Finite Group Databasebeta

← C2×#22622319 #33964441

Subgroups — C2×#22622319 #33964441

Order 18688.

1942 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989294C211yesmaximal, characteristic
2#22622319C2116yesmaximal, direct product factor
3C2×#17988990C211yesmaximal, characteristic, complemented
4C2×#17988922C212yesmaximal, complemented
5C2×#17988920C212yesmaximal, complemented
6C2×#17988882C211yesmaximal, characteristic, complemented
7C2×#17988879C211yesmaximal, characteristic, complemented
8C2×#17988826C212yesmaximal, complemented
9C2×C4×#11924018C211yesmaximal, characteristic, complemented
10C2×D4×#10596831C211yesmaximal, characteristic
11C22×#14556524C212yesmaximal, complemented
12C2×#554×C73C211yesmaximal, characteristic
13C22×#11924018C2211yescharacteristic
14C23×#10596831C2212yes—
15#17989294—24no—
16#17989294C2218yes—
17C2×#14556591C2211yescharacteristic
18C2×#14556582—21no—
19C2×#14556570C2212yes—
20C2×#14556568—22no—
21C2×C4×Dic146C2211yescharacteristic
22C2×D4×Dic73—21no—
23C22×C4⋊Dic73C2212yes—
24#17988990C2218yescomplemented
25#17988922C22116yescomplemented
26#17988920C22116yescomplemented
27#17988882C2218yescomplemented
28#17988879C2218yescomplemented
29#17988826C22116yescomplemented
30C4×#11924018C2218yescomplemented
31D4×#10596831C2218yes—
32C2×#14556524C2212yes—
33C2×#14556524C2212yes—
34C2×#14556524C2212yes—
35C2×#14556524C2212yes—
36C2×#14556524C2211yescharacteristic
37C2×#14556524C22116yescomplemented
38C2×C4⋊#10596831C2211yescharacteristic
39C2×#14556515C2212yes—
40C2×#14556515C2211yescharacteristic
41C2×#14556514C2212yes—
42C2×#14556514C2211yescharacteristic
43C2×C22⋊#10596831C2212yes—
44C2×C4×#10596831C2211yescharacteristic
45#554×C73—24no—
46#554×C73C2218yes—
47C2×C4⋊Q8×C73C2211yescharacteristic
48C2×#125×C73—21no—
49C2×C22⋊Q8×C73—22no—
50C2×C22⋊Q8×C73C2212yes—
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×D4×C73—21no—
53C2×C4×D4×C73C2211yescharacteristic
54C22×C4⋊C4×C73C2212yes—
55C22×Dic146—21no—
56C22×Dic146C2311yescharacteristic
57C23×Dic73—22no—
58C2×#11924018—24no—
59C2×#11924018C2314yes—
60C22×#10596831—28no—
61C22×#10596831C2312yes—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2318yes—
64C22×#10596831C2311yescharacteristic
65#14556591—22no—
66#14556591C2314yes—
67#14556582—28no—
68#14556570—24no—
69#14556570C2318yes—
70#14556568—216no—
71C4×Dic146—22no—
72C4×Dic146C2314yes—
73D4×Dic73—28no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—24no—
76C2×C4⋊Dic73—22no—
77C2×C4⋊Dic73C2312yes—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2318yes—
80C2×#11924007—21no—
81C2×#11924007—22no—
82C2×C22⋊Dic73—22no—
83C2×C4×Dic73—21no—
84#14556524C2314yes—
85#14556524C2318yes—
86#14556524C2318yes—
87#14556524C2318yes—
88#14556524C2318yes—
89C4⋊#10596831C2314yes—
90#14556515C2318yes—
91#14556515C2314yes—
92#14556514C2318yes—
93#14556514C2314yes—
94C22⋊#10596831C2318yes—
95C4×#10596831C2314yes—
96C22×Q8×C73—21no—
97C22×Q8×C73C2311yescharacteristic
98C22×D4×C73D411yescharacteristic
99C23×C4×C73—22no—
100C23×C4×C73C2312yes—
101C4⋊Q8×C73—22no—
102C4⋊Q8×C73C2314yes—
103#125×C73—28no—
104C22⋊Q8×C73—24no—
