Finite Group Databasebeta

← C2×#22624027 #33994207

Subgroups — C2×#22624027 #33994207

Order 18688.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624027C2132yesmaximal, direct product factor
2C2×#17989316C211yesmaximal, characteristic, complemented
3C2×#17989315C213yesmaximal, complemented
4C2×#17989310C213yesmaximal, complemented
5C2×#17989272C216yesmaximal, complemented
6C2×Q8×D292C211yesmaximal, characteristic, complemented
7C2×Q8×C22⋊D73C212yesmaximal, complemented
8C2×#17989195C216yesmaximal, complemented
9C2×#17989144C213yesmaximal, complemented
10C2×#17989121C213yesmaximal
11C2×#17989063C212yesmaximal, complemented
12C22×C4×Q8×C73C211yesmaximal, characteristic, complemented
13C22×Q8×D73C2212yes—
14#17989316C22116yescomplemented
15#17989315C22148yescomplemented
16#17989310C22148yescomplemented
17#17989272C22196yescomplemented
18Q8×D292C22116yescomplemented
19Q8×C22⋊D73C22132yescomplemented
20#17989195C22196yescomplemented
21#17989144C22148yescomplemented
22C2×#14556591C2213yes—
23C2×#14556591C2213yes—
24C2×#14556589C2216yes—
25#17989121C22148yes—
26C2×#14556586C22112yes—
27C2×#14556585C2216yes—
28C2×#14556583C2213yes—
29C2×#14556583C2216yes—
30C2×#14556582C2216yes—
31C2×#14556581C22112yes—
32C2×#14556579C2213yes—
33C2×#14556574C2216yes—
34C2×#14556571C2216yes—
35C2×#14556570C2216yes—
36C2×#14556570C2213yes—
37C2×#14556567C2216yes—
38C2×#14556565C2211yescharacteristic
39#17989063C22132yescomplemented
40C2×Q8×Dic73C2212yes—
41C2×C4×Dic146C2213yes—
42C2×C4×Dic146C2216yes—
43C2×C4×Dic146C2216yes—
44C2×#14556556C2216yes—
45C2×C4×D292C2213yes—
46C2×C4×C22⋊D73C2216yes—
47C2×#14556550C2216yes—
48C2×#14556549C2216yes—
49C2×#14556548C2216yes—
50C23×Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2213yes—
52C2×C4×Q8×C73C2211yescharacteristic
53C2×C4×Q8×C73C2213yes—
54C2×C4×Q8×C73C2211yescharacteristic
55C2×C4×Q8×C73C22116yescomplemented
56C22×C4⋊C4×C73C2213yes—
57C22×C42×C73C2213yes—
58C22×Dic146C2316yes—
59C22×Dic146C2313yes—
60C2×Q8×D73—216no—
61C2×Q8×D73C23116yes—
62C22×D292C2311yescharacteristic
63C22×C22⋊D73C2312yes—
64C22×C4×D73C2316yes—
65#14556591C23124yes—
66#14556591C23124yes—
67#14556589C23148yes—
68#14556586C23196yes—
69#14556585C23148yes—
70#14556583C23124yes—
71#14556583C23148yes—
72#14556582C23148yes—
73#14556581C23196yes—
74#14556579C23124yes—
75#14556574C23148yes—
76#14556571C23148yes—
77#14556570C23124yes—
78#14556570C23148yes—
79#14556567C23148yes—
80#14556565C2318yes—
81Q8×Dic73C23116yes—
82C4×Dic146C23124yes—
