Finite Group Databasebeta

← #29576308

Subgroups — #29576308

Order 15616.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448284C211yesmaximal, characteristic, complemented
2#21446648C211yesmaximal, characteristic, complemented
3#21444314C211yesmaximal, characteristic, complemented
4#21444244C211yesmaximal, characteristic, complemented
5#21443721C211yesmaximal, characteristic, complemented
6#21443651C211yesmaximal, characteristic, complemented
7C61×#3100C211yesmaximal, characteristic
8C2×#14519736—21no—
9#16839815C2211yescharacteristic
10#16839786C2211yescharacteristic
11C2×#14519704—21no—
12C2×#14519704—21no—
13#16839665—21no—
14#16839661—21no—
15#16839655—22no—
16C4×C4⋊Dic61C2211yescharacteristic
17C4×C22⋊Dic61—21no—
18C42×Dic61C2211yescharacteristic
19#16839496—22no—
20D4⋊#92418—22no—
21C61×#532C2211yescharacteristic
22C2×C4×D4×C61C2211yescharacteristic
23C61×#480—21no—
24C61×#474C412yescomplemented
25C61×#459C412yescomplemented
26C61×#447—21no—
27C4×D8×C61—22no—
28C61×#431C2211yescharacteristic
29C2×#11758194—21no—
30#14519736—42no—
31#14519736—42no—
32C2×C4⋊Dic61—22no—
33C2×C4⋊Dic61—21no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61—22no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61—21no—
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44#14519704—42no—
45#14519704—42no—
46C22⋊#92418—21no—
47C4×#92418—21no—
48#14519684—22no—
49#14519684D412yes—
50C2×#41×C61—21no—
51C22×D4×C61D411yescharacteristic
52C23×C4×C61—21no—
53C2×D8×C61—22no—
54C4⋊Q8×C61C2×C411yescharacteristic
55#127×C61C2×C411yescharacteristic
56#124×C61—21no—
57#121×C61—21no—
58#121×C61—21no—
59C4×D4×C61—21no—
60C4×D4×C61—21no—
61C4×D4×C61—21no—
62C4×D4×C61—21no—
63C4×D4×C61C2×C412yes—
64C4×D4×C61D412yes—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C22⋊C4×C61—21no—
67C2×C42×C61C2311yescharacteristic
68#107×C61—22no—
69C4⋊C8×C61C2×C412yes—
70#102×C61—22no—
71#102×C61—22no—
72C22⋊C8×C61—21no—
73C4×C8×C61—21no—
74C24×C61—21no—
75#41×C61—41no—
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79C2×Q8×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—42no—
85C2×D4×C61—41no—
86C2×D4×C61C22⋊C412yes—
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—41no—
91C22×C4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—22no—
101D8×C61—44no—
102C2×C8×C61—22no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61C22⋊C412yes—
107C22⋊C4×C61—22no—
108C22⋊C4×C61—22no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61C22×C411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61#4111yescharacteristic
114C42×C61#4111yescharacteristic
115C22⋊Dic61—44no—
116C22⋊Dic61—44no—
117C4⋊Dic61—44no—
118C4⋊Dic61—42no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121C4×Dic61—21no—
122C4×Dic61—42no—
123C4×Dic61—42no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126C4×Dic61D812yescomplemented
127#11758194—41no—
128#11758194—41no—
129C2×#92418—41no—
130C2×#92418—41no—
131C23×C61—22no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61#12011yescharacteristic
138Q8×C61—41no—
139Q8×C61—41no—
140D4×C61—22no—
141D4×C61—22no—
142D4×C61—42no—
143D4×C61—42no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—81no—
147D4×C61—81no—
148D4×C61—81no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12311yescharacteristic
167C2×C4×C61C2×D811yescharacteristic
168C2×C4×C61#13511yescharacteristic
169C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
171C2×Dic61—22no—
172C2×Dic61—22no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C8×C61—42no—
181#92418—81no—
182#92418—81no—
183#9237—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61—82no—
190C22×C61—82no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61#38611yescharacteristic
195C22×C61#44711yescharacteristic
196C22×C61#44911yescharacteristic
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—42no—
201C4×C61—81no—
202C4×C61#42112yes—
203C4×C61C4×D811yescharacteristic
204C4×C61#44211yescharacteristic
205C4×C61#45911yescharacteristic
206C4×C61#46311yescharacteristic
207Dic61—42no—
208Dic61—42no—
209Dic61—82no—