105C22⋊Q8×C73—216no—
106C22⋊Q8×C73C2318yes—
107C4×Q8×C73—22no—
108C4×Q8×C73C2314yes—
109C4×D4×C73—28no—
110C4×D4×C73D414yes—
111C4×D4×C73C2314yes—
112C2×C4⋊C4×C73—22no—
113C2×C4⋊C4×C73—24no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—22no—
117C2×C4⋊C4×C73D411yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C4⋊C4×C73C2312yes—
120C2×C4⋊C4×C73C2318yes—
121C2×C22⋊C4×C73—22no—
122C2×C22⋊C4×C73D412yes—
123C2×C42×C73—21no—
124C2×C42×C73C2311yescharacteristic
125C2×Dic146—22no—
126C2×Dic146—22no—
127C2×Dic146—21no—
128C2×Dic146—24no—
129C2×Dic146—82no—
130C2×Dic146C2412yes—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—22no—
134C22×Dic73—28no—
135C22×Dic73—84no—
136#11924018—28no—
137C2×#10596831—24no—
138C2×#10596831—22no—
139C2×#10596831—216no—
140C2×#10596831#4114yes—
141C2×#10596831C2412yes—
142C2×#10596831C2414yes—
143C2×#10596831C2412yes—
144C4⋊Dic73—21no—
145C4⋊Dic73—28no—
146C4⋊Dic73—22no—
147C4⋊Dic73—24no—
148C4⋊Dic73C2412yes—
149C4⋊Dic73C2414yes—
150#11924007—24no—
151#11924007—28no—
152C22⋊Dic73—28no—
153C4×Dic73—24no—
154C24×C73C2×D412yes—
155C2×Q8×C73—21no—
156C2×Q8×C73—24no—
157C2×Q8×C73—22no—
158C2×Q8×C73—22no—
159C2×Q8×C73—82no—
160C2×Q8×C73C2412yes—
161C2×D4×C73Q1618yescomplemented
162C2×D4×C73C2×D414yes—
163C22×C4×C73—24no—
164C22×C4×C73—24no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—22no—
168C22×C4×C73—28no—
169C22×C4×C73—84no—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D412yes—
172C22×C4×C73C2×D414yes—
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2414yes—
175C22×C4×C73C2411yescharacteristic
176C4⋊C4×C73—21no—
177C4⋊C4×C73—28no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—24no—
180C4⋊C4×C73—24no—
181C4⋊C4×C73—28no—
182C4⋊C4×C73C2×D414yes—
183C4⋊C4×C73C2412yes—
184C4⋊C4×C73C2414yes—
185C22⋊C4×C73—28no—
186C22⋊C4×C73C2×D418yes—
187C42×C73—24no—
188C42×C73C2×D412yes—
189C42×C73C2412yes—
190C23×C73—24no—
191C23×C73—24no—
192C23×C73C2×Q1618yes—
193C23×C73C22×D414yes—
194C23×C73C22×D411yescharacteristic
195Q8×C73—22no—
196Q8×C73—22no—
197Q8×C73—42no—
198Q8×C73—84no—
199D4×C73C2×Q16116yescomplemented
200C2×C4×C73—24no—
201C2×C4×C73—22no—
202C2×C4×C73—28no—
203C2×C4×C73—24no—
204C2×C4×C73—24no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—24no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—44no—
211C2×C4×C73—88no—
212C2×C4×C73C2×Q1614yes—
213C2×C4×C73C22×D412yes—
214C2×C4×C73C22×D414yes—
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×D412yes—
217C2×C4×C73C2×#4112yes—
218C2×C4×C732−1+412yes—
219C2×C4×C73C2511yescharacteristic
220C2×Dic73—22no—
221C2×Dic73—22no—
222C2×Dic73—24no—
223C2×Dic73—42no—
224C2×Dic73—42no—
225C2×Dic73—44no—
226C2×Dic73—88no—
227Dic146—22no—
228Dic146—22no—
229Dic146—42no—
230Dic146—84no—
231#10596831—24no—
232#10596831—22no—
233#10596831C2×#4114yes—
234C22×C73—216no—
235C22×C73—24no—
236C22×C73—24no—
237C22×C73#54712yes—
238C22×C73C22×Q1618yes—
239C22×C73C22×Q1612yes—
240C22×C73#57612yes—
241C22×C73C23×D411yescharacteristic
242C4×C73—24no—
243C4×C73—22no—
244C4×C73—42no—
245C4×C73—42no—
246C4×C73—44no—
247C4×C73#54712yes—
248C4×C73C22×Q1614yes—
249C4×C73C22×#4111yescharacteristic
