Finite Group Databasebeta

← C2×#22622841 #33966473

Subgroups — C2×#22622841 #33966473

Order 18688.

1854 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic146C211yesmaximal, characteristic, complemented
2#22622841C2116yesmaximal, direct product factor
3C2×#17988997C212yesmaximal, complemented
4C2×#17988990C211yesmaximal, characteristic, complemented
5C2×#17988891C212yesmaximal, complemented
6C2×#17988879C211yesmaximal, characteristic, complemented
7C2×#17988877C211yesmaximal, characteristic
8C2×#17988867C212yesmaximal
9C2×#17988718C211yesmaximal, characteristic, complemented
10C2×#17988687C212yesmaximal
11C2×C4×#11924016C211yesmaximal, characteristic
12C2×#554×C73C211yesmaximal, characteristic
13C22×#11924016C2211yescharacteristic
14Q8×Dic146—24no—
15Q8×Dic146C2218yescomplemented
16C2×#14556591C2211yescharacteristic
17C2×#14556591C2212yes—
18C2×#14556590—21no—
19C2×#14556590—22no—
20C2×Q8×Dic73—21no—
21C2×C4×Dic146C2211yescharacteristic
22C2×C4×Dic146C2212yes—
23#17988997C22116yescomplemented
24#17988990C2218yescomplemented
25#17988891C22116yescomplemented
26#17988879C2218yescomplemented
27#17988877C2218yes—
28#17988867C22116yes—
29#17988718C2218yescomplemented
30#17988687C22116yes—
31C4×#11924016C2218yes—
32C2×#14556524C2212yes—
33C2×#14556524C2212yes—
34C2×#14556524C2212yes—
35C2×#14556522C2212yes—
36C2×#14556522C2211yescharacteristic
37C2×C4⋊#10596831C2212yes—
38C2×C4⋊#10596831C2211yescharacteristic
39C2×#14556514C2212yes—
40C2×#14556514C2211yescharacteristic
41C2×#14556512C2211yescharacteristic
42C2×#14556512C2212yes—
43C2×#14556502C2212yes—
44C2×C4×#10596831C2211yescharacteristic
45#554×C73—24no—
46#554×C73C2218yes—
47C2×C4⋊Q8×C73C2211yescharacteristic
48C2×#125×C73C2212yes—
49C2×C22⋊Q8×C73—21no—
50C2×C22⋊Q8×C73—22no—
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×D4×C73C2212yes—
53C2×C4×D4×C73C2211yescharacteristic
54C22×C4⋊C4×C73—21no—
55C22×Dic146—21no—
56C22×Dic146—22no—
57C22×Dic146C2311yescharacteristic
58C2×#11924016Q818yescomplemented
59C2×#11924016C2314yes—
60C22×#10596831C2312yes—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556591—24no—
64#14556591—22no—
65#14556591C2318yes—
66#14556591C2314yes—
67#14556590—28no—
68#14556590—216no—
69Q8×Dic73—28no—
70C4×Dic146—24no—
71C4×Dic146—22no—
72C4×Dic146C2314yes—
73C4×Dic146C2318yes—
74C2×C4⋊Dic73—22no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73C2312yes—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—22no—
80C2×C4×Dic73—22no—
81C2×C4×Dic73—21no—
82#14556524C2318yes—
83#14556524C2318yes—
84#14556524C2318yes—
85#14556522C2318yes—
86#14556522C2314yes—
87C4⋊#10596831C2318yes—
88C4⋊#10596831C2314yes—
89#14556514C2318yes—
90#14556514C2314yes—
91#14556512C2314yes—
92#14556512C2318yes—
93#14556502C2318yes—
94C4×#10596831C2314yes—
95C22×Q8×C73—21no—
96C22×Q8×C73D411yescharacteristic
97C22×D4×C73C2311yescharacteristic
98C23×C4×C73—21no—
99C23×C4×C73—22no—
100C4⋊Q8×C73—22no—
101C4⋊Q8×C73C2314yes—
102#125×C73—24no—
103#125×C73C2318yes—
104C22⋊Q8×C73—28no—
