Finite Group Databasebeta

← C2×#22618530 #33949656

Subgroups — C2×#22618530 #33949656

Order 18688.

2766 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556599C211yesmaximal, characteristic
2C2×#17989068C211yesmaximal, characteristic
3C2×#17989067C211yesmaximal, characteristic
4#22618530C2116yesmaximal, direct product factor
5C2×#17988629C211yesmaximal, characteristic, complemented
6C2×#17988628C212yesmaximal, complemented
7C2×#17988626C212yesmaximal, complemented
8C2×#17988623C211yesmaximal, characteristic, complemented
9C2×#17988622C212yesmaximal, complemented
10C22×#14556524C211yesmaximal, characteristic, complemented
11C2×#107×D73C212yesmaximal, complemented
12C2×#428×C73C211yesmaximal, characteristic, complemented
13C22×#11924028C2211yescharacteristic
14C2×#14556599C4116yescomplemented
15C2×#14556599C2218yes—
16C22×#11924011C2212yes—
17C23×#10596831C2211yescharacteristic
18C22×C8×D73C2212yes—
19#17989068—24no—
20#17989068C2218yes—
21#17989067—24no—
22#17989067C2218yes—
23C2×C4×Dic146—21no—
24C2×#14556556—21no—
25C2×C4×D292—21no—
26C2×#14556553—21no—
27C2×C4×C22⋊D73—22no—
28C2×#14556551—22no—
29C2×#14556550C2211yescharacteristic
30C2×#14556549C2212yes—
31C2×#14556549C2212yes—
32C22×C4⋊Dic73C2211yescharacteristic
33C2×C4⋊C4×D73C2212yes—
34C2×C4⋊C4×D73C2212yes—
35#17988629C2218yescomplemented
36#17988628C22116yescomplemented
37#17988626C22116yescomplemented
38#17988623C2218yescomplemented
39#17988622C22116yescomplemented
40C2×#14556525C2212yes—
41C2×#14556524C2211yescharacteristic
42C2×#14556524C2212yes—
43C2×#14556524C2212yes—
44C2×#14556524C2211yescharacteristic
45C2×#14556524C2218yescomplemented
46#107×D73C22116yescomplemented
47C2×#14556523C2212yes—
48C2×#14556522C2211yescharacteristic
49C2×#14556522C2211yescharacteristic
50C22×M16×C73C2211yescharacteristic
51C2×#117×C73C2211yescharacteristic
52C22×C4⋊C4×C73C2211yescharacteristic
53#428×C73C2218yescomplemented
54C2×#107×C73C2212yes—
55C2×#106×C73C2212yes—
56C2×#11924028C2×C418yes—
57C2×#11924028D414yes—
58C2×#11924028Q818yes—
59C2×#11924028C2314yes—
60C22×Dic146D411yescharacteristic
61C22×D292D411yescharacteristic
62C22×C22⋊D73D412yes—
63C23×Dic73—21no—
64C22×C4×D73—22no—
65C22×C4×D73—22no—
66C22×C4×D73C2312yes—
67#14556599C2×C4164yescomplemented
68C2×#11924011C2×C4116yescomplemented
69C2×#11924011C2318yes—
70C22×#10596831C2×C418yescomplemented
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2314yes—
74C2×C8×D73C2×C4116yescomplemented
75C2×C8×D73C2318yes—
76C4×Dic146—28no—
77#14556556—28no—
78C4×D292—28no—
79#14556553—28no—
80C4×C22⋊D73—216no—
81#14556551—216no—
82#14556550—22no—
83#14556550C2314yes—
84#14556549—24no—
85#14556549—24no—
86#14556549C2318yes—
87#14556549C2318yes—
88C2×C4⋊Dic73—21no—
