Finite Group Databasebeta

← #29833345

Subgroups — #29833345

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840201C211yesmaximal, characteristic
2C4×#14519796C211yesmaximal, characteristic
3C4×Q16×D61C211yesmaximal, characteristic, complemented
4#21447691C211yesmaximal, characteristic, complemented
5#21447688C211yesmaximal, characteristic, complemented
6#21447683C211yesmaximal, characteristic, complemented
7#21447677C211yesmaximal, characteristic, complemented
8#21447675C211yesmaximal, characteristic, complemented
9#21447674C211yesmaximal, characteristic, complemented
10SD32×Dic61C211yesmaximal, characteristic, complemented
11#21447604C211yesmaximal, characteristic, complemented
12#21447537C211yesmaximal, characteristic, complemented
13#21447528C211yesmaximal, characteristic
14#21447412C211yesmaximal, characteristic, complemented
15C61×#2175C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18C4×#11758224—21no—
19C4×Q8×D61—21no—
20#16840201C418yescomplemented
21C2×Dic488C2211yescharacteristic
22C2×#14519723C2211yescharacteristic
23C2×#14519721C2211yescharacteristic
24C2×#14519716C2211yescharacteristic
25C2×#14519716C2211yescharacteristic
26C4×Dic244C2211yescharacteristic
27C4×Dic244C2211yescharacteristic
28Q16×Dic61C2211yescharacteristic
29#16839856C2211yescharacteristic
30#16839856C2211yescharacteristic
31C4×#11758202—21no—
32#16839854—21no—
33#16839853—21no—
34C4×#11758200—21no—
35D8×Dic61C2211yescharacteristic
36#16839799C2211yescharacteristic
37#107×D61C2211yescharacteristic
38#16839755—21no—
39#103×D61—21no—
40C4×C8×D61C2211yescharacteristic
41#16839590C2211yescharacteristic
42#16839589C2211yescharacteristic
43#16839587C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839567C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839565C2211yescharacteristic
49#16839563C2211yescharacteristic
50#16839538C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C16×Dic61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C4×Q16×C61C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic
56C61×#366C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C61×#346C2211yescharacteristic
60C2×#11758224—21no—
61C2×Q8×D61—21no—
62#14519796—22no—
63#14519796C2×C414yes—
64C2×Dic244C2311yescharacteristic
65C2×Dic244C2311yescharacteristic
66C2×#11758202—21no—
67Q16×D61—22no—
68Q16×D61C2×C414yescomplemented
69C2×#11758200—21no—
70C2×C8×D61D411yescharacteristic
71C2×C8×D61C2311yescharacteristic
72Q8×Dic61—21no—
73C4×Dic122—21no—
74C4×Dic122—21no—
75C4×Dic122—41no—
76#14519742—21no—
77#14519742—21no—
78#14519742—41no—
79D4×Dic61—21no—
80C4×C22⋊D61—41no—
81#14519737—41no—
82#14519735—21no—
83#14519733—41no—
84C4⋊C4×D61—21no—
85C4⋊C4×D61—41no—
86C2×C4×Dic61—41no—
87C42×D61—41no—
88C42×D61D411yescharacteristic
89Dic488C2×C414yescomplemented
90#14519723C2×C414yescomplemented
91#14519721C2×C414yescomplemented
92#14519716—22no—
93#14519716C2×C414yescomplemented
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519708—21no—
100#14519702—21no—
101#14519702—21no—
102#14519701—21no—
103#14519700—21no—
104#14519700—21no—
105#14519698—21no—
106C4×#92418D411yescharacteristic
107C8×Dic61D411yescharacteristic
108C8×Dic61C2311yescharacteristic
109C2×Q16×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C4×Q8×C61—21no—
112C4×D4×C61—21no—
113SD32×C61C2×C414yescomplemented
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#103×C61—21no—
119#102×C61—21no—
120C4×C8×C61C2311yescharacteristic
121C2×Q8×C61—21no—
122C2×D4×C61—21no—
123C22×C4×C61—41no—
124C2×C22⋊D61—41no—
125C22×Dic61—41no—
126C22×Dic61—41no—
127Q8×D61—24no—
128Q8×D61—42no—
129#11758224—24no—
130#11758224—42no—
131C2×C4×D61—41no—
132C2×C4×D61—41no—
133C2×C4×D61D811yescharacteristic
134C2×C4×D61D811yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×Dic122—21no—
