—232no—
392C2×Q8—212no—
393C2×Q8—7312no—
394C2×Q8—14612no—
395C2×Q8#1192402818yescomplemented
396C2×Q8C23×D73112yes—
397C2×Q8C23×D7314yes—
398C2×D4—29212no—
399C2×D4—7324no—
400C2×D4—7324no—
401C2×D4—7324no—
402C2×D4—7312no—
403C2×D4—14624no—
404C2×D4—14624no—
405C2×D4—14612no—
406C22×C4—212no—
407C22×C4—46no—
408C22×C4—29212no—
409C22×C4—7324no—
410C22×C4—732no—
411C22×C4—732no—
412C22×C4—736no—
413C22×C4—7324no—
414C22×C4—7324no—
415C22×C4—14624no—
416C22×C4—14624no—
417C22×C4#1192402918yescomplemented
418C22×C4C23×D7313yes—
419C22×C4C23×D7313yes—
420C22×C4C23×D7313yes—
421C22×C4C23×D73112yes—
422C22×C4C23×D73112yes—
423C22×C4C23×D7313yes—
424C22×C4C23×D7311yescharacteristic
425C22×C4C23×D7314yes—
426C22×C4C23×D7311yescharacteristic
427C4⋊C4—7324no—
428C4⋊C4—7324no—
429C4⋊C4—734no—
430C4⋊C4—734no—
431C4⋊C4—7324no—
432C4⋊C4—7324no—
433C4⋊C4C23×D73112yes—
434C4⋊C4C23×D73112yes—
435C4⋊C4C23×D73124yes—
436C22⋊C4—738no—
437C22⋊C4—7324no—
438C22⋊C4—7348no—
439C22⋊C4—7324no—
440C22⋊C4—7348no—
441C42—738no—
442C42—7324no—
443C42—7324no—
444C42C23×D73112yes—
445C42C23×D73112yes—
446C42C23×D73112yes—
447C42C23×D73112yes—
448C23—22no—
449C23—22no—
450C23—7312no—
451C23—14612no—
452C23—14612no—
453C23—14612no—
454C23C2×#1192402914yes—
455C23C24×D7312yes—
456C23C24×D7311yescharacteristic
457Q8—224no—
458Q8—14624no—
459Q8C2×#11924028116yescomplemented
460D4—29224no—
461D4—14648no—
462D4—14648no—
463D4—14624no—
464C2×C4—26no—
465C2×C4—26no—
466C2×C4—26no—
467C2×C4—224no—
468C2×C4—412no—
469C2×C4—29224no—
470C2×C4—734no—
471C2×C4—734no—
472C2×C4—7312no—
473C2×C4—1464no—
474C2×C4—14612no—
475C2×C4—1464no—
476C2×C4—14648no—
477C2×C4—14648no—
478C2×C4C2×#11924028112yes—
479C2×C4C2×#1192402914yes—
480C2×C4C2×#1192402914yes—
481C2×C4C2×#11924029116yescomplemented
482C2×C4C24×D7316yes—
483C2×C4C24×D7316yes—
484C2×C4C24×D7316yes—
485C2×C4C24×D7316yes—
486C2×C4C24×D7312yes—
487C2×C4C24×D7312yes—
488C22—28no—
489C22—22no—
490C22—22no—
491C22—29224no—
492C22—14612no—
493C22—14612no—
494C22—14612no—
495C22C22×#1192402812yes—
496C22C22×#1192402914yes—
497C22C22×#1192402912yes—
498C22#1798942312yes—
499C22C25×D7311yescharacteristic
500C4—26no—
501C4—26no—
502C4—26no—
503C4—1464no—
504C4—14612no—
505C4—1464no—
506C4C22×#11924028112yes—
507C4C22×#1192402914yes—
508C4C22×#1192402914yes—
509C2—24no—
510C2—29212no—
511C2#2262352614yesdirect product factor
512C2C23×#1192402811yescharacteristic
513C2C23×#1192402911yescharacteristic
514C2C2×#1798942311yescharacteristic
subgroupG/H
centerC22×C4C23×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC22×C4×Q8×C73C2
socleC23×C73C25
hypercenterC22×C4×Q8D73
hypocenterC73C2×#3480
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22623526C2direct (N × H)
2. C2×#17989322C2semidirect
3. C2×#17989316C2semidirect
4. C2×#17989310C2semidirect
5. C2×#17989285C2semidirect
6. C2×#17989259C2semidirect
7. C2×#17989248C2semidirect
8. C2×#17989137C2semidirect
9. C2×#17989124C2semidirect
10. C2×C4×#11924029C2semidirect
11. C2×C4×#11924028C2semidirect
12. C22×C4×Q8×C73C2semidirect
15. #17989322C22semidirect
16. #17989316C22semidirect
17. #17989310C22semidirect
18. #17989285C22semidirect
19. #17989259C22semidirect
20. #17989248C22semidirect
21. #17989137C22semidirect
24. #17989124C22semidirect
34. C4×#11924029C22semidirect
35. C4×#11924028C22semidirect
55. C2×C4×Q8×C73C22semidirect
168. C2×Q8×C73#41semidirect
173. C22×C4×C73#41semidirect
200. Q8×C73C2×#41semidirect
209. C2×C4×C73C2×#41semidirect
253. C2×C73#3480semidirect
269. C22×C4×Q8D73semidirect
270. C73C2×#3480semidirect
302. C2×C4×Q8C2×D73semidirect
395. C2×Q8#11924028semidirect
417. C22×C4#11924029semidirect
459. Q8C2×#11924028semidirect
481. C2×C4C2×#11924029semidirect
511. C2#22623526direct (N × H)

Maximal subgroups

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

subgroupindex
#226235262
C2×#179893222
C2×#179893162
C2×#179893102
C2×#179892852
C2×#179892592
C2×#179892482
C2×#179891372
C2×#179891242
C2×C4×#119240292
C2×C4×#119240282
C22×C4×Q8×C732
C2×#348073