83C4×Dic146C23148yes—
84C4×Dic146C23148yes—
85#14556556C23148yes—
86C4×D292C23124yes—
87C4×C22⋊D73C23148yes—
88#14556550C23148yes—
89#14556549C23148yes—
90#14556548C23148yes—
91C2×C4⋊Dic73C2313yes—
92C2×C4⋊Dic73C2316yes—
93C2×C4⋊Dic73C2316yes—
94C2×C4⋊Dic73C2316yes—
95C2×C4⋊Dic73C2311yescharacteristic
96C2×#11924007C2316yes—
97C2×#11924007C23112yes—
98C2×#11924007C2316yes—
99C2×#11924007C2316yes—
100C2×#11924007C2316yes—
101C2×C22⋊Dic73C2316yes—
102C2×#11924005C2312yes—
103C2×#11924005C2316yes—
104C2×#11924005C2316yes—
105C2×#11924005C2316yes—
106C2×C4×Dic73C2316yes—
107C2×C4×Dic73C23112yes—
108C22×Q8×C73—28no—
109C22×Q8×C73C2313yes—
110C22×Q8×C73C2318yes—
111C22×Q8×C73C2311yescharacteristic
112C23×C4×C73C2311yescharacteristic
113C23×C4×C73C2313yes—
114C23×C4×C73C2313yes—
115C4×Q8×C73C23124yes—
116C4×Q8×C73C2318yes—
117C4×Q8×C73C23124yes—
118C4×Q8×C73C2318yes—
119C2×C4⋊C4×C73C2313yes—
120C2×C4⋊C4×C73C2313yes—
121C2×C4⋊C4×C73C2316yes—
122C2×C4⋊C4×C73C23124yes—
123C2×C42×C73C23124yes—
124C2×C42×C73C2313yes—
125C2×C42×C73C2313yes—
126C2×C42×C73C2313yes—
127C2×C42×C73C2313yes—
128C23×D73C2412yes—
129C2×Dic146—224no—
130C2×Dic146—212no—
131C2×Dic146C24124yes—
132C2×Dic146C24112yes—
133Q8×D73—264no—
134C2×D292—24no—
135C2×D292C2414yes—
136C2×C22⋊D73—28no—
137C2×C22⋊D73C2418yes—
138C22×Dic73C2416yes—
139C22×Dic73C2412yes—
140C22×Dic73C2416yes—
141C2×C4×D73—224no—
142C2×C4×D73C24124yes—
143C4⋊Dic73C24124yes—
144C4⋊Dic73C24124yes—
145C4⋊Dic73C24124yes—
146C4⋊Dic73C24112yes—
147C4⋊Dic73C2414yes—
148#11924007C24124yes—
149#11924007C24124yes—
150#11924007C24124yes—
151#11924007C24148yes—
152#11924007C24124yes—
153C22⋊Dic73C24124yes—
154#11924005C2418yes—
155#11924005C24124yes—
156#11924005C24124yes—
157#11924005C24124yes—
158C4×Dic73C24124yes—
159C4×Dic73C24148yes—
160C24×C73C2411yescharacteristic
161C2×Q8×C73—232no—
162C2×Q8×C73—212no—
163C2×Q8×C73#4118yes—
164C2×Q8×C73C24112yes—
165C2×Q8×C73C2414yes—
166C22×C4×C73—212no—
167C22×C4×C73—212no—
168C22×C4×C73—24no—
169C22×C4×C73C2411yescharacteristic
170C22×C4×C73C2414yes—
171C22×C4×C73C2413yes—
172C22×C4×C73C2413yes—
173C22×C4×C73C24112yes—
174C22×C4×C73C2413yes—
175C22×C4×C73C24112yes—
176C22×C4×C73C2413yes—
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73C24112yes—
179C4⋊C4×C73C24112yes—
180C4⋊C4×C73C24124yes—
181C42×C73C24112yes—
182C42×C73C24112yes—
183C42×C73C24112yes—
184C42×C73C24112yes—
185C23×C73—22no—
186C23×C73—22no—
187C23×C732−1+414yes—
188C23×C73C2512yes—