137C2×Dic122—21no—
138C2×Dic122—41no—
139Q16×C61—21no—
140Q16×C61C22×C412yes—
141D8×C61—21no—
142D8×C61C22×C412yes—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2411yescharacteristic
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C42×C61—41no—
150C42×C61C2×D411yescharacteristic
151C22⋊Dic61—41no—
152#11758202—24no—
153#11758200—24no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—41no—
158#11758196—41no—
159#11758196—41no—
160#11758196—41no—
161#11758196—41no—
162#11758196—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61D811yescharacteristic
170C4×Dic61D811yescharacteristic
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174Dic244—21no—
175Dic244—21no—
176Dic244C22×C412yes—
177Dic244C22×C412yes—
178C8×D61—22no—
179C8×D61C22×C412yes—
180C8×D61#4112yes—
181C16×C61—21no—
182C16×C61C22×C412yes—
183#11758185—21no—
184#11758185C22×C412yes—
185C23×C61—41no—
186C22×D61C2×D811yescharacteristic
187Q8×C61—22no—
188Q8×C61—41no—
189D4×C61—22no—
190D4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—82no—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—44no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—44no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—82no—
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208C4×D61—22no—
209C4×D61—41no—
210C4×D61—44no—
211C4×D61—82no—
212C4×D61C4×D412yes—
213C4×D61#13512yes—
214Dic122—22no—
215Dic122—22no—
216Dic122—41no—
217Dic122—41no—
218Dic122—44no—
219C8×C61—21no—
220C8×C61C23×C411yescharacteristic
221C8×C61C2×#4111yescharacteristic
222#92418—21no—
223#92418C4×D412yes—
224C4×#510—611nomaximal, Sylow
225C22×C61—42no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61C22×D811yescharacteristic
229C2×D61—21no—
230C2×D61—21no—
231C2×D61C4×D812yes—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61—81no—
236C4×C61—81no—
237C4×C61C2×C4×D411yescharacteristic
238C4×C61C2×#13511yescharacteristic
239Dic61—21no—
240Dic61—42no—
241Dic61—42no—
242Dic61—81no—
243Dic61—81no—
244Dic61C4×D812yes—
245Dic61#51012yescomplemented
246C2×#510—611no—
247C4×#135—611no—
248C2×C4×Q16—611no—
249#2176—611no—
250#2176—611no—
251#2175—611no—
252#2175D6111yescharacteristic, complemented
253C4×Q32—611no—
254C4×Q32—611no—
255C4×SD32—611no—
256C4×SD32—611no—
257#2159—611no—
258#2141—611no—
259#2139—611no—
260C4×M32—611no—
261C2×C61—82no—
262C2×C61C2×C4×D811yescharacteristic
263C2×C61C2×#51011yescharacteristic
264C2×C61#341411yescharacteristic
265D61—22no—
266C2×#135—611no—
267C22×Q16—611no—
268C4×#41—1221no—
269C2×C4×Q8—1221no—
270#510—618no—
271C2×Q32—611no—
272C2×Q32—611no—
273C2×SD32—611no—
274C2×SD32C2×D6111yescharacteristic
275C2×M32—611no—
276C2×M32—611no—
277C4×Q16—611no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×Q16—611no—
281C4×Q16—611no—
282C4×Q16—1221no—
283C4×Q16—1221no—
284C4×Q16C2×D6111yescharacteristic
285C4×SD16—1221no—
286C4×SD16—1221no—
287C4×D8—611no—
288C4×D8C2×D6111yescharacteristic
289#427—611no—
290C2×#107—611no—
291#416—1221no—
292C2×#103—1221no—
293C2×C4×C8—611no—
294#367—611no—
295#367—611no—
296#366—611no—
297#366C2×D6111yescharacteristic
298#358—611no—
299#358—611no—
300#358—611no—
301#358—611no—
302#358—611no—
303#358C2×D6111yescharacteristic
304#357—611no—
305#357C2×D6111yescharacteristic
306#346—611no—
307#346C2×D6111yescharacteristic
308C4×C16—611no—
309C4×C16—611no—
310C61C4×#51011yescharacteristic, complemented, Sylow
311C2×#41—1221no—
312C22×Q8—1221no—
313#135—614no—
314#135—1222no—
315C2×Q16—614no—
316C2×Q16—611no—
317C2×Q16—611no—
318C2×Q16—1222no—
319C2×Q16—1221no—
320C2×Q16C22×D6111yescharacteristic
321C2×SD16—1221no—
322C2×D8C22×D6111yescharacteristic
323C22×C8—611no—
324C22×C8—611no—
325C4×Q8—21no—
326C4×Q8—1221no—
327C4×Q8—1221no—
328C4×Q8—1221no—
329C4×Q8—1221no—