210#3100D6111yescharacteristic
211#2293—611no—
212#1962—611no—
213#1502—611no—
214#1486—611no—
215#1303—611no—
216#1276—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—82no—
220C2×C61—161no—
221C2×C61#187911yescharacteristic
222C2×C61#189711yescharacteristic
223C2×C61#190611yescharacteristic
224#532C2×D6111yescharacteristic
225C2×C4×D4C2×D6111yescharacteristic
226C2×#117—1221no—
227#480—21no—
228#474Dic6112yescomplemented
229#459Dic6112yescomplemented
230#447—21no—
231C4×D8—22no—
232#431—611no—
233#431C2×D6111yescharacteristic
234#423—611no—
235C2×#104—1221no—
236C2×#104—1221no—
237#390—1221no—
238#389—1221no—
239#388—1222no—
240C4×C4⋊C4—611no—
241C4×C22⋊C4—1221no—
242C43—611no—
243#327—1222no—
244D4⋊C8—1222no—
245C61#923711yescharacteristic, complemented, Sylow
246C2×#41—21no—
247C22×D4C22⋊D6111yescharacteristic
248C23×C4—21no—
249C2×D8—22no—
250C2×M16—1221no—
251C4⋊Q8C2×Dic6111yescharacteristic
252#127C2×Dic6111yescharacteristic
253#124—21no—
254#121—21no—
255#121—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×Dic6112yes—
261C4×D4C22⋊D6112yes—
262#117—2442no—
263#117—2442no—
264C2×C4⋊C4—1222no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4C22⋊D6111yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1222no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—1221no—
273C2×C42—1221no—
274C2×C42—1221no—
275C2×C42C22×D6111yescharacteristic
276#107—22no—
277C4⋊C8—1221no—
278C4⋊C8—1221no—
279C4⋊C8—1221no—
280C4⋊C8C2×Dic6112yes—
281#104—2442no—
282#104—2442no—
283#102—22no—
284#102—22no—
285C22⋊C8—21no—
286C22⋊C8—1221no—
287C4×C8—21no—
288C4×C8—1221no—
289#95—612no—
290#95—1222no—
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4C22⋊Dic6112yes—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1222no—
315C22×C4D4×D6111yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318D8—44no—
319M16—2441no—
320M16—2441no—
321C2×C8—22no—
322C2×C8—2441no—
323C2×C8—2441no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—2442no—
328C4⋊C4—2444no—
329C4⋊C4C22⋊Dic6112yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—2444no—
335C22⋊C4—2444no—
336C42—612no—
337C42—1221no—
338C42—1221no—
339C42—1221no—
340C42—2442no—
341C42—2442no—
342C42—2441no—
343C42—2441no—
344C42#1175822411yescharacteristic
345C42#1175822411yescharacteristic
346C42C22×Dic6111yescharacteristic
347C42C2×C22⋊D6111yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23#1451974511yescharacteristic
355Q8—41no—
356Q8—41no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4C2×C22⋊Dic6111yescharacteristic
388C2×C4D4×Dic6111yescharacteristic
389C2×C4D4×Dic6111yescharacteristic
390C2×C4#1451974911yescharacteristic
391C2×C4#1451974911yescharacteristic
392C2×C4#1451975811yescharacteristic
393C2×C4D8×D6111yescharacteristic
394C2×C4#1451979611yescharacteristic
395C2×C4#1451980111yescharacteristic
396C2×C4#1451980111yescharacteristic
397C8—42no—
398C8—4881no—
399C8—4881no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#1683964211yescharacteristic
411C22#1683989411yescharacteristic
412C22#1683990411yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—4882no—
419C4—2442no—
420C4—2442no—
421C4#1683977912yes—
422C4D8×Dic6111yescharacteristic
423C4#1683987411yescharacteristic
424C4#1683995511yescharacteristic
425C4#1683997511yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2#2144616911yescharacteristic
431C2#2144626811yescharacteristic
432C2#2144632211yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3100C2
socleC22×C61#386
cosocleC2×C42×C61C23
hypercenter#3100D61
hypocenterC61#9237
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21448284C2semidirect
2. #21446648C2semidirect
3. #21444314C2semidirect
4. #21444244C2semidirect
5. #21443721C2semidirect
6. #21443651C2semidirect
24. C61×#474C4semidirect
25. C61×#459C4semidirect
126. C4×Dic61D8semidirect
228. #474Dic61semidirect
229. #459Dic61semidirect
245. C61#9237semidirect

Maximal subgroups

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

subgroupindex
#214482842
#214466482
#214443142
#214442442
#214437212
#214436512
C61×#31002
#923761