Finite Group Databasebeta

← C2×#22626390 #34030549

Subgroups — C2×#22626390 #34030549

Order 18688.

2750 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#22626390C2116yesmaximal, direct product factor
2C2×(C2×#64408)⋊Q8C212yesmaximal
3C2×#17989538C212yesmaximal
4C2×#17989536C214yesmaximal
5C2×C4×C4⋊#64408C212yesmaximal
6C2×C4×C4⋊#64408C211yesmaximal, characteristic
7C2×#17989523C212yesmaximal
8C2×C42×#64408C211yesmaximal, characteristic
9C2×C4×Q8×D73C211yesmaximal, characteristic
10C22×C4⋊#64408C2214yes—
11C22×C4⋊#64408C2212yes—
12C22×C4×#64408C2211yescharacteristic
13C22×C4×#64408C2212yes—
14C22×C4×#64408C2212yes—
15C22×C4×#64408C2212yes—
16C22×C4×#64408C2211yescharacteristic
17(C2×#64408)⋊Q8C22116yes—
18#17989538C22116yes—
19#17989536C22132yes—
20C4×C4⋊#64408C2218yes—
21C4×C4⋊#64408C22116yes—
22#17989523C22116yes—
23C2×(C2×#64408)⋊C4C2214yes—
24C2×(C2×#64408)⋊C4C2214yes—
25C2×(C2×#64408)⋊C4C2212yes—
26C2×(C2×#64408)⋊C4C2214yes—
27C42×#64408C2218yes—
28C22×Q8×D73C2211yescharacteristic
29C2×Q8×Dic73C411yescharacteristic, complemented
30C2×C4×Dic146C412yescomplemented
31C2×C4×Dic146C411yescharacteristic, complemented
32C2×#14556556C412yescomplemented
33C2×#14556556C411yescharacteristic, complemented
34C4×Q8×D73C418yescomplemented
35C4×Q8×D73C2218yes—
36C2×C4⋊C4×D73C2212yes—
37C2×C4⋊C4×D73C2211yescharacteristic
38C2×C42×D73C2211yescharacteristic
39C2×C42×D73C2212yes—
40C2×C4×Q8×C73C411yescharacteristic, complemented
41C23×#64408C2314yes—
42C23×#64408C2312yes—
43C23×#64408C2312yes—
44C2×C4⋊#64408—216no—
45C2×C4⋊#64408—28no—
46C2×C4⋊#64408C23116yes—
47C2×C4⋊#64408C2318yes—
48C2×C4×#64408—28no—
49C2×C4×#64408—28no—
50C2×C4×#64408—24no—
51C2×C4×#64408—28no—
52C2×C4×#64408Q818yescomplemented
53C2×C4×#64408C2314yes—
54C2×C4×#64408C2318yes—
55C2×C4×#64408C2318yes—
56C2×C4×#64408C2318yes—
57C2×C4×#64408C2314yes—
58(C2×#64408)⋊C4C23116yes—
59(C2×#64408)⋊C4C23116yes—
60(C2×#64408)⋊C4C2318yes—
61(C2×#64408)⋊C4C23116yes—
62C22×Dic146C2×C412yes—
63C22×Dic146C2×C411yescharacteristic
64C2×Q8×D73—28no—
65C2×Q8×D73C2×C414yes—
66C2×Q8×D73C2314yes—
67C22×C4×D73C2311yescharacteristic
68C22×C4×D73C2312yes—
69C22×C4×D73C2312yes—
70C22×C4×D73C2311yescharacteristic
71C22×C4×D73C2311yescharacteristic
72Q8×Dic73C2×C418yescomplemented
73C4×Dic146C2×C4116yescomplemented
74C4×Dic146C2×C418yescomplemented
75#14556556C2×C4116yescomplemented
76#14556556C2×C418yescomplemented
77C2×C4⋊Dic73C2×C412yes—
78C2×C4⋊Dic73C2×C411yescharacteristic
79C2×#11924007C2×C412yes—
80C2×#11924007C2×C412yes—
