Finite Group Databasebeta

← C2×#22620482 #33955639

Subgroups — C2×#22620482 #33955639

Order 18688.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989142C211yesmaximal, characteristic
2C2×#125×D73C211yesmaximal, characteristic
3#22620482C2116yesmaximal, direct product factor
4C2×#17988997C211yesmaximal, characteristic, complemented
5C2×#17988996C211yesmaximal, characteristic, complemented
6C2×#17988993C211yesmaximal, characteristic, complemented
7C2×#17988988C212yesmaximal, complemented
8C2×#17988987C212yesmaximal, complemented
9C2×#17988628C212yesmaximal
10C2×#17988626C212yesmaximal
11C2×#17988539C211yesmaximal, characteristic
12C2×#501×C73C211yesmaximal, characteristic, complemented
13C22×#11924011C2212yes—
14#17989142—24no—
15#17989142C2218yes—
16C2×#14556591C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556582C2211yescharacteristic
19C2×#14556582C2212yes—
20C2×#14556582C2211yescharacteristic
21C2×#14556581—21no—
22C2×#14556581—21no—
23#125×D73—24no—
24#125×D73C2218yes—
25C2×#14556571—21no—
26C2×#14556567—21no—
27C2×Q8×Dic73—21no—
28C2×#14556556—21no—
29C2×#14556549C2212yes—
30C2×C4⋊C4×D73—21no—
31C2×C4⋊C4×D73—21no—
32C2×C4⋊C4×D73C2211yescharacteristic
33C2×C4⋊C4×D73C2211yescharacteristic
34C2×C42×D73C2211yescharacteristic
35#17988997C2218yescomplemented
36#17988996C2218yescomplemented
37#17988993C2218yescomplemented
38#17988988C22116yescomplemented
39#17988987C22116yescomplemented
40#17988628C22116yes—
41#17988626C22116yes—
42C2×#14556525C2212yes—
43C2×#14556524C2212yes—
44C2×#14556524C2212yes—
45C2×#14556523C2212yes—
46C2×#14556522C2212yes—
47C2×#14556522C2212yes—
48#17988539C2218yes—
49C2×#14556502C2211yescharacteristic
50C2×C8×Dic73C2212yes—
51C2×C4⋊Q8×C73C2211yescharacteristic
52C2×#125×C73C2211yescharacteristic
53#501×C73C2218yescomplemented
54C2×#107×C73C2212yes—
55C2×#106×C73C2212yes—
56C2×#97×C73C2211yescharacteristic
57C22×Dic146—21no—
58C22×C4×D73—21no—
59C22×C4×D73—21no—
60C22×C4×D73D411yescharacteristic
61C22×C4×D73D411yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C2×#11924011Q8116yescomplemented
64C2×#11924011C2318yes—
65C22×#10596831C2312yes—
66#14556591—22no—
67#14556591C2314yes—
68#14556583—22no—
69#14556583C2314yes—
70#14556582—24no—
71#14556582—22no—
72#14556582—22no—
73#14556582C2318yes—
74#14556582C2314yes—
75#14556582C2314yes—
76#14556581—28no—
77#14556581—28no—
78#14556571—28no—
79#14556567—28no—
80Q8×Dic73—28no—
81#14556556—28no—
82#14556549—24no—
83#14556549C2318yes—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73—21no—
86C2×C4⋊Dic73—22no—
87C2×C4⋊Dic73—21no—
88C2×C4⋊Dic73C2311yescharacteristic
89C2×C4⋊Dic73C2312yes—
90C2×C4⋊Dic73C2311yescharacteristic
91C2×#11924007—21no—
92C2×#11924007—21no—
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—22no—
96C2×#11924007—21no—
97C2×#11924007—21no—
98C4⋊C4×D73—28no—
99C4⋊C4×D73—28no—
100C4⋊C4×D73—22no—
101C4⋊C4×D73—22no—
102C4⋊C4×D73C2314yes—
103C4⋊C4×D73C2314yes—
104C2×#11924005—22no—
105C2×C4×Dic73—22no—
106C2×C4×Dic73D411yescharacteristic
107C2×C4×Dic73D411yescharacteristic
