Finite Group Databasebeta

← #29598352

Subgroups — #29598352

Order 15616.

452 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455460C211yesmaximal, characteristic
2#21447490C211yesmaximal, characteristic
3#21447473C211yesmaximal, characteristic, complemented
4#21447468C211yesmaximal, characteristic
5#21445863C211yesmaximal, characteristic, complemented
6#21445621C211yesmaximal, characteristic, complemented
7C61×#2129C211yesmaximal, characteristic, complemented
8C22×#11758201C411yescharacteristic, complemented
9C2×#14519784C2211yescharacteristic
10#16840326C2211yescharacteristic
11#16840010—21no—
12#16840010—21no—
13#16839993—21no—
14#16839993—21no—
15#16839798—21no—
16#16839798C411yescharacteristic, complemented
17#16839771C411yescharacteristic, complemented
18#16839766C411yescharacteristic, complemented
19#16839746C2211yescharacteristic
20#16839736C2211yescharacteristic
21#16839718—21no—
22#16839554—22no—
23#16839552—22no—
24#16839548—22no—
25#16839532—21no—
26#16839518—21no—
27C61×#536C2211yescharacteristic
28C61×#410C2211yescharacteristic
29C61×#410C2211yescharacteristic
30C61×#354—22no—
31C22×D244C2×C411yescharacteristic
32C2×#11758201—21no—
33C2×#11758201—21no—
34C2×#11758201—41no—
35C2×#11758201—41no—
36#14519784—41no—
37#14519784—41no—
38C2×#11758194—21no—
39C2×#11758194C2×C411yescharacteristic
40C2×#11758194D411yescharacteristic
41C2×#11758194D411yescharacteristic
42C22×#92418C2×C411yescharacteristic
43C22×#92418D411yescharacteristic
44C22×#92418D411yescharacteristic
45#14519773—21no—
46#14519766—21no—
47#14519751—21no—
48#14519751—21no—
49#14519745—21no—
50#14519745—21no—
51#14519736—21no—
52#14519736C2×C411yescharacteristic
53#14519710—21no—
54#14519710—41no—
55#14519708—21no—
56#14519708—41no—
57#14519701—21no—
58#14519701—21no—
59#14519697—21no—
60#14519697—21no—
61#14519696—21no—
62#14519694D412yes—
63#14519692—21no—
64#14519692—21no—
65#14519688—41no—
66C4×#92418—41no—
67C2×#41×C61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C4⋊Q8×C61—21no—
70#124×C61—21no—
71#123×C61—21no—
72#123×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75#117×C61—21no—
76#117×C61—21no—
77#117×C61C2×C411yescharacteristic
78#99×C61—21no—
79#99×C61—21no—
80#99×C61—21no—
81#99×C61D412yes—
82C23×D61—21no—
83#41×C61—41no—
84#41×C61—41no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—41no—
88C2×Q8×C61—41no—
89C2×Q8×C61#4111yescharacteristic
90C2×D4×C61C2×D411yescharacteristic
91C2×D4×C61C2×D411yescharacteristic
92C2×D4×C61#4111yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61C22×C411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—41no—
102C2×D244—41no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊C4×C61—21no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61—21no—
114C42×C61—41no—
115C42×C61—41no—
116C22⋊Dic61—21no—
117C22⋊Dic61—41no—
118#11758201—42no—
119#11758201—41no—
120#11758201—41no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—41no—
125C4×Dic61—21no—
126C4×Dic61—41no—
127#11758194—22no—
128#11758194—41no—
129#11758194—41no—
130#11758194—41no—
131#11758194—41no—
132C2×#92418—21no—
133C2×#92418—21no—
134C2×#92418—21no—
135C2×#92418—21no—
136C2×#92418—21no—
137C2×#92418—21no—
138C2×#92418—41no—
139C2×#92418C22⋊C411yescharacteristic
140C2×#92418C22⋊C411yescharacteristic
141C2×#92418C22⋊C411yescharacteristic
142C2×#92418C22⋊C411yescharacteristic
143C23×C61C4×D411yescharacteristic
144C23×C61#12111yescharacteristic
145C23×C61#12711yescharacteristic
146C22×D61—41no—
147C22×D61—81no—
148C22×D61—161no—
149Q8×C61—41no—
150Q8×C61—41no—
151Q8×C61—41no—
152Q8×C61—81no—
153D4×C61—41no—
154D4×C61—41no—
155D4×C61—41no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61C2×C22⋊C411yescharacteristic
168C2×C4×C61C4×D411yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12411yescharacteristic
171C22⋊D61—161no—
172C22⋊D61—161no—
173C2×Dic61—41no—
174C2×Dic61—81no—
175D244—41no—
176D244—41no—
177D244—81no—
178#92418—21no—
179#92418—21no—
180#92418—21no—
181#92418—42no—
182#92418#9911yescharacteristic
183#92418#9911yescharacteristic
184#13229—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61#39011yescharacteristic
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—162no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—81no—
200C4×C61—82no—
201C4×C61—81no—
202C4×C61—161no—
203C4×C61C2×#9911yescharacteristic
204C4×C61#41011yescharacteristic
205Dic61—81no—