189C23×C73C2511yescharacteristic
190C22×D73—24no—
191C22×D73—24no—
192C22×D73—24no—
193C22×D73C2514yes—
194Q8×C73—224no—
195Q8×C73C2×#41116yes—
196C2×C4×C73—26no—
197C2×C4×C73—224no—
198C2×C4×C73—26no—
199C2×C4×C73—26no—
200C2×C4×C73—22no—
201C2×C4×C73—224no—
202C2×C4×C73—28no—
203C2×C4×C73C2×#41112yes—
204C2×C4×C732−1+414yes—
205C2×C4×C73C2512yes—
206C2×C4×C73C2516yes—
207C2×C4×C73C2516yes—
208C2×C4×C73C2516yes—
209C2×C4×C73C2516yes—
210C2×C4×C73C2512yes—
211C22⋊D73—216no—
212C2×Dic73—212no—
213C2×Dic73—24no—
214C2×Dic73—212no—
215C2×Dic73C25112yes—
216C2×Dic73C2514yes—
217C2×Dic73C25112yes—
218D292—28no—
219C4×D73—248no—
220Dic146—248no—
221Dic146—224no—
222C22×C73—22no—
223C22×C73—22no—
224C22×C73—28no—
225C22×C73C22×#4112yes—
226C22×C73C2×2−1+414yes—
227C22×C73C2×2−1+412yes—
228C22×C73C2×2−1+412yes—
229C22×C73C2611yescharacteristic
230C2×D73—24no—
231C2×D73—24no—
232C2×D73—24no—
233C2×D73—48no—
234C4×C73—26no—
235C4×C73—26no—
236C4×C73—26no—
237C4×C73—22no—
238C4×C73C22×#41112yes—
239C4×C73C2×2−1+414yes—
240Dic73—212no—
241Dic73—212no—
242Dic73—24no—
243C2×#3523—731nomaximal, Sylow
244C2×C73—24no—
245C2×C73#352314yescomplemented
246C2×C73C23×#4111yescharacteristic
247C2×C73C22×2−1+411yescharacteristic
248C2×C73C22×2−1+411yescharacteristic
249D73—44no—
250#3523—7332no—
251C2×#557—733no—
252C2×#557—731no—
253C2×#556—733no—
254C2×#552—736no—
255C2×D4×Q8—732no—
256C2×D4×Q8—731no—
257C2×#541—736no—
258C2×#533—733no—
259C2×#529—733no—
260C2×#519—732no—
261C22×C4×Q8D7311yescharacteristic, complemented
262C73C2×#352311yescharacteristic, complemented, Sylow
263C23×Q8—732no—
264C23×Q8C2×D7311yescharacteristic
265#557—7348no—
266#557—7316no—
267#556—7348no—
268#552—7396no—
269D4×Q8—7332no—
270D4×Q8—7316no—
271#541—7396no—
272#533—7348no—
273C2×C4⋊Q8—736no—
274C2×C4⋊Q8—733no—
275C2×C4⋊Q8—733no—
276#529—7348no—
277C2×#126—7312no—
278C2×#126—736no—
279C2×#125—7312no—
280C2×#125—736no—
281C2×#125—733no—
282C2×#125—736no—
283C2×#124—733no—
284C2×#123—736no—
285C2×C22⋊Q8—736no—
286C2×C22⋊Q8—733no—
287C2×C22⋊Q8—736no—
288C2×C22⋊Q8—736no—
289C2×#121—731no—
290#519—7332no—
291C2×C4×Q8—732no—
292C2×C4×Q8—736no—
293C2×C4×Q8—736no—
294C2×C4×Q8—733no—
295C2×C4×Q8—736no—
296C2×C4×Q8C2×D7311yescharacteristic
297C2×C4×Q8C2×D7313yes—
298C2×C4×Q8C2×D7311yescharacteristic
299C2×C4×Q8C2×D7313yes—
300C2×C4×Q8C2×D73116yescomplemented
301C2×C4×D4—736no—
302C2×C4×D4—733no—
303C2×#117—736no—
304C2×#117—736no—
305C2×#117—736no—
306C22×C4⋊C4C2×D7313yes—
307C22×C42C2×D7313yes—