108C2×C4×Dic73C2311yescharacteristic
109C42×D73D414yes—
110C42×D73C2314yes—
111#14556525C2318yes—
112#14556524C2318yes—
113#14556524C2318yes—
114#14556523C2318yes—
115#14556522C2318yes—
116#14556522C2318yes—
117#14556502C2314yes—
118C8×Dic73C2318yes—
119C22×Q8×C73—21no—
120C22×C8×C73C2312yes—
121C4⋊Q8×C73—22no—
122C4⋊Q8×C73C2314yes—
123#125×C73—22no—
124#125×C73C2314yes—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73—21no—
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2312yes—
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C42×C73C2311yescharacteristic
132#107×C73C2318yes—
133#106×C73C2318yes—
134#97×C73C2314yes—
135C23×D73C2×D411yescharacteristic
136C2×Dic146—24no—
137C2×Dic146—44no—
138C22×Dic73—22no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C2×C4×D73—24no—
145C2×C4×D73—24no—
146C2×C4×D73—42no—
147C2×C4×D73—42no—
148C2×C4×D73—44no—
149C2×C4×D73—44no—
150C2×C4×D73C2×D414yes—
151C2×C4×D73C2×D414yes—
152C2×C4×D73C2×D412yes—
153C2×C4×D73C2×Q818yes—
154C2×C4×D73C2412yes—
155#11924011C2×Q8132yescomplemented
156C2×#10596831C2×Q818yes—
157C2×#10596831C2414yes—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—24no—
161C4⋊Dic73—22no—
162C4⋊Dic73—24no—
163C4⋊Dic73—24no—
164C4⋊Dic73—28no—
165C4⋊Dic73C2412yes—
166C4⋊Dic73C2414yes—
167C4⋊Dic73C2412yes—
168#11924007—24no—
169#11924007—24no—
170#11924007—24no—
171#11924007—24no—
172#11924007—24no—
173#11924007—24no—
174#11924007—28no—
175#11924005—28no—
176C4×Dic73—28no—
177C4×Dic73C2×D412yes—
178C4×Dic73C2×D414yes—
179C4×Dic73C2×D414yes—
180C4×Dic73C2412yes—
181C2×Q8×C73—24no—
182C2×Q8×C73—44no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—22no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C2×C8×C73C2×Q818yes—
190C2×C8×C73C2414yes—
191C4⋊C4×C73—24no—
192C4⋊C4×C73—24no—
193C4⋊C4×C73—24no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—22no—
197C4⋊C4×C73C2412yes—
198C4⋊C4×C73C2414yes—
199C4⋊C4×C73C2412yes—
200C42×C73C2×D412yes—
201C42×C73C2412yes—
202C23×C73C22×D411yescharacteristic
203C22×D73—22no—
204C22×D73—22no—
205C22×D73C4⋊Q814yes—
206C22×D73C22×D412yes—
207Q8×C73—48no—
208C2×C4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—24no—
213C2×C4×C73—44no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73C22×D412yes—
217C2×C4×C73C22×D412yes—
218C2×C4×C73C22×D411yescharacteristic
219C2×C4×C73C22×Q814yes—
220C2×C4×C73C2511yescharacteristic
221C2×Dic73—22no—
222C2×Dic73—22no—
223C2×Dic73—24no—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—42no—
227C2×Dic73—42no—
228C2×Dic73—44no—
229C2×Dic73C4⋊Q814yes—
230C2×Dic73C22×D412yes—
231C2×Dic73C22×D412yes—
232C2×Dic73C22×D412yes—
233C4×D73—44no—
234C4×D73—44no—
235C4×D73—48no—
236C4×D73—48no—
237C4×D73C4⋊Q818yes—
238C4×D73C22×Q818yes—
239Dic146—48no—
240C8×C73C22×Q818yes—
241#10596831C22×Q818yes—
242C22×C73C2×C4⋊Q812yes—
243C22×C73#57512yes—
244C22×C73#57512yes—
245C22×C73C23×D411yescharacteristic
246C2×D73—28no—
247C2×D73—22no—
248C2×D73—22no—
249C2×D73C2×C4⋊Q814yes—
250C4×C73—22no—
251C4×C73—22no—
252C4×C73—42no—
253C4×C73—42no—
254C4×C73—44no—