330C4×Q8—1221no—
331C4×Q8—2441no—
332C4×Q8—2441no—
333C4×D4—21no—
334C4×D4—1221no—
335C4×D4—2441no—
336C4×D4—2441no—
337#117—1221no—
338#117—2441no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—2441no—
341C2×C42—611no—
342C2×C42—2441no—
343C2×C42—2441no—
344Q32—614no—
345Q32—614no—
346SD32—614no—
347SD32C4×D6114yescomplemented
348M32—614no—
349M32—1222no—
350C2×C16—611no—
351C2×C16—611no—
352C2×C16C22×D6111yescharacteristic
353C2×C16C22×D6111yescharacteristic
354#107—611no—
355#107—611no—
356#107—1221no—
357#107C22×D6111yescharacteristic
358#107C22×D6111yescharacteristic
359#106—1221no—
360#103—21no—
361#103—1221no—
362#103—1221no—
363#103—1221no—
364#103—1221no—
365#103—1221no—
366#102—21no—
367#102—1221no—
368C4×C8—611no—
369C4×C8—611no—
370C4×C8—611no—
371C4×C8C22×D6111yescharacteristic
372#41—1224no—
373#41—2442no—
374C2×Q8—21no—
375C2×Q8—1224no—
376C2×Q8—1221no—
377C2×Q8—1221no—
378C2×Q8—2441no—
379C2×Q8—2442no—
380C2×D4—21no—
381C2×D4—2441no—
382C22×C4—41no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4—2441no—
390Q16—21no—
391Q16—612no—
392Q16—612no—
393Q16—1224no—
394Q16—1221no—
395Q16—1221no—
396Q16C2×C4×D6112yes—
397SD16—1224no—
398D8—21no—
399D8C2×C4×D6112yes—
400C2×C8—611no—
401C2×C8—612no—
402C2×C8—612no—
403C2×C8—611no—
404C2×C8—1222no—
405C2×C8D4×D6111yescharacteristic
406C2×C8C23×D6111yescharacteristic
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C22⋊C4—41no—
419C22⋊C4—2441no—
420C22⋊C4—2441no—
421C42—41no—
422C42—611no—
423C42—611no—
424C42—611no—
425C42—2441no—
426C42—2441no—
427C42—2441no—
428C42—2441no—
429C42—2441no—
430C42—2441no—
431C42D4×D6111yescharacteristic
432C16—21no—
433C16—612no—
434C16—1221no—
435C16C2×C4×D6112yes—
436C23—41no—
437C23—611no—
438Q8—22no—
439Q8—41no—
440Q8—1222no—
441Q8—1222no—
442Q8—2444no—
443Q8—2441no—
444Q8—2441no—
445D4—22no—
446D4—41no—
447D4—2444no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—612no—
455C2×C4—612no—
456C2×C4—611no—
457C2×C4—1222no—
458C2×C4—4882no—
459C2×C4—4882no—
460C2×C4—2441no—
461C2×C4—2444no—
462C2×C4—2444no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4D8×D6111yescharacteristic
468C2×C4D8×D6111yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C8—21no—
471C8—612no—
472C8—1221no—
473C8C22×C4×D6111yescharacteristic
474C8#41×D6111yescharacteristic
475C22—42no—
476C22—81no—
477C22—81no—
478C22—612no—
479C22—1221no—
480C22—1221no—
481C22C2×D8×D6111yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—612no—
488C4—612no—
489C4—1221no—
490C4—4881no—
491C4—4881no—
492C4—2442no—
493C4—2442no—
494C4C4×D4×D6111yescharacteristic
495C4#135×D6111yescharacteristic
496C2—82no—
497C2—1222no—
498C2C4×D8×D6111yescharacteristic
499C2#510×D6111yescharacteristic
500C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2175C2
socleC22×C61C22×D8
prime radicalC2×#14519796C22
cosocleC2×C8×C61C24
hypercenter#2175D61
hypocenterC61C4×#510
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
3. C4×Q16×D61C2semidirect
4. #21447691C2semidirect
5. #21447688C2semidirect
6. #21447683C2semidirect
7. #21447677C2semidirect
8. #21447675C2semidirect
9. #21447674C2semidirect
10. SD32×Dic61C2semidirect
11. #21447604C2semidirect
12. #21447537C2semidirect
14. #21447412C2semidirect
15. C61×#2175C2semidirect
20. #16840201C4semidirect
68. Q16×D61C2×C4semidirect
89. Dic488C2×C4semidirect
90. #14519723C2×C4semidirect
91. #14519721C2×C4semidirect
93. #14519716C2×C4semidirect
113. SD32×C61C2×C4semidirect
245. Dic61#510semidirect
252. #2175D61semidirect
310. C61C4×#510semidirect
347. SD32C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402012
C4×#145197962
C4×Q16×D612
#214476912
#214476882
#214476832
#214476772
#214476752
#214476742
SD32×Dic612
#214476042
#214475372
#214475282
#214474122
C61×#21752
C4×#51061