Finite Group Databasebeta

← C2×#22626726 #34031485

Subgroups — C2×#22626726 #34031485

Order 18688.

1938 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626726C2116yesmaximal, direct product factor
2C2×#17989639C212yesmaximal, complemented
3C2×#17989638C211yesmaximal, characteristic, complemented
4C2×#17989637C211yesmaximal, characteristic, complemented
5C2×#17989637C212yesmaximal, complemented
6C2×#17989636C211yesmaximal, characteristic, complemented
7C2×Q8×#10596833C211yesmaximal, characteristic, complemented
8C2×#17989593C212yesmaximal
9C22×C4⋊#10596833C211yesmaximal, characteristic
10C2×#17989551C212yesmaximal
11C2×C4×#11924037C211yesmaximal, characteristic
12C2×#17989139C211yesmaximal, characteristic
13C22×#11924037C2211yescharacteristic
14C22×#11924037C2211yescharacteristic
15C23×#10596833C2212yes—
16#17989639C22116yescomplemented
17#17989638C2218yescomplemented
18#17989637C2218yescomplemented
19#17989637C22116yescomplemented
20#17989636C2218yescomplemented
21Q8×#10596833C2218yescomplemented
22#17989593C22116yes—
23C2×#14556658C2212yes—
24C2×#14556658C2212yes—
25C2×C4⋊#10596833C2211yescharacteristic
26C2×C4⋊#10596833C2211yescharacteristic
27C2×C4⋊#10596833C2212yes—
28C2×C4⋊#10596833C2218yes—
29C2×C4⋊#10596833C2212yes—
30C2×C4⋊#10596833C2212yes—
31#17989551C22116yes—
32C4×#11924037C2218yes—
33C2×#14556646C2211yescharacteristic
34C2×#14556646C2212yes—
35C2×#14556646C2212yes—
36C2×#14556646C2211yescharacteristic
37C2×C4×#10596833C2211yescharacteristic
38C2×C4×#10596833C2212yes—
39C2×C4×#10596833C2212yes—
40C2×C4×#10596833C2211yescharacteristic
41#17989139C418yes—
42#17989139C2218yes—
43C2×#14556590C411yescharacteristic
44C2×#14556589C411yescharacteristic
45C2×#14556581C412yes—
46C2×#14556570C411yescharacteristic
47C2×#14556568C412yes—
48C2×Q8×Dic73C2211yescharacteristic
49C2×#14556556C2211yescharacteristic
50C2×#14556556C2212yes—
51C2×#14556547C2212yes—
52C22×C4×Dic73C2211yescharacteristic
53C2×C22⋊Q8×C73C411yescharacteristic
54C2×#11924037—24no—
55C2×#11924037Q818yescomplemented
56C2×#11924037C2314yes—
57C2×#11924037C2314yes—
58C22×#10596833—28no—
59C22×#10596833C2312yes—
60C22×#10596833C2312yes—
61C22×#10596833C2318yes—
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66#14556658C2318yes—
67#14556658C2318yes—
68C4⋊#10596833C2314yes—
69C4⋊#10596833C2318yes—
70C4⋊#10596833C2318yes—
71C4⋊#10596833C2314yes—
72C4⋊#10596833C2318yes—
73#14556646C2314yes—
74#14556646C2318yes—
75#14556646C2318yes—
76#14556646C2314yes—
77C4×#10596833C2314yes—
78C4×#10596833C2318yes—
79C4×#10596833C2318yes—
80C4×#10596833C2314yes—
81C22×Dic146C2×C411yescharacteristic
82C22×Dic146C2×C412yes—
83C23×Dic73C2×C411yescharacteristic
84C23×Dic73C2311yescharacteristic
85#14556590C2×C418yes—
86#14556589C2×C418yes—
87#14556581C2×C4116yes—
88#14556570C2×C418yes—
89#14556568C2×C4116yes—
90Q8×Dic73C2×C414yes—
91Q8×Dic73C2314yes—
92#14556556C2×C418yes—
93#14556556C2×C414yes—
94#14556556C2314yes—
95#14556556C2318yes—
96#14556547C2×C418yes—
97#14556547C2318yes—
98C2×C4⋊Dic73C2×C411yescharacteristic
99C2×C4⋊Dic73C2×C412yes—
100C2×#11924007C2×C412yes—
