Finite Group Databasebeta

← #29832982

Subgroups — #29832982

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840576C211yesmaximal, characteristic
2#21458921C211yesmaximal, characteristic, complemented
3#21454427C211yesmaximal, characteristic, complemented
4#21447568C211yesmaximal, characteristic, complemented
5#21447565C214yesmaximal, complemented
6C2×#16839579C211yesmaximal, characteristic, complemented
7C2×#16839577C211yesmaximal, characteristic, complemented
8#362×D61C214yesmaximal, complemented
9C61×#2148C211yesmaximal, characteristic, complemented
10C2×#14519824—21no—
11#16840576—22no—
12#16840576—22no—
13#16840576C414yes—
14#16840576C414yes—
15C2×#14519807—21no—
16C2×#14519806—22no—
17C2×#14519796C2212yes—
18C22×Dic244C2211yescharacteristic
19C2×Q16×D61C2212yes—
20#16840528—21no—
21C2×#14519783C2211yescharacteristic
22#16840180—22no—
23#16840178—22no—
24C2×#14519717C2211yescharacteristic
25M32×D61C2214yes—
26#16839810—21no—
27#16839809—22no—
28C2×#14519713C2211yescharacteristic
29#108×D61C2214yes—
30#16839579C2214yes—
31#16839578C2212yes—
32#16839577C2214yes—
33#16839575C2212yes—
34C2×#135×C61C2211yescharacteristic
35C2×M32×C61C2211yescharacteristic
36C2×#108×C61C2211yescharacteristic
37C61×#362C2212yes—
38C61×#361C2212yes—
39#14519824—24no—
40#14519824—24no—
41#14519824—42no—
42#14519824—42no—
43C2×#11758222—41no—
44C2×#11758222D411yescharacteristic
45C2×#11758224—22no—
46C2×#11758224—42no—
47C22×Dic122—21no—
48C22×Dic122—41no—
49C2×Q8×D61—22no—
50#14519807—24no—
51#14519807—24no—
52#14519806—28no—
53#14519806—28no—
54#14519796—24no—
55#14519796C2×C418yes—
56C2×Dic244—21no—
57C2×Dic244—21no—
58C2×Dic244—21no—
59C2×Dic244C2×C412yes—
60C2×Dic244C2×C412yes—
61C2×Dic244C2312yes—
62C2×#11758202—22no—
63Q16×D61—24no—
64Q16×D61C2×C418yes—
65C2×#11758200—22no—
66#14519785—42no—
67#14519783—42no—
68#14519783C2×C412yes—
69#14519783C2×C412yes—
70#14519783D412yes—
71#14519783D412yes—
72C2×#11758194—21no—
73C2×#11758194D411yescharacteristic
74C2×#11758189D412yes—
75M16×D61—24no—
76C2×C8×D61C2312yes—
77#14519717—21no—
78#14519717C2312yes—
79#14519716—24no—
80C2×#11758185—21no—
81C16×D61—24no—
82#14519713—21no—
83#14519713C2312yes—
84#14519712—24no—
85C2×#41×C61—21no—
86#135×C61—21no—
87#135×C61—21no—
88#135×C61C2×C412yes—
89#135×C61C2×C412yes—
90C2×Q16×C61C2311yescharacteristic
91C2×SD16×C61—21no—
92C2×D8×C61C2311yescharacteristic
93C2×M16×C61—21no—
94C22×C8×C61C2311yescharacteristic
95M32×C61—21no—
96M32×C61C2312yes—
97C2×C16×C61—21no—
98#108×C61—21no—
99#108×C61C2312yes—
100#41×C61—22no—
101#41×C61—22no—
102#41×C61—41no—
103#41×C61—41no—
104C2×Q8×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C2×C22⋊D61—42no—
110C2×C22⋊D61SD1612yes—
111C22×Dic61—41no—
112C22×Dic61—42no—
113Q8×D61—28no—
114Q8×D61—44no—
115#11758224—28no—
116#11758224—48no—
117#11758224—48no—
118#11758224—44no—
119#11758222—44no—
120#11758222—44no—
121#11758222C22⋊C412yes—
122#11758222C22⋊C412yes—
123#11758222D812yes—
124#11758222D812yes—
125C2×D244SD1611yescharacteristic
126C2×C4×D61—42no—
127C2×C4×D61C2×D412yes—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—41no—
132C2×Dic122—41no—
133C2×Dic122—41no—
134C2×Dic122—42no—
135C2×Dic122—44no—
136C2×Dic122—44no—
137C2×Dic122SD1611yescharacteristic
138Q16×C61—21no—
139Q16×C61C22×C412yes—
140SD16×C61—22no—
141SD16×C61—22no—
142D8×C61—21no—
143D8×C61C22×C412yes—
144M16×C61—22no—
145M16×C61—41no—
146C2×C8×C61—41no—
147C2×C8×C61C22×C411yescharacteristic
148C2×C8×C61C22×C411yescharacteristic
149C2×C8×C61C2×D411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2×D411yescharacteristic