105C22⋊Q8×C73—216no—
106C4×Q8×C73D414yes—
107C4×Q8×C73C2314yes—
108C4×D4×C73—24no—
109C4×D4×C73—22no—
110C4×D4×C73C2318yes—
111C4×D4×C73C2314yes—
112C2×C4⋊C4×C73—22no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—22no—
115C2×C4⋊C4×C73—28no—
116C2×C4⋊C4×C73D411yescharacteristic
117C2×C4⋊C4×C73D412yes—
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2312yes—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—22no—
122C2×C42×C73C2312yes—
123C2×C42×C73C2311yescharacteristic
124C2×Dic146—21no—
125C2×Dic146—22no—
126C2×Dic146—24no—
127C2×Dic146—28no—
128C2×Dic146—84no—
129C2×Dic146—82no—
130C2×Dic146C2×Q814yes—
131C2×Dic146C2412yes—
132C22×Dic73—21no—
133C22×Dic73—22no—
134C22×Dic73—21no—
135#11924016C2×Q8116yescomplemented
136C2×#10596831—24no—
137C2×#10596831—22no—
138C2×#10596831C2×Q814yes—
139C2×#10596831C2412yes—
140C2×#10596831C2414yes—
141C2×#10596831C2412yes—
142C4⋊Dic73—28no—
143C4⋊Dic73—24no—
144C4⋊Dic73—22no—
145C4⋊Dic73—21no—
146C4⋊Dic73C2412yes—
147C4⋊Dic73C2414yes—
148#11924007—24no—
149#11924007—28no—
150C4×Dic73—28no—
151C4×Dic73—24no—
152C24×C73—21no—
153C2×Q8×C73—24no—
154C2×Q8×C73—24no—
155C2×Q8×C73—82no—
156C2×Q8×C73C2×D414yes—
157C2×D4×C73—21no—
158C2×D4×C73—22no—
159C2×D4×C73C2×Q814yes—
160C2×D4×C73C2412yes—
161C22×C4×C73—21no—
162C22×C4×C73—22no—
163C22×C4×C73—24no—
164C22×C4×C73—28no—
165C22×C4×C73—44no—
166C22×C4×C73—84no—
167C22×C4×C73C2×D412yes—
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D412yes—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2411yescharacteristic
172C4⋊C4×C73—28no—
173C4⋊C4×C73—24no—
174C4⋊C4×C73—28no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73C2×D414yes—
178C4⋊C4×C73C2×D418yes—
179C4⋊C4×C73C2412yes—
180C4⋊C4×C73C2414yes—
181C22⋊C4×C73—24no—
182C22⋊C4×C73—28no—
183C42×C73C2×D414yes—
184C42×C73C2×D412yes—
185C42×C73C2414yes—
186C42×C73C2412yes—
187C23×C73—24no—
188C23×C73—22no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73C22×D411yescharacteristic
192Q8×C73—28no—
193Q8×C73—84no—
194D4×C73—24no—
195D4×C73—22no—
196D4×C73C22×Q814yes—
197C2×C4×C73—24no—
198C2×C4×C73—24no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—24no—
202C2×C4×C73—42no—
203C2×C4×C73—44no—
204C2×C4×C73—48no—
205C2×C4×C73—88no—
206C2×C4×C73#13714yes—
207C2×C4×C73C22×D414yes—
208C2×C4×C73C22×D412yes—
209C2×C4×C73C22×D414yes—
210C2×C4×C73C22×D411yescharacteristic
211C2×C4×C73C22×D412yes—
212C2×C4×C73C22×Q812yes—
213C2×C4×C732−1+412yes—
214C2×C4×C73C2511yescharacteristic
215C2×Dic73—24no—
216C2×Dic73—22no—
217C2×Dic73—24no—
218C2×Dic73—22no—
219C2×Dic73—42no—
220C2×Dic73—44no—
221Dic146—24no—
222Dic146—22no—
223Dic146—88no—
224Dic146—84no—
225Dic146C22×Q814yes—
226#10596831—22no—
227#10596831—24no—
228#10596831C22×Q814yes—
229C22×C73—24no—
230C22×C73—48no—
231C22×C73—42no—
232C22×C73—42no—
233C22×C73D4×Q812yes—
234C22×C73C2×#13712yes—
235C22×C73#57612yes—
236C22×C73C23×D411yescharacteristic
237C4×C73—24no—
238C4×C73—24no—
239C4×C73—22no—
240C4×C73—42no—
241C4×C73—44no—
242C4×C73D4×Q812yes—