101C2×#11924007C2×C412yes—
102C2×#11924007C2×C412yes—
103C2×#11924007C2×C412yes—
104C2×#11924007C2×C412yes—
105C2×C22⋊Dic73C2×C412yes—
106C2×C4×Dic73C2×C411yescharacteristic
107C2×C4×Dic73C2×C414yes—
108C2×C4×Dic73C2×C411yescharacteristic
109C2×C4×Dic73C2314yes—
110C2×C4×Dic73C2311yescharacteristic
111C2×C4×Dic73C2312yes—
112C2×C4×Dic73C2312yes—
113C2×C4×Dic73C2311yescharacteristic
114C22×Q8×C73C2×C411yescharacteristic
115C23×C4×C73C2×C411yescharacteristic
116C22⋊Q8×C73C2×C418yes—
117C2×C4⋊C4×C73C2×C411yescharacteristic
118C2×C4⋊C4×C73C2×C412yes—
119C2×C22⋊C4×C73C2×C412yes—
120#11924037—28no—
121#11924037C2×Q8116yescomplemented
122C2×#10596833—24no—
123C2×#10596833—24no—
124C2×#10596833—216no—
125C2×#10596833—22no—
126C2×#10596833—22no—
127C2×#10596833C2×Q814yes—
128C2×#10596833C2×Q814yes—
129C2×#10596833C2414yes—
130C2×#10596833C2414yes—
131C2×#10596833C2412yes—
132C2×#10596833C2412yes—
133C2×#10596833C2412yes—
134C2×#10596833C2412yes—
135C2×Dic146—24no—
136C2×Dic146—44no—
137C2×Dic146C22×C414yes—
138C2×Dic146C22×C418yes—
139C22×Dic73—42no—
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C22×Dic73C22×C412yes—
143C22×Dic73C22×C412yes—
144C22×Dic73C22×C414yes—
145C22×Dic73C22×C411yescharacteristic
146C22×Dic73C22×C412yes—
147C22×Dic73C2×Q814yes—
148C22×Dic73#4114yes—
149C22×Dic73C2411yescharacteristic
150C22×Dic73C2412yes—
151C4⋊Dic73C22×C418yes—
152C4⋊Dic73C22×C414yes—
153#11924007C22×C418yes—
154#11924007C22×C418yes—
155#11924007C22×C418yes—
156#11924007C22×C418yes—
157#11924007C22×C418yes—
158C22⋊Dic73C22×C418yes—
159C4×Dic73C22×C412yes—
160C4×Dic73C22×C412yes—
161C4×Dic73C22×C414yes—
162C4×Dic73C22×C414yes—
163C4×Dic73C22×C414yes—
164C4×Dic73C22×C414yes—
165C4×Dic73C2412yes—
166C4×Dic73C2412yes—
167C4×Dic73C2414yes—
168C4×Dic73C2414yes—
169C24×C73C22×C411yescharacteristic
170C2×Q8×C73—24no—
171C2×Q8×C73C22×C414yes—
172C22×C4×C73—24no—
173C22×C4×C73C22×C412yes—
174C22×C4×C73C22×C412yes—
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C22×C414yes—
177C22×C4×C73C22×C411yescharacteristic
178C4⋊C4×C73C22×C418yes—
179C4⋊C4×C73C22×C414yes—
180C22⋊C4×C73C22×C418yes—
181C23×C73—22no—
182C23×C73—22no—
183C23×C73C4×Q814yes—
184C23×C73C23×C412yes—
185C23×C73C23×C411yescharacteristic
186Q8×C73—28no—
187C2×C4×C73—22no—
188C2×C4×C73—24no—
189C2×C4×C73—24no—
190C2×C4×C73—28no—
191C2×C4×C73#13114yes—
192C2×C4×C73C23×C414yes—
193C2×C4×C73C23×C414yes—
194C2×C4×C73C23×C412yes—
195C2×C4×C73C23×C412yes—
196C2×Dic73—24no—
197C2×Dic73—22no—
198C2×Dic73—22no—
199C2×Dic73—24no—
200C2×Dic73—44no—
201C2×Dic73C4×Q818yes—
202C2×Dic73C4×Q814yes—
203C2×Dic73C23×C412yes—
204C2×Dic73C23×C412yes—
205C2×Dic73C23×C414yes—
206C2×Dic73C23×C414yes—
207C2×Dic73C23×C411yescharacteristic
208C2×Dic73C23×C412yes—
209C2×Dic73C22×Q812yes—
210C2×Dic73C22×Q814yes—
211C2×Dic73C2×#4112yes—
212C2×Dic73C2×#4114yes—
213C2×Dic73C2511yescharacteristic
214Dic146—28no—
215Dic146—48no—
216#10596833—24no—
217#10596833—24no—
218#10596833—22no—
219#10596833—22no—
220#10596833C22×Q814yes—
221#10596833C22×Q814yes—
222C22×C73—28no—
223C22×C73—22no—
224C22×C73—22no—