152C2×C8×C61C2411yescharacteristic
153#11758202—28no—
154#11758200—28no—
155#11758194—22no—
156#11758194—41no—
157#11758194C22⋊C412yes—
158#11758194C22⋊C412yes—
159C2×#92418—41no—
160C2×#92418C2×D412yes—
161Dic244—22no—
162Dic244—22no—
163Dic244—22no—
164Dic244C22×C414yes—
165#11758189—44no—
166#11758189C22⋊C414yes—
167#11758189C22⋊C414yes—
168C8×D61—44no—
169C8×D61C22×C414yes—
170C8×D61C2×D414yes—
171C16×C61—22no—
172#11758185—22no—
173C23×C61—41no—
174C23×C61C2×SD1611yescharacteristic
175C22×D61C2×SD1612yes—
176Q8×C61—22no—
177Q8×C61—41no—
178D4×C61—22no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61C2×C22⋊C411yescharacteristic
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C2×C4×C61C2×D811yescharacteristic
188C2×C4×C61C2×D811yescharacteristic
189C2×C4×C61C2×SD1611yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C22⋊D61—48no—
192C22⋊D61#10214yes—
193C22⋊D61#10214yes—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—48no—
198C2×Dic61C2×SD1612yes—
199D244#10212yes—
200D244#10212yes—
201C4×D61—48no—
202C4×D61C2×C22⋊C414yes—
203C4×D61C2×D814yes—
204Dic122—24no—
205Dic122—42no—
206Dic122—42no—
207Dic122—42no—
208Dic122—48no—
209Dic122#10212yes—
210Dic122#10212yes—
211C8×C61—42no—
212C8×C61C2×C22⋊C411yescharacteristic
213C8×C61C2×C22⋊C411yescharacteristic
214C8×C61C23×C411yescharacteristic
215C8×C61C22×D411yescharacteristic
216#92418—42no—
217#92418C2×C22⋊C414yes—
218#33393—611nomaximal, Sylow
219C22×C61—41no—
220C22×C61—42no—
221C22×C61—82no—
222C22×C61C2×#10211yescharacteristic
223C22×C61C2×#10211yescharacteristic
224C22×C61C22×SD1611yescharacteristic
225C2×D61—42no—
226C2×D61C2×#10214yes—
227C4×C61—42no—
228C4×C61C2×#10211yescharacteristic
229C4×C61C2×#10211yescharacteristic
230C4×C61C22×C22⋊C411yescharacteristic
231C4×C61C22×D811yescharacteristic
232Dic61—44no—
233Dic61C2×#10214yes—
234C2×#578—611no—
235#3407—611no—
236#2916—611no—
237#2148—614no—
238#2148—611no—
239#2148D6111yescharacteristic, complemented
240C2×#362—611no—
241C2×#362—611no—
242C2×#362—614no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—82no—
247C2×C61C22×#10211yescharacteristic
248D61—24no—
249C2×2−1+4—1221no—
250#578—614no—
251#578—614no—
252#578—1222no—
253#578—1222no—
254C2×#137—1222no—
255C2×#137—1221no—
256C2×#135—612no—
257C2×#135C2×D6111yescharacteristic
258C22×Q16—612no—
259C22×Q16—611no—
260#568—1221no—
261C2×#131—611no—
262#504—1222no—
263#504—1222no—
264C2×M32—611no—
265C2×M32—614no—
266C2×M32C2×D6111yescharacteristic
267#430—1222no—
268#430—1221no—
269C2×#108—611no—
270C2×#108—614no—
271C2×#108C2×D6111yescharacteristic
272#362—612no—
273#362—614no—
274#362—614no—
275#362C2×D6112yes—
276#361—612no—
277#361C2×D6112yes—
278C61#3339311yescharacteristic, complemented, Sylow
2792−1+4—1224no—
2802−1+4—1224no—
2812−1+4—2442no—
2822−1+4—2442no—
283C2×#41—21no—
284C2×#41—611no—
285C2×#41—1222no—
286C2×#41—2441no—
287C2×#41—2442no—
288C22×Q8—1222no—
289C22×Q8—1221no—
290C22×Q8—2441no—
291#137—1228no—
292#137—1228no—
293#137—1224no—
294#137—1224no—
295#135—21no—
296#135—21no—
297#135—618no—
298#135—1224no—
299#135C4×D6112yes—
300#135C4×D6112yes—
301C2×Q16—618no—
302C2×Q16—612no—
303C2×Q16—612no—
304C2×Q16—612no—
305C2×Q16—1224no—
306C2×Q16—1222no—
307C2×Q16—1221no—
308C2×Q16—1221no—
309C2×Q16—1221no—
310C2×Q16C22×D6111yescharacteristic
311C2×SD16—21no—
312C2×SD16—1222no—
313C2×D8C22×D6111yescharacteristic
314#131—612no—
315#131—612no—
316#131—612no—
317#131—612no—
318#131—2442no—
319#131—2442no—
320C2×M16—21no—
321C2×M16—611no—
322C2×M16—612no—
323C2×M16—1224no—
324C2×M16—1221no—