81C2×#11924007C2×C412yes—
82C4⋊C4×D73C2×C418yes—
83C4⋊C4×D73C2×C414yes—
84C4⋊C4×D73C2318yes—
85C4⋊C4×D73C2314yes—
86C2×C4×Dic73C2×C412yes—
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73C2×C412yes—
89C2×C4×Dic73C2×C411yescharacteristic
90C2×C4×Dic73C2×C412yes—
91C2×C4×Dic73C2×C411yescharacteristic
92C42×D73C2×C418yes—
93C42×D73C2×C414yes—
94C42×D73C2314yes—
95C42×D73C2318yes—
96C22×Q8×C73C2×C411yescharacteristic
97C4×Q8×C73C2×C418yescomplemented
98C2×C4⋊C4×C73C2×C412yes—
99C2×C4⋊C4×C73C2×C411yescharacteristic
100C2×C42×C73C2×C412yes—
101C2×C42×C73C2×C411yescharacteristic
102C22×#64408—24no—
103C22×#64408—24no—
104C22×#64408—24no—
105C22×#64408—24no—
106C22×#64408—28no—
107C22×#64408—28no—
108C22×#64408—28no—
109C22×#64408C2×Q818yes—
110C22×#64408#4118yes—
111C22×#64408C2418yes—
112C22×#64408C2414yes—
113C22×#64408C2414yes—
114C4⋊#64408—232no—
115C4⋊#64408—216no—
116C4×#64408—216no—
117C4×#64408—216no—
118C4×#64408—28no—
119C4×#64408—216no—
120C4×#64408C2×Q8116yescomplemented
121C23×D73C2411yescharacteristic
122C2×Dic146—24no—
123C2×Dic146—28no—
124C2×Dic146C22×C414yes—
125C2×Dic146C22×C418yes—
126Q8×D73—216no—
127Q8×D73—216no—
128C22×Dic73C22×C411yescharacteristic
129C22×Dic73C22×C412yes—
130C22×Dic73C22×C412yes—
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C2×C4×D73—28no—
134C2×C4×D73—24no—
135C2×C4×D73—28no—
136C2×C4×D73C22×C412yes—
137C2×C4×D73C22×C414yes—
138C2×C4×D73C22×C414yes—
139C2×C4×D73C22×C412yes—
140C2×C4×D73C22×C412yes—
141C2×C4×D73C2×Q814yes—
142C2×C4×D73#4118yes—
143C2×C4×D73#4114yes—
144C2×C4×D73C2412yes—
145C2×C4×D73C2414yes—
146C2×C4×D73C2414yes—
147C2×C4×D73C2412yes—
148C2×C4×D73C2412yes—
149C4⋊Dic73C22×C418yes—
150C4⋊Dic73C22×C414yes—
151#11924007C22×C418yes—
152#11924007C22×C418yes—
153#11924007C22×C418yes—
154C4×Dic73C22×C418yes—
155C4×Dic73C22×C414yes—
156C4×Dic73C22×C418yes—
157C4×Dic73C22×C414yes—
158C4×Dic73C22×C418yes—
159C4×Dic73C22×C414yes—
160C2×Q8×C73—24no—
161C2×Q8×C73C22×C414yes—
162C22×C4×C73C22×C411yescharacteristic
163C22×C4×C73C22×C412yes—
164C22×C4×C73C22×C412yes—
165C22×C4×C73C22×C411yescharacteristic
166C22×C4×C73C22×C411yescharacteristic
167C4⋊C4×C73C22×C418yes—
168C4⋊C4×C73C22×C414yes—
169C42×C73C22×C418yes—
170C42×C73C22×C414yes—
171C23×C73C23×C411yescharacteristic
172C22×D73C23×C411yescharacteristic
173C22×D73C22×Q812yes—
174C22×D73C2×#4114yes—
175C22×D73C2×#4112yes—
176C22×D73C2×#4114yes—
177C22×D73C2511yescharacteristic
178C2×#64408—216no—
179C2×#64408—216no—
180C2×#64408—24no—
181C2×#64408—24no—
182C2×#64408—24no—