225C22×C73C2×C4×Q814yes—
226C22×C73C2×C4×Q812yes—
227C22×C73C2×#13112yes—
228C22×C73#56812yes—
229C22×C73C24×C411yescharacteristic
230C4×C73—22no—
231C4×C73—24no—
232C4×C73—24no—
233C4×C73C2×#13114yes—
234Dic73—24no—
235Dic73—22no—
236Dic73—24no—
237Dic73—22no—
238Dic73C2×C4×Q812yes—
239Dic73C2×C4×Q814yes—
240Dic73C23×Q811yescharacteristic
241Dic73C22×#4111yescharacteristic
242C2×#2962—731nomaximal, Sylow
243C2×C73—24no—
244C2×C73#296214yescomplemented
245C2×C73C22×C4×Q811yescharacteristic
246C2×C73C22×#13111yescharacteristic
247C2×C73C2×#56811yescharacteristic
248C2×#533—731no—
249#2962—7316no—
250C2×#446—731no—
251C2×#446—732no—
252C2×#446—731no—
253C2×#446—732no—
254C2×C8×Q8—731no—
255C2×C8×Q8—731no—
256C2×#424—732no—
257C22×C4⋊C8—731no—
258C2×#405—732no—
259C2×C4×M16—731no—
260C73C2×#296211yescharacteristic, complemented, Sylow
261C22×M16—731no—
262C22×M16—731no—
263C23×C8—732no—
264#533—738no—
265#533—738no—
266C2×C4⋊Q8—731no—
267C2×C4⋊Q8—731no—
268C2×#125—732no—
269C2×C22⋊Q8—731no—
270C2×C22⋊Q8—732no—
271C2×C22⋊Q8#6440811yescharacteristic
272C2×C4×Q8—731no—
273C2×C4×Q8—731no—
274C2×C4×Q8—732no—
275C2×#117—732no—
276C22×C42—731no—
277#446—738no—
278#446—7316no—
279#446—738no—
280#446—7316no—
281C8×Q8—738no—
282C8×Q8—738no—
283#424—7316no—
284C2×C4⋊C8—731no—
285C2×C4⋊C8—731no—
286C2×C4⋊C8—732no—
287C2×C4⋊C8—738no—
288C2×C4⋊C8—732no—
289C2×C4⋊C8—732no—
290C2×C4⋊C8—732no—
291C2×C4⋊C8—732no—
292#405—7316no—
293C4×M16—738no—
294C2×#97—731no—
295C2×#97—731no—
296C2×#97—732no—
297C2×#97—732no—
298C2×C4×C8—731no—
299C2×C4×C8—731no—
300C2×C4×C8—732no—
301C2×C4×C8—732no—
302C22×Q8—732no—
303C22×Q8—731no—
304C22×Q8C2×#6440811yescharacteristic
305C23×C4—731no—
306C23×C4—731no—
307C23×C4C2×#6440811yescharacteristic
308C2×M16—738no—
309C2×M16—734no—
310C2×M16—734no—
311C2×M16—1464no—
312C22×C8—731no—
313C22×C8—731no—
314C22×C8—732no—
315C22×C8—732no—
316C22×C8—738no—
317C22×C8—731no—
318C22×C8—731no—
319C22×C8—1468no—
320C4⋊Q8—738no—
321C4⋊Q8—738no—
322#125—7316no—
323C22⋊Q8—738no—
324C22⋊Q8—7316no—
325C22⋊Q8C2×#6440818yes—
326C4×Q8—734no—
327C4×Q8—734no—
328C4×Q8—734no—
329C4×Q8—738no—
330C4×Q8—734no—
331C4×Q8—738no—
332#117—738no—
333#117—738no—
334C2×C4⋊C4—732no—
335C2×C4⋊C4—732no—
336C2×C4⋊C4—732no—
337C2×C4⋊C4—732no—
338C2×C4⋊C4—732no—
339C2×C4⋊C4—732no—
340C2×C4⋊C4—731no—
341C2×C4⋊C4C2×#6440812yes—
342C2×C4⋊C4C2×#6440811yescharacteristic
343C2×C22⋊C4—732no—
344C2×C22⋊C4C2×#6440812yes—
345C2×C42—732no—
346C2×C42—732no—
347C2×C42—731no—
348C2×C42—731no—
349C2×C42—731no—
350C2×C42—731no—
351C2×C42—734no—
352C2×C42—734no—
353C4⋊C8—734no—
354C4⋊C8—734no—
355C4⋊C8—738no—
356C4⋊C8—738no—
357C4⋊C8—738no—
358C4⋊C8—738no—
359C4⋊C8—738no—
360#97—734no—
361#97—734no—
362#97—738no—
363#97—738no—
364C4×C8—734no—
365C4×C8—734no—
366C4×C8—738no—
367C4×C8—738no—
368C24C22×#6440811yescharacteristic
369C2×Q8—24no—
370C2×Q8—2924no—
371C2×Q8—738no—
372C2×Q8—734no—
373C2×Q8—1464no—
374C2×Q8C22×#6440814yes—
375C22×C4—24no—
376C22×C4—2922no—
377C22×C4—734no—
378C22×C4—734no—
379C22×C4—734no—
380C22×C4—732no—
381C22×C4—732no—
382C22×C4—731no—