89C2×C4⋊Dic73—22no—
90C2×C4⋊Dic73C2311yescharacteristic
91C2×C4⋊Dic73C2311yescharacteristic
92C2×C4⋊Dic73C2312yes—
93C2×C4⋊Dic73C2314yes—
94C2×#11924007—22no—
95C2×#11924007—22no—
96C4⋊C4×D73—24no—
97C4⋊C4×D73—24no—
98C4⋊C4×D73C2318yes—
99C4⋊C4×D73C2318yes—
100C2×C22⋊Dic73—21no—
101C2×#11924005—22no—
102C2×#11924005—22no—
103C2×C4×Dic73—22no—
104C2×C4×Dic73—22no—
105C2×C4×Dic73—21no—
106#14556525C2318yes—
107#14556524C2314yes—
108#14556524C2318yes—
109#14556524C2318yes—
110#14556524C2314yes—
111#14556523C2318yes—
112#14556522C2314yes—
113#14556522C2314yes—
114C23×C4×C73—21no—
115C23×C4×C73C2311yescharacteristic
116C2×M16×C73C2×C418yescomplemented
117C2×M16×C73C2314yes—
118C22×C8×C73C2312yes—
119#117×C73—22no—
120#117×C73C2314yes—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73—22no—
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2312yes—
126C2×C4⋊C4×C73C2314yes—
127C2×C22⋊C4×C73—21no—
128C2×C42×C73—21no—
129#107×C73C2318yes—
130#106×C73C2318yes—
131C23×D73C2×D412yes—
132#11924028C4⋊C4132yes—
133#11924028C22×C4116yes—
134#11924028C2×Q8116yes—
135C2×Dic146C4⋊C418yes—
136C2×Dic146C2×D414yes—
137C2×D292C4⋊C418yes—
138C2×D292C2×D414yes—
139C2×C22⋊D73C4⋊C4116yes—
140C2×C22⋊D73C2×D418yes—
141C22×Dic73—21no—
142C22×Dic73—22no—
143C22×Dic73—21no—
144C22×Dic73—24no—
145C22×Dic73—82no—
146C22×Dic73C2×D412yes—
147C2×C4×D73—28no—
148C2×C4×D73—28no—
149C2×C4×D73—84no—
150C2×C4×D73—84no—
151C2×C4×D73C22×C418yes—
152C2×C4×D73C2×D414yes—
153C2×C4×D73C2×Q818yes—
154C2×C4×D73C2414yes—
155#11924011C22×C4132yescomplemented
156C2×#10596831C22×C414yes—
157C2×#10596831C22×C414yes—
158C2×#10596831C22×C4116yescomplemented
159C2×#10596831C2412yes—
160C2×#10596831C2412yes—
161C8×D73C22×C4132yescomplemented
162C4⋊Dic73—21no—
163C4⋊Dic73—22no—
164C4⋊Dic73—24no—
165C4⋊Dic73—21no—
166C4⋊Dic73C2412yes—
167C4⋊Dic73C2412yes—
168C4⋊Dic73C2414yes—
169#11924007—28no—
170#11924007—28no—
171C22⋊Dic73—24no—
172#11924005—28no—
173#11924005—28no—
174C4×Dic73—24no—
175C4×Dic73—28no—
176C4×Dic73—28no—
177C24×C73C2×D411yescharacteristic
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—22no—
181C22×C4×C73—24no—
182C22×C4×C73—82no—
183C22×C4×C73C22×C414yes—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D412yes—
186C22×C4×C73C2×Q814yes—
187C22×C4×C73C2411yescharacteristic
188C22×C4×C73C2412yes—
189M16×C73C22×C4116yescomplemented
190C2×C8×C73C22×C418yes—
191C2×C8×C73C2414yes—
192C4⋊C4×C73—22no—
193C4⋊C4×C73—24no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73C2412yes—
197C4⋊C4×C73C2414yes—
198C4⋊C4×C73C2412yes—
199C22⋊C4×C73—24no—
200C42×C73—24no—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73C2×C4⋊C414yes—
204C23×C73C22×D412yes—
205C23×C73C22×D411yescharacteristic
206C22×D73—24no—
207C22×D73—24no—