183C2×#64408—24no—
184C2×#64408—28no—
185C2×#64408—28no—
186C2×#64408—28no—
187C2×#64408—416no—
188C2×#64408C22×Q818yes—
189C2×#64408C2×#4118yes—
190Q8×C73—28no—
191C2×C4×C73—24no—
192C2×C4×C73—22no—
193C2×C4×C73—24no—
194C2×C4×C73#11714yes—
195C2×C4×C73C4×Q814yes—
196C2×C4×C73C23×C412yes—
197C2×C4×C73C23×C414yes—
198C2×C4×C73C23×C414yes—
199C2×C4×C73C23×C412yes—
200C2×C4×C73C23×C412yes—
201C2×Dic73—24no—
202C2×Dic73—22no—
203C2×Dic73—24no—
204C2×Dic73#11714yes—
205C2×Dic73C4×Q814yes—
206C2×Dic73C23×C412yes—
207C2×Dic73C23×C414yes—
208C2×Dic73C23×C414yes—
209C2×Dic73C23×C412yes—
210C2×Dic73C23×C412yes—
211C4×D73—28no—
212C4×D73—24no—
213C4×D73—28no—
214C4×D73—28no—
215C4×D73—24no—
216C4×D73—28no—
217C4×D73#11718yes—
218C4×D73C4×Q818yes—
219C4×D73C22×Q814yes—
220C4×D73C2×#4118yes—
221C4×D73C2×#4114yes—
222Dic146—28no—
223Dic146—216no—
224C22×C73C2×#11712yes—
225C22×C73C2×C4×Q812yes—
226C22×C73C4×#4112yes—
227C22×C73C24×C411yescharacteristic
228C2×D73C2×#11712yes—
229C2×D73C2×C4×Q812yes—
230C2×D73C4×#4112yes—
231C2×D73#52918yes—
232C2×D73#53318yes—
233C2×D73C23×Q811yescharacteristic
234C2×D73C22×#4111yescharacteristic
235C2×D73C22×#4112yes—
236C2×D73C22×#4112yes—
237#64408—28no—
238#64408—28no—
239#64408—48no—
240C4×C73—24no—
241C4×C73—22no—
242C4×C73—24no—
243C4×C73C2×#11714yes—
244C4×C73C2×C4×Q814yes—
245Dic73—24no—
246Dic73—22no—
247Dic73—24no—
248Dic73C2×#11714yes—
249Dic73C2×C4×Q814yes—
250C2×#2295—731nomaximal, Sylow
251C2×C73#229514yescomplemented
252C2×C73C22×#11711yescharacteristic
253C2×C73C22×C4×Q811yescharacteristic
254C2×C73C2×C4×#4111yescharacteristic
255D73#229514yescomplemented
256D73C2×#52912yes—
257D73C2×#53312yes—
258C22×C4×Q8—731no—
259#2295—7316no—
260C2×#391—732no—
261C2×#389—732no—
262C2×#387—734no—
263C2×C4×C4⋊C4—732no—
264C2×C4×C4⋊C4—731no—
265C2×#376—732no—
266C2×C43—731no—
267C73C2×#229511yescharacteristic, complemented, Sylow
268C23×Q8—731no—
269C2×C4×Q8—731no—
270C2×C4×Q8—738no—
271C2×C4×Q8—738no—
272C2×C4×Q8—732no—
273C2×C4×Q8—731no—
274C2×C4×Q8—732no—
275C2×C4×Q8—731no—
276C2×C4×Q8#6440811yescharacteristic, complemented
277C22×C4⋊C4—732no—
278C22×C4⋊C4—731no—
279C22×C4⋊C4—734no—
280C22×C4⋊C4—732no—
281C22×C42—731no—
282C22×C42—731no—
283C22×C42—732no—
284C22×C42—731no—
285C22×C42—732no—
286C22×C42—732no—
287C22×C42—732no—
288#391—7316no—
289#389—7316no—
290#387—7332no—
291C4×C4⋊C4—738no—
292C4×C4⋊C4—7316no—
293#376—7316no—
294C2×#95—734no—
295C2×#95—734no—
296C2×#95—732no—
297C2×#95—734no—
298C43—738no—
299C22×Q8—734no—
300C22×Q8—734no—
301C22×Q8—732no—