383C22×C4—731no—
384C22×C4—731no—
385C22×C4—731no—
386C22×C4—732no—
387C22×C4—732no—
388C22×C4C22×#6440814yes—
389C22×C4C22×#6440811yescharacteristic
390C22×C4C22×#6440812yes—
391C22×C4C22×#6440812yes—
392C22×C4C22×#6440811yescharacteristic
393M16—7316no—
394M16—1468no—
395C2×C8—734no—
396C2×C8—734no—
397C2×C8—732no—
398C2×C8—732no—
399C2×C8—734no—
400C2×C8—732no—
401C2×C8—732no—
402C2×C8—734no—
403C2×C8—1464no—
404C2×C8—1464no—
405C2×C8—14616no—
406C2×C8—1462no—
407C2×C8—1462no—
408C4⋊C4—738no—
409C4⋊C4—738no—
410C4⋊C4—738no—
411C4⋊C4—738no—
412C4⋊C4—738no—
413C4⋊C4—738no—
414C4⋊C4—734no—
415C4⋊C4C22×#6440818yes—
416C4⋊C4C22×#6440814yes—
417C22⋊C4—738no—
418C22⋊C4C22×#6440818yes—
419C42—732no—
420C42—732no—
421C42—734no—
422C42—734no—
423C42—734no—
424C42—734no—
425C42—732no—
426C42—734no—
427C42—734no—
428C42—732no—
429C23—22no—
430C23—22no—
431C23Q8×#6440814yes—
432C23C23×#6440812yes—
433C23C23×#6440811yescharacteristic
434Q8—28no—
435Q8—2928no—
436Q8—1468no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—2924no—
442C2×C4—732no—
443C2×C4—732no—
444C2×C4—734no—
445C2×C4—734no—
446C2×C4—738no—
447C2×C4—734no—
448C2×C4—731no—
449C2×C4—731no—
450C2×C4—732no—
451C2×C4—732no—
452C2×C4—732no—
453C2×C4—734no—
454C2×C4—734no—
455C2×C4—1464no—
456C2×C4—1462no—
457C2×C4—1462no—
458C2×C4—1464no—
459C2×C4#1455667814yes—
460C2×C4C23×#6440812yes—
461C2×C4C23×#6440814yes—
462C2×C4C23×#6440814yes—
463C2×C4C23×#6440812yes—
464C8—734no—
465C8—734no—
466C8—1464no—
467C8—1464no—
468C8—1462no—
469C8—1462no—
470C22—22no—
471C22—22no—
472C22—28no—
473C22C2×Q8×#6440814yes—
474C22C2×Q8×#6440812yes—
475C22C2×#1455667812yes—
476C22#1798966612yes—
477C22C24×#6440811yescharacteristic
478C4—22no—
479C4—24no—
480C4—24no—
481C4—731no—
482C4—731no—
483C4—732no—
484C4—734no—
485C4—1464no—
486C4—1462no—
487C4—1462no—
488C4—1464no—
489C4C2×#1455667814yes—
490C2—24no—
491C2#2262672614yesdirect product factor
492C2C22×Q8×#6440811yescharacteristic
493C2C22×#1455667811yescharacteristic
494C2C2×#1798966611yescharacteristic
subgroupG/H
centerC23C23×#64408
commutatorC22×C73C24×C4
FrattiniC22C24×#64408
FittingC2×C22⋊Q8×C73C4
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
cosocleC2×Dic73C25
hypercenterC2×C22⋊Q8#64408
hypocenterC73C2×#2962
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626726C2direct (N × H)
2. C2×#17989639C2semidirect
3. C2×#17989638C2semidirect
4. C2×#17989637C2semidirect
5. C2×#17989637C2semidirect
6. C2×#17989636C2semidirect
7. C2×Q8×#10596833C2semidirect
16. #17989639C22semidirect
17. #17989638C22semidirect
18. #17989637C22semidirect
19. #17989637C22semidirect
20. #17989636C22semidirect
21. Q8×#10596833C22semidirect
55. C2×#11924037Q8semidirect
121. #11924037C2×Q8semidirect
244. C2×C73#2962semidirect
260. C73C2×#2962semidirect
491. C2#22626726direct (N × H)

Maximal subgroups

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

subgroupindex
#226267262
C2×#179896392
C2×#179896382
C2×#179896372
C2×#179896362
C2×Q8×#105968332
C2×#179895932
C22×C4⋊#105968332
C2×#179895512
C2×C4×#119240372
C2×#179891392
C2×#296273