325C22×C8—612no—
326C22×C8C22×D6111yescharacteristic
327M32—21no—
328M32—612no—
329M32—1224no—
330M32—1221no—
331M32C22×D6112yes—
332C2×C16—21no—
333C2×C16—1224no—
334C2×C16—1221no—
335#108—21no—
336#108—612no—
337#108—1224no—
338#108—1221no—
339#108C22×D6112yes—
340#41—22no—
341#41—22no—
342#41—41no—
343#41—41no—
344#41—612no—
345#41—612no—
346#41—612no—
347#41—612no—
348#41—1228no—
349#41—2444no—
350#41—2444no—
351#41—2444no—
352#41—2448no—
353#41—2448no—
354C2×Q8—21no—
355C2×Q8—611no—
356C2×Q8—1228no—
357C2×Q8—1222no—
358C2×Q8—1222no—
359C2×Q8—1222no—
360C2×Q8—2442no—
361C2×Q8—2444no—
362C2×Q8—2444no—
363C2×Q8—2444no—
364C2×Q8—2441no—
365C2×Q8—2441no—
366C2×Q8—2441no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—612no—
370C2×D4—611no—
371C2×D4—2442no—
372C22×C4—41no—
373C22×C4—612no—
374C22×C4—2442no—
375C22×C4—2442no—
376C22×C4—2441no—
377C22×C4D4×D6111yescharacteristic
378Q16—21no—
379Q16—614no—
380Q16—1228no—
381Q16—1222no—
382Q16—1222no—
383Q16—1222no—
384Q16C2×C4×D6112yes—
385SD16—22no—
386SD16—22no—
387SD16—1228no—
388D8—21no—
389D8C2×C4×D6112yes—
390M16—22no—
391M16—41no—
392M16—612no—
393M16—612no—
394M16—614no—
395M16—614no—
396M16—1222no—
397M16—2444no—
398M16—2441no—
399C2×C8—41no—
400C2×C8—612no—
401C2×C8—614no—
402C2×C8—614no—
403C2×C8—2444no—
404C2×C8—2441no—
405C2×C8C2×C4×D6111yescharacteristic
406C2×C8C2×C4×D6111yescharacteristic
407C2×C8D4×D6111yescharacteristic
408C2×C8D4×D6111yescharacteristic
409C2×C8D4×D6111yescharacteristic
410C2×C8C23×D6111yescharacteristic
411C16—22no—
412C16—1222no—
413C23—41no—
414C23—612no—
415C23SD16×D6111yescharacteristic
416Q8—22no—
417Q8—41no—
418Q8—612no—
419Q8—612no—
420Q8—1224no—
421Q8—2448no—
422Q8—2442no—
423Q8—2442no—
424Q8—2442no—
425D4—22no—
426D4—42no—
427D4—42no—
428D4—41no—
429D4—614no—
430D4—614no—
431D4—612no—
432D4—612no—
433D4—2448no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—612no—
438C2×C4—614no—
439C2×C4—614no—
440C2×C4—2448no—
441C2×C4—2448no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4C22⋊C4×D6111yescharacteristic
446C2×C4C22⋊C4×D6111yescharacteristic
447C2×C4D8×D6111yescharacteristic
448C2×C4D8×D6111yescharacteristic
449C2×C4SD16×D6111yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C8—42no—
452C8—614no—
453C8—2442no—
454C8C22⋊C4×D6111yescharacteristic
455C8C22⋊C4×D6111yescharacteristic
456C8C22×C4×D6111yescharacteristic
457C8C2×D4×D6111yescharacteristic
458C22—41no—
459C22—42no—
460C22—82no—
461C22—614no—
462C22—2442no—
463C22#102×D6111yescharacteristic
464C22#102×D6111yescharacteristic
465C22C2×SD16×D6111yescharacteristic
466C4—42no—
467C4—614no—
468C4—2444no—
469C4#102×D6111yescharacteristic
470C4#102×D6111yescharacteristic
471C4C2×C22⋊C4×D6111yescharacteristic
472C4C2×D8×D6111yescharacteristic
473C2—21no—
474C2—21no—
475C2—21no—
476C2—82no—
477C2—1224no—
478C2C2×#102×D6111yescharacteristic
subgroupG/H
centerC2C2×#102×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2148C2
socleC2×C61C22×#102
prime radicalC2×#16840576C2
cosocleC2×C8×C61C24
hypercenter#2148D61
hypocenterC61#33393
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. #21458921C2semidirect
3. #21454427C2semidirect
4. #21447568C2semidirect
5. #21447565C2semidirect
6. C2×#16839579C2semidirect
7. C2×#16839577C2semidirect
8. #362×D61C2semidirect
9. C61×#2148C2semidirect
239. #2148D61semidirect
278. C61#33393semidirect

Maximal subgroups

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

subgroupindex
C2×#168405762
#214589212
#214544272
#214475682
#214475652
C2×#168395792
C2×#168395772
#362×D612
C61×#21482
#3339361