208C22×D73C2×C4⋊C418yes—
209C22×D73C22×D414yes—
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×C73—84no—
217C2×C4×C73C2×C4⋊C414yes—
218C2×C4×C73C2×C4⋊C418yes—
219C2×C4×C73C23×C412yes—
220C2×C4×C73C23×C414yes—
221C2×C4×C73C22×D412yes—
222C2×C4×C73C22×D411yescharacteristic
223C2×C4×C73C22×Q812yes—
224C2×C4×C73C22×Q814yes—
225C2×C4×C73C2511yescharacteristic
226C22⋊D73C2×C4⋊C4132yes—
227C2×Dic73—22no—
228C2×Dic73—24no—
229C2×Dic73—22no—
230C2×Dic73—42no—
231C2×Dic73—44no—
232C2×Dic73—42no—
233C2×Dic73—84no—
234C2×Dic73C2×C4⋊C418yes—
235C2×Dic73C22×D414yes—
236D292C2×C4⋊C4116yes—
237C4×D73—88no—
238C4×D73—88no—
239C4×D73C2×C4⋊C4116yes—
240C4×D73C23×C418yes—
241C4×D73C22×Q818yes—
242Dic146C2×C4⋊C4116yes—
243C8×C73C23×C418yes—
244#10596831C23×C414yes—
245#10596831C23×C414yes—
246C22×C73—28no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73C22×C4⋊C414yes—
250C22×C73C22×C4⋊C412yes—
251C22×C73#57612yes—
252C22×C73#57812yes—
253C22×C73C23×D411yescharacteristic
254C2×D73—216no—
255C2×D73—24no—
256C2×D73—24no—
257C2×D73C22×C4⋊C418yes—
258C4×C73—42no—
259C4×C73—44no—
260C4×C73—42no—
261C4×C73C22×C4⋊C414yes—
262C4×C73C22×C4⋊C412yes—
263C4×C73C24×C411yescharacteristic
264C4×C73C23×Q811yescharacteristic
265Dic73—42no—
266Dic73—44no—
267Dic73—42no—
268Dic73C22×C4⋊C418yes—
269C2×#2913—731nomaximal, Sylow
270C2×C73—24no—
271C2×C73#291314yescomplemented
272C2×C73C23×C4⋊C411yescharacteristic
273C2×C73C2×#57611yescharacteristic
274C2×C73C2×#57811yescharacteristic
275D73—28no—
276C22×#131—731no—
277C2×#520—731no—
278C2×#520—731no—
279#2913—7316no—
280C2×#428—732no—
281C2×#428—732no—
282C2×#428—731no—
283C2×#428D7311yescharacteristic, complemented
284C2×#427—732no—
285C2×#427—731no—
286C22×#107—732no—
287C22×#107—731no—
288C73C2×#291311yescharacteristic, complemented, Sylow
289C22×#41—731no—
290C2×#131—7316no—
291C2×#131—738no—
292C22×M16—732no—
293C22×M16C2×D7311yescharacteristic
294C23×C8—731no—
295C23×C8—732no—
296#520—738no—
297#520—738no—
298#520—1464no—
299#520—1464no—
300C2×C4×Q8—1461no—
301C2×C4×Q8—1461no—
302C2×C4×D4—1461no—
303C2×C4×D4—1462no—
304C2×C4×D4—1461no—
305C2×C4×D4—1462no—
306C2×#117—731no—
307C2×#117—732no—
308C2×#117—732no—
309C2×#117C2×D7311yescharacteristic
310C22×C4⋊C4—731no—
311C22×C4⋊C4—732no—
312C22×C4⋊C4—732no—
313C22×C4⋊C4C2×D7311yescharacteristic
314#428—738no—
315#428—7316no—
316#428—7316no—
317#428C2×D7318yescomplemented
318#427—738no—
319#427—7316no—
320C2×#107—732no—
321C2×#107—731no—
322C2×#107—732no—
323C2×#107—732no—
324C2×#107—731no—
325C2×#107—738no—
326C2×#107—7316no—
327C2×#107C2×D7312yes—
328C2×#106—732no—
329C2×#106—731no—
330C2×#106—731no—
331C2×#106C2×D7312yes—
332C2×#41—738no—
333C2×#41—734no—
334C2×#41—738no—
335C2×#41—734no—