243C4×C73C2×#13714yes—
244C4×C73C23×Q811yescharacteristic
245C4×C73C2×2−1+411yescharacteristic
246Dic73—24no—
247Dic73—42no—
248Dic73—44no—
249C2×#3385—731nomaximal, Sylow
250C2×C73—44no—
251C2×C73#338514yescomplemented
252C2×C73C2×D4×Q811yescharacteristic
253C2×C73C22×#13711yescharacteristic
254C2×C73C2×#57611yescharacteristic
255C2×Q82—731no—
256C2×#554D7311yescharacteristic
257#3385—7316no—
258C2×#501—732no—
259C2×#500—731no—
260C2×#479—732no—
261C2×#477—731no—
262C2×#476—731no—
263C2×#474—732no—
264C2×#446—731no—
265C2×#440—732no—
266C2×C4×SD16—731no—
267C73C2×#338511yescharacteristic, complemented, Sylow
268C22×SD16—731no—
269Q82—738no—
270Q82—1464no—
271#554—24no—
272#554C2×D7318yes—
273C2×C4⋊Q8—731no—
274C2×C4⋊Q8—732no—
275C2×C4⋊Q8—1461no—
276C2×C4⋊Q8—1462no—
277C2×C4⋊Q8C2×D7311yescharacteristic
278C2×#125C2×D7312yes—
279C2×C22⋊Q8—21no—
280C2×C22⋊Q8—22no—
281C2×C4×Q8—731no—
282C2×C4×Q8—732no—
283C2×C4×Q8—1461no—
284C2×C4×Q8C2×D7311yescharacteristic
285C2×C4×D4C2×D7311yescharacteristic
286C2×C4×D4C2×D7312yes—
287C22×C4⋊C4—21no—
288#501—7316no—
289#500—738no—
290#479—7316no—
291#477—738no—
292#476—738no—
293#474—7316no—
294#446—738no—
295#440—7316no—
296C4×SD16—738no—
297C2×#107—732no—
298C2×#107—732no—
299C2×#107—732no—
300C2×#106—732no—
301C2×#106—731no—
302C2×C4⋊C8—731no—
303C2×C4⋊C8—732no—
304C2×#103—732no—
305C2×#103—731no—
306C2×#102—731no—
307C2×#102—732no—
308C2×#97—732no—
309C2×C4×C8—731no—
310C22×Q8—21no—
311C22×Q8—731no—
312C22×Q8—1461no—
313C22×Q8—1462no—
314C22×Q8C22⋊D7311yescharacteristic
315C22×D4C22×D7311yescharacteristic
316C23×C4—21no—
317C23×C4—22no—
318C2×SD16—738no—
319C2×SD16—734no—
320C22×C8—731no—
321C22×C8—731no—
322C22×C8—732no—
323C4⋊Q8—22no—
324C4⋊Q8—738no—
325C4⋊Q8—734no—
326C4⋊Q8—1468no—
327C4⋊Q8—14616no—
328C4⋊Q8—1464no—
329C4⋊Q8—1462no—
330C4⋊Q8C22×D7314yes—
331#125—24no—
332#125C22×D7318yes—
333C22⋊Q8—28no—
334C22⋊Q8—216no—
335C4×Q8—734no—
336C4×Q8—738no—
337C4×Q8—1468no—
338C4×Q8—1464no—
339C4×Q8—1462no—
340C4×Q8C22⋊D7314yes—
341C4×Q8C22×D7314yes—
342C4×D4—24no—
343C4×D4—22no—
344C4×D4C22×D7314yes—
345C4×D4C22×D7318yes—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—731no—
351C2×C4⋊C4—732no—
352C2×C4⋊C4—1461no—
353C2×C4⋊C4—1462no—
354C2×C4⋊C4—1462no—
355C2×C4⋊C4—1461no—
356C2×C4⋊C4C22⋊D7311yescharacteristic
357C2×C4⋊C4C22⋊D7312yes—
358C2×C4⋊C4C22×D7311yescharacteristic
359C2×C4⋊C4C22×D7312yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C42—1462no—
363C2×C42—1461no—
364C2×C42C22×D7312yes—
365C2×C42C22×D7311yescharacteristic
366#107—738no—
367#107—738no—
368#107—738no—
369#106—734no—
370#106—738no—
371C4⋊C8—738no—
372C4⋊C8—734no—
373#103—734no—
374#103—738no—
375#102—734no—
376#102—738no—
377#97—738no—
378C4×C8—734no—
379C24—21no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—82no—
383C2×Q8—5844no—
384C2×Q8—5842no—
385C2×Q8—734no—
386C2×Q8—732no—
387C2×Q8—1464no—
388C2×Q8—1468no—
389C2×Q8—1461no—
390C2×Q8—1462no—
391C2×Q8C2×C22⋊D7314yes—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4Q8×D7314yes—