302C22×Q8—731no—
303C22×Q8—1468no—
304C22×Q8C2×#6440811yescharacteristic
305C23×C4—731no—
306C23×C4—731no—
307C23×C4—732no—
308C23×C4—732no—
309C23×C4—732no—
310C23×C4—731no—
311C23×C4—732no—
312C23×C4—734no—
313C4×Q8—738no—
314C4×Q8—7316no—
315C4×Q8—738no—
316C4×Q8—7316no—
317C4×Q8—738no—
318C4×Q8C2×#6440818yescomplemented
319C2×C4⋊C4—732no—
320C2×C4⋊C4—732no—
321C2×C4⋊C4—732no—
322C2×C4⋊C4—732no—
323C2×C4⋊C4—731no—
324C2×C4⋊C4—734no—
325C2×C4⋊C4—738no—
326C2×C4⋊C4—738no—
327C2×C4⋊C4—734no—
328C2×C4⋊C4—7316no—
329C2×C4⋊C4—738no—
330C2×C4⋊C4—1468no—
331C2×C4⋊C4—14616no—
332C2×C4⋊C4C2×#6440812yes—
333C2×C4⋊C4C2×#6440811yescharacteristic
334C2×C42—734no—
335C2×C42—738no—
336C2×C42—731no—
337C2×C42—732no—
338C2×C42—731no—
339C2×C42—732no—
340C2×C42—731no—
341C2×C42—732no—
342C2×C42—734no—
343C2×C42—738no—
344C2×C42—734no—
345C2×C42—738no—
346C2×C42—734no—
347C2×C42—738no—
348C2×C42—738no—
349C2×C42—738no—
350C2×C42—1468no—
351C2×C42—1464no—
352C2×C42—1468no—
353C2×C42—1468no—
354C2×C42C2×#6440812yes—
355C2×C42C2×#6440811yescharacteristic
356#95—7316no—
357#95—7316no—
358#95—738no—
359#95—7316no—
360C24—731no—
361C2×Q8—24no—
362C2×Q8—738no—
363C2×Q8—734no—
364C2×Q8—1468no—
365C2×Q8—14616no—
366C2×Q8—14616no—
367C2×Q8—1464no—
368C2×Q8C22×#6440814yes—
369C22×C4—731no—
370C22×C4—731no—
371C22×C4—734no—
372C22×C4—734no—
373C22×C4—732no—
374C22×C4—732no—
375C22×C4—732no—
376C22×C4—732no—
377C22×C4—738no—
378C22×C4—738no—
379C22×C4—738no—
380C22×C4—734no—
381C22×C4—734no—
382C22×C4—732no—
383C22×C4—731no—
384C22×C4—732no—
385C22×C4—734no—
386C22×C4—732no—
387C22×C4—734no—
388C22×C4—734no—
389C22×C4—732no—
390C22×C4—734no—
391C22×C4—738no—
392C22×C4—1464no—
393C22×C4—1468no—
394C22×C4—1468no—
395C22×C4—1468no—
396C22×C4—1468no—
397C22×C4—1468no—
398C22×C4—1464no—
399C22×C4—1464no—
400C22×C4—1464no—
401C22×C4—1464no—
402C22×C4C22×#6440812yes—
403C22×C4C22×#6440811yescharacteristic
404C22×C4C22×#6440812yes—
405C22×C4C22×#6440811yescharacteristic
406C22×C4C22×#6440811yescharacteristic
407C4⋊C4—738no—
408C4⋊C4—738no—
409C4⋊C4—738no—
410C4⋊C4—738no—
411C4⋊C4—734no—
412C4⋊C4—14616no—
413C4⋊C4—14632no—
414C4⋊C4C22×#6440818yes—
415C4⋊C4C22×#6440814yes—
416C42—7316no—
417C42—734no—
418C42—738no—
419C42—734no—
420C42—738no—
421C42—734no—
422C42—738no—
423C42—14616no—
424C42—14616no—
425C42—1468no—
426C42—14616no—
427C42C22×#6440818yes—
428C42C22×#6440814yes—
429C23—734no—
430C23—732no—
431C23—732no—
432C23—734no—
433C23—731no—
434C23—731no—
435C23C23×#6440811yescharacteristic
436Q8—28no—