308C22×Q8—28no—
309C22×Q8—7316no—
310C22×Q8—733no—
311C22×Q8—736no—
312C22×Q8—14616no—
313C22×Q8C22×D7313yes—
314C22×Q8C22×D7318yes—
315C22×Q8C22×D7311yescharacteristic
316C22×D4—732no—
317C22×D4—731no—
318C23×C4—736no—
319C23×C4C22×D7313yes—
320C23×C4C22×D7313yes—
321C23×C4C22×D7311yescharacteristic
322C4⋊Q8—7348no—
323C4⋊Q8—7324no—
324C4⋊Q8—7324no—
325#126—7396no—
326#126—7348no—
327#125—7396no—
328#125—7348no—
329#125—7324no—
330#125—7348no—
331#124—7324no—
332#123—7348no—
333C22⋊Q8—7348no—
334C22⋊Q8—7324no—
335C22⋊Q8—7348no—
336C22⋊Q8—7348no—
337#121—738no—
338C4×Q8—7316no—
339C4×Q8—7348no—
340C4×Q8—7348no—
341C4×Q8—7324no—
342C4×Q8—7348no—
343C4×Q8C22×D7318yes—
344C4×Q8C22×D73124yes—
345C4×Q8C22×D7318yes—
346C4×Q8C22×D73124yes—
347C4×D4—7348no—
348C4×D4—7324no—
349#117—7348no—
350#117—7348no—
351#117—7348no—
352C2×C4⋊C4—736no—
353C2×C4⋊C4—736no—
354C2×C4⋊C4—736no—
355C2×C4⋊C4—736no—
356C2×C4⋊C4—736no—
357C2×C4⋊C4—733no—
358C2×C4⋊C4—736no—
359C2×C4⋊C4—731no—
360C2×C4⋊C4—7312no—
361C2×C4⋊C4—736no—
362C2×C4⋊C4C22×D7313yes—
363C2×C4⋊C4C22×D7313yes—
364C2×C4⋊C4C22×D7316yes—
365C2×C4⋊C4C22×D73124yes—
366C2×C22⋊C4—736no—
367C2×C22⋊C4—732no—
368C2×C22⋊C4—736no—
369C2×C22⋊C4—736no—
370C2×C22⋊C4—736no—
371C2×C42—7312no—
372C2×C42—736no—
373C2×C42C22×D7313yes—
374C2×C42C22×D7313yes—
375C2×C42C22×D7313yes—
376C2×C42C22×D7313yes—
377C2×C42C22×D73124yes—
378C24—732no—
379C24C23×D7311yescharacteristic
380C2×Q8—232no—
381C2×Q8—212no—
382C2×Q8—7312no—
383C2×Q8—7324no—
384C2×Q8—14664no—
385C2×Q8—14624no—
386C2×Q8—14612no—
387C2×Q8#1192402818yes—
388C2×Q8C23×D73112yes—
389C2×Q8C23×D7314yes—
390C2×D4—738no—
391C2×D4—734no—
392C2×D4—1468no—
393C2×D4—1464no—
394C22×C4—212no—
395C22×C4—212no—
396C22×C4—24no—
397C22×C4—736no—
398C22×C4—732no—
399C22×C4—736no—
400C22×C4—7324no—
401C22×C4—14624no—
402C22×C4C23×D7314yes—
403C22×C4C23×D73112yes—
404C22×C4C23×D73112yes—
405C22×C4C23×D7313yes—
406C22×C4C23×D7311yescharacteristic
407C22×C4C23×D7313yes—
408C22×C4C23×D7313yes—
409C22×C4C23×D7313yes—
410C22×C4C23×D7311yescharacteristic
411C4⋊C4—7324no—
412C4⋊C4—7324no—
413C4⋊C4—7324no—
414C4⋊C4—7348no—
415C4⋊C4—7324no—
416C4⋊C4—7324no—
417C4⋊C4—7324no—
418C4⋊C4—7324no—
419C4⋊C4—7312no—
420C4⋊C4—734no—
421C4⋊C4C23×D73112yes—
422C4⋊C4C23×D73112yes—
423C4⋊C4C23×D73124yes—
424C22⋊C4—7324no—
425C22⋊C4—738no—
426C22⋊C4—7324no—
427C22⋊C4—7324no—
428C22⋊C4—7324no—
429C42—7348no—
430C42—7324no—
431C42C23×D73112yes—
432C42C23×D73112yes—
433C42C23×D73112yes—