255C4×C73C2×C4⋊Q812yes—
256C4×C73C23×Q812yes—
257Dic73—22no—
258Dic73—22no—
259Dic73—44no—
260Dic73—42no—
261Dic73—42no—
262Dic73C2×C4⋊Q814yes—
263C2×#3166—731nomaximal, Sylow
264C2×C73#316614yescomplemented
265C2×C73C22×C4⋊Q811yescharacteristic
266C2×C73C2×#57511yescharacteristic
267C2×C73C2×#57511yescharacteristic
268D73—24no—
269C2×#533—731no—
270C22×#125—731no—
271#3166—7316no—
272C2×#501—731no—
273C2×#501—731no—
274C2×#501—731no—
275C2×#501D7311yescharacteristic, complemented
276C2×#499—732no—
277C2×#499—732no—
278C2×#428—732no—
279C2×#428—732no—
280C2×C4×M16—731no—
281C73C2×#316611yescharacteristic, complemented, Sylow
282C22×M16—732no—
283#533—738no—
284#533—1464no—
285C2×C4⋊Q8—731no—
286C2×C4⋊Q8C2×D7311yescharacteristic
287C2×#125—738no—
288C2×#125—731no—
289C2×#125—731no—
290C2×#125—732no—
291C2×#125—731no—
292C2×#125—1461no—
293C2×#125—1461no—
294C2×#125—1464no—
295C2×#125C2×D7311yescharacteristic
296C2×C22⋊Q8—1461no—
297C2×C22⋊Q8—1461no—
298C2×C4×Q8—1461no—
299C2×C4×Q8—1461no—
300C2×#117—732no—
301C22×C4⋊C4—731no—
302C22×C4⋊C4—731no—
303C22×C4⋊C4—1461no—
304C22×C4⋊C4—1461no—
305C22×C42—731no—
306#501—738no—
307#501—738no—
308#501—738no—
309#501C2×D7318yescomplemented
310#499—7316no—
311#499—7316no—
312#428—7316no—
313#428—7316no—
314C2×#107—732no—
315C2×#107—732no—
316C2×#107—732no—
317C2×#107C2×D7312yes—
318C2×#106—732no—
319C2×#106—732no—
320C2×#106—732no—
321C2×#106C2×D7312yes—
322C4×M16—738no—
323C2×#97—731no—
324C2×#97C2×D7311yescharacteristic
325C2×C4×C8—732no—
326C22×Q8—21no—
327C22×Q8—1461no—
328C23×C4—731no—
329C23×C4—731no—
330C23×C4—731no—
331C23×C4—1461no—
332C23×C4—1461no—
333C2×M16—7316no—
334C2×M16—738no—
335C22×C8—732no—
336C22×C8C22×D7312yes—
337C4⋊Q8—22no—
338C4⋊Q8—734no—
339C4⋊Q8—1462no—
340C4⋊Q8C22×D7314yes—
341#125—22no—
342#125—734no—
343#125—734no—
344#125—738no—
345#125—734no—
346#125—1468no—
347#125—1468no—
348#125—1462no—
349#125—1462no—
350#125—1464no—
351#125—1462no—
352#125C22×D7314yes—
353C22⋊Q8—1468no—
354C22⋊Q8—1468no—
355C4×Q8—1468no—
356C4×Q8—1468no—
357#117—738no—
358#117—1464no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—731no—
363C2×C4⋊C4—731no—
364C2×C4⋊C4—732no—
365C2×C4⋊C4—734no—
366C2×C4⋊C4—734no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1462no—
369C2×C4⋊C4—1468no—
370C2×C4⋊C4—1468no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4—1462no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4C22×D7311yescharacteristic
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7312yes—
385C2×C22⋊C4—1462no—
386C2×C42—731no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—734no—
390C2×C42—734no—
391C2×C42—1462no—
392C2×C42C22×D7311yescharacteristic
393#107—738no—
394#107—738no—
395#107—738no—
396#107C22×D7318yes—
397#106—738no—
398#106—738no—
399#106—738no—
400#106C22×D7318yes—
401#97—734no—
402#97C22×D7314yes—
403C4×C8—738no—
404C24—731no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—2924no—
408C2×Q8—1464no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—2922no—
413C22×C4—2922no—
414C22×C4—2924no—
415C22×C4—2924no—