206Dic61—161no—
207#3057—611no—
208#2129D6111yescharacteristic, complemented
209#2128—611no—
210#2123—611no—
211#2122—611no—
212#1812—611no—
213#1757—611no—
214C2×C61—21no—
215C2×C61—41no—
216C2×C61—41no—
217C2×C61—41no—
218C2×C61#196411yescharacteristic
219D61—161no—
220C22×SD16—611no—
221C2×#131—611no—
222#536C2×D6111yescharacteristic
223#535—611no—
224#468—1221no—
225#468—1221no—
226#465—1221no—
227#465—1221no—
228#427—611no—
229#427—1221no—
230#419—611no—
231#418—611no—
232#413—611no—
233#410—611no—
234#410C2×D6111yescharacteristic
235#410C2×D6111yescharacteristic
236#405—1221no—
237#354—22no—
238#353—1222no—
239#352—1222no—
240AΣL(1,𝔽16)—1222no—
241#344—1221no—
242H(3,ℤ/4)—1221no—
243C61#1322911yescharacteristic, complemented, Sylow
244C2×#41C22×D6111yescharacteristic
245C22×Q8C4×D6111yescharacteristic
246C22×D4—611no—
247C2×SD16—1221no—
248C2×SD16—1221no—
249C2×SD16—2441no—
250C2×SD16—2441no—
251#131—2441no—
252#131—2441no—
253C2×M16—611no—
254C2×M16—611no—
255C2×M16—611no—
256C2×M16—1221no—
257C22×C8—611no—
258C22×C8—611no—
259C22×C8—611no—
260C4⋊Q8—21no—
261#127—1221no—
262#124—21no—
263#124—1221no—
264#123—21no—
265#123—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—1221no—
269#121—1221no—
270#120—1221no—
271#120—1221no—
272#117—21no—
273#117—21no—
274#117—611no—
275#117—1221no—
276#117C4×D6111yescharacteristic
277#107—1221no—
278#107—2441no—
279#106—1221no—
280#106—2441no—
281#103—1221no—
282#103—1221no—
283#102—1221no—
284#102—1221no—
285#101—1221no—
286#100—612no—
287#99—21no—
288#99—21no—
289#99—21no—
290#99—1221no—
291#99—1221no—
292#99C22⋊D6112yes—
293#97—2441no—
294C4×C8—2441no—
295C24—1221no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8#1175822211yescharacteristic
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—2441no—
306C2×D4—2441no—
307C2×D4—2441no—
308C2×D4—2441no—
309C2×D4D4×D6111yescharacteristic
310C2×D4#1175822411yescharacteristic
311C2×D4C2×C22⋊D6111yescharacteristic
312C22×C4—21no—
313C22×C4C2×C4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316SD16—2441no—
317SD16—2441no—
318SD16—2442no—
319M16—1222no—
320M16—2441no—
321M16—2441no—
322M16—2441no—
323M16—2441no—
324C2×C8—611no—
325C2×C8—611no—
326C2×C8—611no—
327C2×C8—611no—
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—1221no—
331C2×C8—1221no—
332C2×C8—1221no—
333C2×C8—1221no—
334C2×C8—2441no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—1221no—
341C4⋊C4—2441no—
342C22⋊C4—21no—
343C22⋊C4—22no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—1221no—
348C22⋊C4—2441no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C42—21no—
352C42—41no—
353C42—41no—
354C42—1221no—
355C42—2441no—
356C23—9761no—
357C23—4881no—
358C23—2441no—
359C23#1451973711yescharacteristic
360C23#1451974911yescharacteristic
361C23#1451977311yescharacteristic
362Q8—41no—
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—9761no—
370D4—9761no—
371D4—4881no—
372D4—2441no—
373D4—2441no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—4881no—
386C2×C4—2441no—
387C2×C4C22⋊C4×D6111yescharacteristic
388C2×C4C4×C22⋊D6111yescharacteristic
389C2×C4#1451975011yescharacteristic
390C2×C4#1451976311yescharacteristic
391C8—611no—
392C8—611no—
393C8—1221no—
394C8—1221no—
395C8—1221no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—9762no—
404C22—4881no—
405C22—4881no—
406C22—4881no—
407C22#1683966211yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—82no—
413C4—81no—
414C4—161no—
415C4—9761no—
416C4—4881no—
417C4#99×D6111yescharacteristic
418C4#1683973411yescharacteristic
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—9761no—
424C2#2144665311yescharacteristic
subgroupG/H
centerC2#21446653
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#2129C2
socleC2×C61#1964
prime radical#16840326C22
cosocleC2×#41×C61C23
hypercenter#2129D61
hypocenterC61#13229
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2129

Complements

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

NH ≅ G/Ntype
3. #21447473C2semidirect
5. #21445863C2semidirect
6. #21445621C2semidirect
7. C61×#2129C2semidirect
8. C22×#11758201C4semidirect
16. #16839798C4semidirect
17. #16839771C4semidirect
18. #16839766C4semidirect
208. #2129D61semidirect
243. C61#13229semidirect

Maximal subgroups

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

subgroupindex
#214554602
#214474902
#214474732
#214474682
#214458632
#214456212
C61×#21292
#1322961