434C42C23×D73112yes—
435C23—22no—
436C23—22no—
437C23—734no—
438C23—1464no—
439C23—1464no—
440C23—1464no—
441C23#1455663814yes—
442C23C24×D7312yes—
443C23C24×D7311yescharacteristic
444Q8—224no—
445Q8—14648no—
446Q8—14624no—
447Q8C2×#11924028116yes—
448D4—14616no—
449D4—1468no—
450C2×C4—26no—
451C2×C4—224no—
452C2×C4—26no—
453C2×C4—26no—
454C2×C4—22no—
455C2×C4—224no—
456C2×C4—28no—
457C2×C4—7312no—
458C2×C4—734no—
459C2×C4—7312no—
460C2×C4—14612no—
461C2×C4—1464no—
462C2×C4—14612no—
463C2×C4—14648no—
464C2×C4C2×#11924028112yes—
465C2×C4#1455663914yes—
466C2×C4C24×D7316yes—
467C2×C4C24×D7312yes—
468C2×C4C24×D7316yes—
469C2×C4C24×D7316yes—
470C2×C4C24×D7316yes—
471C2×C4C24×D7312yes—
472C22—28no—
473C22—22no—
474C22—22no—
475C22—2928no—
476C22—1464no—
477C22—1464no—
478C22—1464no—
479C22C22×#1192402812yes—
480C22C2×#1455663814yes—
481C22C2×#1455663812yes—
482C22C2×#1455663912yes—
483C22C25×D7311yescharacteristic
484C4—26no—
485C4—26no—
486C4—26no—
487C4—22no—
488C4—14612no—
489C4—1464no—
490C4—14612no—
491C4C22×#11924028112yes—
492C4C2×#1455663914yes—
493C2—24no—
494C2—2924no—
495C2#2262402714yesdirect product factor
496C2C23×#1192402811yescharacteristic
497C2C22×#1455663811yescharacteristic
498C2C22×#1455663911yescharacteristic
subgroupG/H
centerC23C24×D73
commutator, cosocleC22×C73C26
FrattiniC22C25×D73
FittingC22×C4×Q8×C73C2
socleC23×C73C25
prime radicalC2×#14556565C22
hypercenterC22×C4×Q8D73
hypocenterC73C2×#3523
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22624027C2direct (N × H)
2. C2×#17989316C2semidirect
3. C2×#17989315C2semidirect
4. C2×#17989310C2semidirect
5. C2×#17989272C2semidirect
6. C2×Q8×D292C2semidirect
7. C2×Q8×C22⋊D73C2semidirect
8. C2×#17989195C2semidirect
9. C2×#17989144C2semidirect
11. C2×#17989063C2semidirect
12. C22×C4×Q8×C73C2semidirect
14. #17989316C22semidirect
15. #17989315C22semidirect
16. #17989310C22semidirect
17. #17989272C22semidirect
18. Q8×D292C22semidirect
19. Q8×C22⋊D73C22semidirect
20. #17989195C22semidirect
21. #17989144C22semidirect
39. #17989063C22semidirect
55. C2×C4×Q8×C73C22semidirect
245. C2×C73#3523semidirect
261. C22×C4×Q8D73semidirect
262. C73C2×#3523semidirect
300. C2×C4×Q8C2×D73semidirect
495. C2#22624027direct (N × H)

Maximal subgroups

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

subgroupindex
#226240272
C2×#179893162
C2×#179893152
C2×#179893102
C2×#179892722
C2×Q8×D2922
C2×Q8×C22⋊D732
C2×#179891952
C2×#179891442
C2×#179891212
C2×#179890632
C22×C4×Q8×C732
C2×#352373