250C4×C73C2×2−1+411yescharacteristic
251Dic73—42no—
252Dic73—44no—
253Dic73—42no—
254C2×#3338—731nomaximal, Sylow
255C2×C73—28no—
256C2×C73#333814yescomplemented
257C2×C73C2×#54711yescharacteristic
258C2×C73C23×Q1611yescharacteristic
259C2×C73C2×#57611yescharacteristic
260C2×#554—731no—
261C2×#554D7311yescharacteristic
262#3338—7316no—
263C2×#500—731no—
264C2×#484—732no—
265C2×#484—732no—
266C2×#477—731no—
267C2×#477—731no—
268C2×#467—732no—
269C2×C4×Q16—731no—
270C2×C8×D4—731no—
271C22×#107—732no—
272C73C2×#333811yescharacteristic, complemented, Sylow
273C22×Q16—731no—
274C23×C8—732no—
275#554—24no—
276#554—738no—
277#554—1464no—
278#554C2×D7318yes—
279C2×C4⋊Q8—731no—
280C2×C4⋊Q8C2×D7311yescharacteristic
281C2×#125—21no—
282C2×#125—1461no—
283C2×C22⋊Q8—22no—
284C2×C22⋊Q8—732no—
285C2×C22⋊Q8—1462no—
286C2×C22⋊Q8C2×D7312yes—
287C2×C4×Q8—731no—
288C2×C4×Q8C2×D7311yescharacteristic
289C2×C4×D4—21no—
290C2×C4×D4—1461no—
291C2×C4×D4C2×D7311yescharacteristic
292C22×C4⋊C4—732no—
293C22×C4⋊C4C2×D7312yes—
294#500—738no—
295#484—7316no—
296#484—7316no—
297#477—738no—
298#477—738no—
299#467—7316no—
300C4×Q16—738no—
301C8×D4—738no—
302C2×#107—732no—
303C2×#107—732no—
304C2×#107—7316no—
305C2×#107—732no—
306C2×#107—732no—
307C2×#107—731no—
308C2×C4⋊C8—731no—
309C2×#103—732no—
310C2×#103—731no—
311C2×#103—732no—
312C2×#103—731no—
313C2×C22⋊C8—732no—
314C2×C4×C8—731no—
315C22×Q8—21no—
316C22×Q8—731no—
317C22×Q8—1461no—
318C22×Q8C22×D7311yescharacteristic
319C22×D4C22⋊D7311yescharacteristic
320C23×C4—22no—
321C23×C4—1462no—
322C23×C4C22×D7312yes—
323C2×Q16—734no—
324C2×Q16—1464no—
325C22×C8—731no—
326C22×C8—732no—
327C22×C8—738no—
328C22×C8—731no—
329C22×C8—1468no—
330C4⋊Q8—22no—
331C4⋊Q8—734no—
332C4⋊Q8—1462no—
333C4⋊Q8C22×D7314yes—
334#125—28no—
335#125—1468no—
336C22⋊Q8—24no—
337C22⋊Q8—216no—
338C22⋊Q8—738no—
339C22⋊Q8—1464no—
340C22⋊Q8—14616no—
341C22⋊Q8C22×D7318yes—
342C4×Q8—22no—
343C4×Q8—734no—
344C4×Q8—1462no—
345C4×Q8C22×D7314yes—
346C4×D4—28no—
347C4×D4—1468no—
348C4×D4C22⋊D7314yes—
349C4×D4C22×D7314yes—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—731no—
356C2×C4⋊C4—732no—
357C2×C4⋊C4—738no—
358C2×C4⋊C4—1462no—
359C2×C4⋊C4—1464no—
360C2×C4⋊C4—1461no—
361C2×C4⋊C4—1461no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4C22⋊D7311yescharacteristic
364C2×C4⋊C4C22×D7311yescharacteristic
365C2×C4⋊C4C22×D7312yes—
366C2×C4⋊C4C22×D7318yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—1462no—
369C2×C22⋊C4C22⋊D7312yes—
370C2×C42—21no—
371C2×C42—1461no—
372C2×C42C22×D7311yescharacteristic
373#107—734no—
374#107—738no—
375#107—738no—
376#107—738no—
377#107—738no—
378C4⋊C8—734no—
379#103—734no—
380#103—738no—
381#103—734no—
382#103—738no—
383C22⋊C8—738no—
384C4×C8—734no—
385C24C2×C22⋊D7312yes—
386C2×Q8—21no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—82no—
391C2×Q8—5842no—
392C2×Q8—732no—
393C2×Q8—1462no—
394C2×Q8—1462no—
395C2×Q8—1461no—
396C2×Q8—1464no—
397C2×Q8C23×D7312yes—
398C2×D4#1192401818yescomplemented
399C2×D4C2×C22⋊D7314yes—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—84no—
407C22×C4—5844no—
408C22×C4—1461no—