416C22×C4—734no—
417C22×C4—734no—
418C22×C4—731no—
419C22×C4—732no—
420C22×C4—738no—
421C22×C4—732no—
422C22×C4—731no—
423C22×C4—731no—
424C22×C4—1464no—
425C22×C4—1464no—
426C22×C4—1462no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4D4×D7311yescharacteristic
430C22×C4D4×D7311yescharacteristic
431C22×C4C23×D7311yescharacteristic
432M16—7332no—
433C2×C8—738no—
434C2×C8—734no—
435C2×C8Q8×D7318yes—
436C2×C8C23×D7314yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—732no—
444C4⋊C4—732no—
445C4⋊C4—734no—
446C4⋊C4—1464no—
447C4⋊C4—1464no—
448C4⋊C4—1464no—
449C4⋊C4—1464no—
450C4⋊C4—1468no—
451C4⋊C4—1464no—
452C4⋊C4—1464no—
453C4⋊C4—1468no—
454C4⋊C4—1464no—
455C4⋊C4—1464no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C4⋊C4—1462no—
459C4⋊C4—1464no—
460C4⋊C4C23×D7312yes—
461C4⋊C4C23×D7314yes—
462C4⋊C4C23×D7312yes—
463C22⋊C4—1468no—
464C42—734no—
465C42—734no—
466C42—732no—
467C42—732no—
468C42—1468no—
469C42D4×D7312yes—
470C42C23×D7312yes—
471C23—734no—
472C23—732no—
473C23—1462no—
474C23—1462no—
475C23C2×D4×D7311yescharacteristic
476Q8—48no—
477Q8—2928no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—2924no—
487C2×C4—2924no—
488C2×C4—2928no—
489C2×C4—2928no—
490C2×C4—2922no—
491C2×C4—2922no—
492C2×C4—2924no—
493C2×C4—734no—
494C2×C4—732no—
495C2×C4—738no—
496C2×C4—738no—
497C2×C4—732no—
498C2×C4—732no—
499C2×C4—1464no—
500C2×C4—1462no—
501C2×C4—1462no—
502C2×C4—1462no—
503C2×C4—1462no—
504C2×C4C2×D4×D7312yes—
505C2×C4C2×D4×D7312yes—
506C2×C4C2×D4×D7311yescharacteristic
507C2×C4C2×Q8×D7314yes—
508C2×C4C24×D7311yescharacteristic
509C8—738no—
510C8C2×Q8×D7318yes—
511C22—734no—
512C22—1468no—
513C22—1462no—
514C22—1462no—
515C22C4⋊Q8×D7312yes—
516C22#1798938312yes—
517C22#1798938612yes—
518C22C22×D4×D7311yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—2922no—
525C4—2922no—
526C4—2924no—
527C4—734no—
528C4—1462no—
529C4—1462no—
530C4C4⋊Q8×D7312yes—
531C4C22×Q8×D7312yes—
532C2—1464no—
533C2#2262048214yesdirect product factor
534C2C2×C4⋊Q8×D7311yescharacteristic
535C2C2×#1798938311yescharacteristic
536C2C2×#1798938611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#501×C73C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
hypercenterC2×#501D73
hypocenterC73C2×#3166
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #22620482C2direct (N × H)
4. C2×#17988997C2semidirect
5. C2×#17988996C2semidirect
6. C2×#17988993C2semidirect
7. C2×#17988988C2semidirect
8. C2×#17988987C2semidirect
12. C2×#501×C73C2semidirect
35. #17988997C22semidirect
36. #17988996C22semidirect
37. #17988993C22semidirect
38. #17988988C22semidirect
39. #17988987C22semidirect
53. #501×C73C22semidirect
63. C2×#11924011Q8semidirect
155. #11924011C2×Q8semidirect
264. C2×C73#3166semidirect
275. C2×#501D73semidirect
281. C73C2×#3166semidirect
309. #501C2×D73semidirect
533. C2#22620482direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179891422
C2×#125×D732
#226204822
C2×#179889972
C2×#179889962
C2×#179889932
C2×#179889882
C2×#179889872
C2×#179886282
C2×#179886262
C2×#179885392
C2×#501×C732
C2×#316673