336C22×Q8—731no—
337C22×D4—732no—
338C22×D4—731no—
339C23×C4—21no—
340C23×C4—732no—
341C23×C4—1462no—
342C23×C4—1462no—
343C23×C4—1461no—
344C23×C4C22×D7311yescharacteristic
345#131—7364no—
346C2×M16—7316no—
347C2×M16—738no—
348C2×M16C4×D7318yescomplemented
349C2×M16C22×D7314yes—
350C22×C8—738no—
351C22×C8—731no—
352C22×C8—731no—
353C22×C8—734no—
354C22×C8—7316no—
355C22×C8—738no—
356C22×C8C22×D7312yes—
357C4×Q8—1468no—
358C4×Q8—1468no—
359C4×D4—1468no—
360C4×D4—14616no—
361C4×D4—1468no—
362C4×D4—14616no—
363#117—22no—
364#117—734no—
365#117—738no—
366#117—738no—
367#117—1464no—
368#117—1464no—
369#117—1462no—
370#117C22×D7314yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—732no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—734no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—738no—
378C2×C4⋊C4—738no—
379C2×C4⋊C4—1464no—
380C2×C4⋊C4—1464no—
381C2×C4⋊C4—1462no—
382C2×C4⋊C4—1462no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1462no—
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7312yes—
387C2×C4⋊C4C22×D7314yes—
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1462no—
391C2×C22⋊C4—1462no—
392C2×C22⋊C4—1461no—
393C2×C42—21no—
394C2×C42—1462no—
395C2×C42—1462no—
396C2×C42—1461no—
397#107—738no—
398#107—734no—
399#107—738no—
400#107—738no—
401#107—734no—
402#107C22×D7318yes—
403#106—738no—
404#106—734no—
405#106—734no—
406#106C22×D7318yes—
407C24—732no—
408C24D4×D7311yescharacteristic
409#41—7332no—
410#41—7316no—
411#41—7316no—
412C2×Q8—738no—
413C2×Q8—734no—
414C2×D4—7316no—
415C2×D4—738no—
416C2×D4—738no—
417C2×D4—734no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—82no—
423C22×C4—5844no—
424C22×C4—5844no—
425C22×C4—5842no—
426C22×C4—732no—
427C22×C4—738no—
428C22×C4—734no—
429C22×C4—738no—
430C22×C4—734no—
431C22×C4—1468no—
432C22×C4—1468no—
433C22×C4—1461no—
434C22×C4—1462no—
435C22×C4—1461no—
436C22×C4—1464no—
437C22×C4C2×C4×D7314yes—
438C22×C4D4×D7312yes—
439C22×C4D4×D7311yescharacteristic
440C22×C4Q8×D7314yes—
441C22×C4C23×D7312yes—
442C22×C4C23×D7311yescharacteristic
443M16—7332no—
444M16C2×C4×D73116yescomplemented
445C2×C8—734no—
446C2×C8—734no—
447C2×C8—7316no—
448C2×C8—732no—
449C2×C8—732no—
450C2×C8—7332no—
451C2×C8C2×C4×D7318yes—
452C2×C8C23×D7314yes—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—732no—
458C4⋊C4—732no—
459C4⋊C4—734no—
460C4⋊C4—1468no—
461C4⋊C4—1468no—
462C4⋊C4—1461no—
463C4⋊C4—1462no—
464C4⋊C4—1464no—
465C4⋊C4—1461no—
466C4⋊C4C23×D7312yes—
467C4⋊C4C23×D7312yes—
468C4⋊C4C23×D7314yes—
469C22⋊C4—24no—
470C22⋊C4—1468no—
471C22⋊C4—1468no—
472C22⋊C4—1464no—
473C42—24no—
474C42—1468no—
475C42—1468no—
476C42—1464no—
477C23—22no—
478C23—22no—
479C23—734no—
480C23—738no—
481C23—1464no—
482C23—1464no—
483C23C4⋊C4×D7314yes—