409C22×C4—1461no—
410C22×C4—1462no—
411C22×C4—1468no—
412C22×C4C2×C22⋊D7314yes—
413C22×C4C2×C22⋊D7311yescharacteristic
414C22×C4C2×C22⋊D7312yes—
415C22×C4C2×C22⋊D7311yescharacteristic
416C22×C4C23×D7314yes—
417C22×C4C23×D7311yescharacteristic
418Q16—1468no—
419C2×C8—734no—
420C2×C8—732no—
421C2×C8—732no—
422C2×C8—734no—
423C2×C8—14616no—
424C2×C8—1464no—
425C2×C8—1462no—
426C4⋊C4—21no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—732no—
433C4⋊C4—734no—
434C4⋊C4—1464no—
435C4⋊C4—1468no—
436C4⋊C4—1461no—
437C4⋊C4—1468no—
438C4⋊C4—1462no—
439C4⋊C4—1464no—
440C4⋊C4C2×C22⋊D7314yes—
441C4⋊C4C23×D7312yes—
442C4⋊C4C23×D7314yes—
443C22⋊C4—28no—
444C22⋊C4—1468no—
445C22⋊C4C2×C22⋊D7318yes—
446C42—24no—
447C42—1464no—
448C42C2×C22⋊D7312yes—
449C42C23×D7312yes—
450C23—24no—
451C23—24no—
452C23C2×#1192401818yes—
453C23C22×C22⋊D7314yes—
454C23C22×C22⋊D7311yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—42no—
458Q8—84no—
459Q8—2922no—
460Q8—5844no—
461Q8—1462no—
462Q8—1462no—
463D4C2×#11924018116yescomplemented
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—88no—
476C2×C4—2922no—
477C2×C4—2924no—
478C2×C4—2922no—
479C2×C4—5848no—
480C2×C4—1462no—
481C2×C4—1462no—
482C2×C4—1464no—
483C2×C4C2×#1192401814yes—
484C2×C4C22×C22⋊D7312yes—
485C2×C4C22×C22⋊D7314yes—
486C2×C4C22×C22⋊D7311yescharacteristic
487C2×C4C22×C22⋊D7312yes—
488C2×C4#41×D7312yes—
489C2×C4#1455663812yes—
490C2×C4C24×D7311yescharacteristic
491C8—734no—
492C8—1464no—
493C8—1462no—
494C22—216no—
495C22—24no—
496C22—24no—
497C22#1798923612yes—
498C22C22×#1192401818yes—
499C22C22×#1192401812yes—
500C22#1798939012yes—
501C22C23×C22⋊D7311yescharacteristic
502C4—24no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—44no—
507C4—2922no—
508C4—2922no—
509C4—2924no—
510C4#1798923612yes—
511C4C22×#1192401814yes—
512C4C2×#41×D7311yescharacteristic
513C4C2×#1455663811yescharacteristic
514C2—28no—
515C2#2262231914yesdirect product factor
516C2C2×#1798923611yescharacteristic
517C2C23×#1192401811yescharacteristic
518C2C2×#1798939011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#554×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenterC2×#554D73
hypocenterC73C2×#3338
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. #22622319C2direct (N × H)
3. C2×#17988990C2semidirect
4. C2×#17988922C2semidirect
5. C2×#17988920C2semidirect
6. C2×#17988882C2semidirect
7. C2×#17988879C2semidirect
8. C2×#17988826C2semidirect
9. C2×C4×#11924018C2semidirect
11. C22×#14556524C2semidirect
24. #17988990C22semidirect
25. #17988922C22semidirect
26. #17988920C22semidirect
27. #17988882C22semidirect
28. #17988879C22semidirect
29. #17988826C22semidirect
30. C4×#11924018C22semidirect
37. C2×#14556524C22semidirect
161. C2×D4×C73Q16semidirect
199. D4×C73C2×Q16semidirect
256. C2×C73#3338semidirect
272. C73C2×#3338semidirect
398. C2×D4#11924018semidirect
463. D4C2×#11924018semidirect
515. C2#22622319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179892942
#226223192
C2×#179889902
C2×#179889222
C2×#179889202
C2×#179888822
C2×#179888792
C2×#179888262
C2×C4×#119240182
C2×D4×#105968312
C22×#145565242
C2×#554×C732
C2×#333873