437Q8—14616no—
438Q8—1468no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—29216no—
443C2×C4—734no—
444C2×C4—734no—
445C2×C4—732no—
446C2×C4—732no—
447C2×C4—738no—
448C2×C4—738no—
449C2×C4—734no—
450C2×C4—734no—
451C2×C4—738no—
452C2×C4—738no—
453C2×C4—738no—
454C2×C4—732no—
455C2×C4—734no—
456C2×C4—734no—
457C2×C4—1464no—
458C2×C4—1468no—
459C2×C4—1468no—
460C2×C4—1468no—
461C2×C4—1464no—
462C2×C4—1468no—
463C2×C4—1468no—
464C2×C4—1468no—
465C2×C4—1468no—
466C2×C4—1464no—
467C2×C4—1464no—
468C2×C4—1464no—
469C2×C4—1464no—
470C2×C4—14616no—
471C2×C4—14616no—
472C2×C4—1464no—
473C2×C4—1462no—
474C2×C4—1464no—
475C2×C4#1455667014yes—
476C2×C4Q8×#6440814yes—
477C2×C4C23×#6440812yes—
478C2×C4C23×#6440814yes—
479C2×C4C23×#6440814yes—
480C2×C4C23×#6440812yes—
481C2×C4C23×#6440812yes—
482C22—738no—
483C22—738no—
484C22—732no—
485C22—732no—
486C22—731no—
487C22—731no—
488C22—732no—
489C22—732no—
490C22—732no—
491C22C2×#1455667012yes—
492C22C2×Q8×#6440812yes—
493C22#41×#6440812yes—
494C22C24×#6440811yescharacteristic
495C4—24no—
496C4—22no—
497C4—24no—
498C4—2928no—
499C4—734no—
500C4—734no—
501C4—1468no—
502C4—1468no—
503C4—1464no—
504C4—1462no—
505C4—1464no—
506C4C2×#1455667014yes—
507C4C2×Q8×#6440814yes—
508C2—732no—
509C2—732no—
510C2—734no—
511C2#2262639014yesdirect product factor
512C2C22×#1455667011yescharacteristic
513C2C22×Q8×#6440811yescharacteristic
514C2C2×#41×#6440811yescharacteristic
subgroupG/H
centerC22×C4C22×#64408
commutatorC22×C73C24×C4
FrattiniC22C24×#64408
FittingC2×C4×Q8×C73C4
socleC23×C73C23×C4
prime radicalC23×D73C24
cosocleC22×D73C25
hypercenterC2×C4×Q8#64408
hypocenterC73C2×#2295
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22626390C2direct (N × H)
29. C2×Q8×Dic73C4semidirect
30. C2×C4×Dic146C4semidirect
31. C2×C4×Dic146C4semidirect
32. C2×#14556556C4semidirect
33. C2×#14556556C4semidirect
34. C4×Q8×D73C4semidirect
40. C2×C4×Q8×C73C4semidirect
52. C2×C4×#64408Q8semidirect
72. Q8×Dic73C2×C4semidirect
73. C4×Dic146C2×C4semidirect
74. C4×Dic146C2×C4semidirect
75. #14556556C2×C4semidirect
76. #14556556C2×C4semidirect
97. C4×Q8×C73C2×C4semidirect
120. C4×#64408C2×Q8semidirect
251. C2×C73#2295semidirect
255. D73#2295semidirect
267. C73C2×#2295semidirect
276. C2×C4×Q8#64408semidirect
318. C4×Q8C2×#64408semidirect
511. C2#22626390direct (N × H)

Maximal subgroups

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

subgroupindex
#226263902
C2×(C2×#64408)⋊Q82
C2×#179895382
C2×#179895362
C2×C4×C4⋊#644082
C2×#179895232
C2×C42×#644082
C2×C4×Q8×D732
C2×#229573