484C23C2×D4×D7312yes—
485C23C2×D4×D7311yescharacteristic
486Q8—7316no—
487D4—7332no—
488D4—7316no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—2922no—
497C2×C4—2924no—
498C2×C4—2922no—
499C2×C4—5848no—
500C2×C4—5848no—
501C2×C4—5844no—
502C2×C4—734no—
503C2×C4—738no—
504C2×C4—738no—
505C2×C4—738no—
506C2×C4—7316no—
507C2×C4—1462no—
508C2×C4—1464no—
509C2×C4—1462no—
510C2×C4C4⋊C4×D7314yes—
511C2×C4C4⋊C4×D7318yes—
512C2×C4C22×C4×D7312yes—
513C2×C4C22×C4×D7314yes—
514C2×C4C2×D4×D7312yes—
515C2×C4C2×D4×D7311yescharacteristic
516C2×C4C2×Q8×D7312yes—
517C2×C4C2×Q8×D7314yes—
518C2×C4C24×D7311yescharacteristic
519C8—734no—
520C8—734no—
521C8C22×C4×D7318yes—
522C22—28no—
523C22—22no—
524C22—22no—
525C22—738no—
526C22—14616no—
527C22—1464no—
528C22—1464no—
529C22C2×C4⋊C4×D7314yes—
530C22C2×C4⋊C4×D7312yes—
531C22#1798939112yes—
532C22#1798940312yes—
533C22C22×D4×D7311yescharacteristic
534C4—44no—
535C4—42no—
536C4—42no—
537C4—2922no—
538C4—2924no—
539C4—2922no—
540C4—738no—
541C4C2×C4⋊C4×D7312yes—
542C4C2×C4⋊C4×D7314yes—
543C4C23×C4×D7311yescharacteristic
544C4C22×Q8×D7311yescharacteristic
545C2—24no—
546C2—1468no—
547C2#2261853014yesdirect product factor
548C2C22×C4⋊C4×D7311yescharacteristic
549C2C2×#1798939111yescharacteristic
550C2C2×#1798940311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC4×C73C24×C4
FrattiniC2×C4C24×D73
FittingC2×#428×C73C2
socleC23×C73C22×D4
prime radicalC22×#11924028C22
cosocleC2×C4×C73C25
hypercenterC2×#428D73
hypocenterC73C2×#2913
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #22618530C2direct (N × H)
5. C2×#17988629C2semidirect
6. C2×#17988628C2semidirect
7. C2×#17988626C2semidirect
8. C2×#17988623C2semidirect
9. C2×#17988622C2semidirect
10. C22×#14556524C2semidirect
11. C2×#107×D73C2semidirect
12. C2×#428×C73C2semidirect
14. C2×#14556599C4semidirect
35. #17988629C22semidirect
36. #17988628C22semidirect
37. #17988626C22semidirect
38. #17988623C22semidirect
39. #17988622C22semidirect
45. C2×#14556524C22semidirect
46. #107×D73C22semidirect
53. #428×C73C22semidirect
67. #14556599C2×C4semidirect
68. C2×#11924011C2×C4semidirect
70. C22×#10596831C2×C4semidirect
74. C2×C8×D73C2×C4semidirect
116. C2×M16×C73C2×C4semidirect
155. #11924011C22×C4semidirect
158. C2×#10596831C22×C4semidirect
161. C8×D73C22×C4semidirect
189. M16×C73C22×C4semidirect
271. C2×C73#2913semidirect
283. C2×#428D73semidirect
288. C73C2×#2913semidirect
317. #428C2×D73semidirect
348. C2×M16C4×D73semidirect
444. M16C2×C4×D73semidirect
547. C2#22618530direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565992
C2×#179890682
C2×#179890672
#226185302
C2×#179886292
C2×#179886282
C2×#179886262
C2×#179886232
C2×#179886222
C22×#145565242
C2×#107×D732
C2×#428×C732
C2×#291373