395C2×D4C23×D7312yes—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—44no—
401C22×C4—84no—
402C22×C4—1462no—
403C22×C4—1461no—
404C22×C4—1461no—
405C22×C4C2×C22⋊D7312yes—
406C22×C4C2×C22⋊D7311yescharacteristic
407C22×C4C2×C22⋊D7312yes—
408C22×C4C2×C22⋊D7311yescharacteristic
409C22×C4C23×D7311yescharacteristic
410SD16—7316no—
411C2×C8—734no—
412C2×C8—732no—
413C2×C8—732no—
414C2×C8—734no—
415C2×C8—1462no—
416C2×C8—1464no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—732no—
423C4⋊C4—734no—
424C4⋊C4—1464no—
425C4⋊C4—1468no—
426C4⋊C4—1461no—
427C4⋊C4—1462no—
428C4⋊C4—1468no—
429C4⋊C4—1464no—
430C4⋊C4C2×C22⋊D7314yes—
431C4⋊C4C2×C22⋊D7318yes—
432C4⋊C4C23×D7312yes—
433C4⋊C4C23×D7314yes—
434C22⋊C4—24no—
435C22⋊C4—28no—
436C42—1468no—
437C42—1464no—
438C42C2×C22⋊D7314yes—
439C42C2×C22⋊D7312yes—
440C42C23×D7312yes—
441C42C23×D7314yes—
442C23—22no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23C22×C22⋊D7311yescharacteristic
447Q8—28no—
448Q8—84no—
449Q8—5848no—
450Q8—5844no—
451Q8—734no—
452Q8—1462no—
453Q8—1464no—
454D4—22no—
455D4—24no—
456D4C2×Q8×D7314yes—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—48no—
465C2×C4—88no—
466C2×C4—2922no—
467C2×C4—2924no—
468C2×C4—1464no—
469C2×C4—1462no—
470C2×C4—1462no—
471C2×C4—1464no—
472C2×C4#1455662114yes—
473C2×C4C22×C22⋊D7314yes—
474C2×C4C22×C22⋊D7312yes—
475C2×C4C22×C22⋊D7314yes—
476C2×C4C22×C22⋊D7312yes—
477C2×C4C22×C22⋊D7311yescharacteristic
478C2×C4C2×Q8×D7312yes—
479C2×C4#1455663912yes—
480C2×C4C24×D7311yescharacteristic
481C8—734no—
482C8—1462no—
483C8—1464no—
484C22—24no—
485C22—48no—
486C22—42no—
487C22—42no—
488C22Q8×C22⋊D7312yes—
489C22C2×#1455662112yes—
490C22#1798939012yes—
491C22C23×C22⋊D7311yescharacteristic
492C4—24no—
493C4—24no—
494C4—22no—
495C4—42no—
496C4—44no—
497C4—2922no—
498C4—2924no—
499C4—1464no—
500C4Q8×C22⋊D7312yes—
501C4C2×#1455662114yes—
502C4C22×Q8×D7311yescharacteristic
503C4C2×#1455663911yescharacteristic
504C2—44no—
505C2#2262284114yesdirect product factor
506C2C2×Q8×C22⋊D7311yescharacteristic
507C2C22×#1455662111yescharacteristic
508C2C2×#1798939011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#554×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenterC2×#554D73
hypocenterC73C2×#3385
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic146C2semidirect
2. #22622841C2direct (N × H)
3. C2×#17988997C2semidirect
4. C2×#17988990C2semidirect
5. C2×#17988891C2semidirect
6. C2×#17988879C2semidirect
9. C2×#17988718C2semidirect
15. Q8×Dic146C22semidirect
23. #17988997C22semidirect
24. #17988990C22semidirect
25. #17988891C22semidirect
26. #17988879C22semidirect
29. #17988718C22semidirect
58. C2×#11924016Q8semidirect
135. #11924016C2×Q8semidirect
251. C2×C73#3385semidirect
267. C73C2×#3385semidirect
505. C2#22622841direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1462
#226228412
C2×#179889972
C2×#179889902
C2×#179888912
C2×#179888792
C2×#179888772
C2×#179888672
C2×#179887182
C2×#179886872
C2×C4×#119240162
C2×#554×C732
C2×#338573