Finite Group Databasebeta

← #30606528

Subgroups — #30606528

Order 15616.

456 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462538C211yesmaximal, characteristic
2#21462071C211yesmaximal, characteristic
3#21461577C211yesmaximal, characteristic, complemented
4#21461576C211yesmaximal, characteristic, complemented
5#21460909C211yesmaximal, characteristic, complemented
6#21460906C211yesmaximal, characteristic, complemented
7#21447457C211yesmaximal, characteristic
8C2×#14519864C2211yescharacteristic
9C22×#11758217C411yescharacteristic, complemented
10#16840797—21no—
11#16840795—21no—
12#16840794—21no—
13#16840793—21no—
14#16840768C411yescharacteristic, complemented
15#16840749C2211yescharacteristic
16#16840742C2211yescharacteristic
17#16840735C2211yescharacteristic
18#16840732C411yescharacteristic, complemented
19#16840728—21no—
20#16840726—21no—
21#16840726C411yescharacteristic, complemented
22#16840665C412yescomplemented
23#16840662C412yescomplemented
24#16840645—21no—
25#16840643—21no—
26#16840268C2211yescharacteristic
27#16839743C2211yescharacteristic
28#16839734C2211yescharacteristic
29#16839553C411yescharacteristic, complemented
30#16839551C411yescharacteristic, complemented
31#16839549C411yescharacteristic, complemented
32C61×AΣL(1,𝔽16)C411yescharacteristic, complemented
33#14519864—41no—
34#14519864—41no—
35C2×#11758213D411yescharacteristic
36C2×#11758217—21no—
37C2×#11758217—21no—
38C2×#11758217—21no—
39C2×#11758217—41no—
40C2×#11758217—41no—
41C2×#11758217C2×C411yescharacteristic
42C2×#11758217C2×C411yescharacteristic
43C2×#11758217D411yescharacteristic
44C2×D61⋊C8D411yescharacteristic
45C22×#92420C2×C411yescharacteristic
46C22×#92420C2×C411yescharacteristic
47C22×#92420D411yescharacteristic
48#14519856—21no—
49#14519844—21no—
50C4⋊#92420—21no—
51#14519838—21no—
52#14519837C2×C412yes—
53#14519835—21no—
54#14519835C2×C412yes—
55#14519834—21no—
56C22⋊#92420—21no—
57C22⋊#92420—21no—
58C22⋊#92420—21no—
59#14519832—21no—
60#14519832—41no—
61#14519831—41no—
62C4×#92420—21no—
63C4×#92420—41no—
64C8×#6923—41no—
65C2×#11758224C2311yescharacteristic
66C23×Dic61C2×C411yescharacteristic
67M16×D61—21no—
68#14519761C2×C411yescharacteristic
69#14519758C2×C411yescharacteristic
70#14519754—21no—
71#14519749—21no—
72#14519746C2×C411yescharacteristic
73D4×Dic61—21no—
74#14519737—21no—
75#14519733—21no—
76#14519733C2×C411yescharacteristic
77#14519695—21no—
78#14519694C2×C411yescharacteristic
79#14519692C2×C411yescharacteristic
80#14519691—21no—
81#120×C61C2×C411yescharacteristic
82#100×C61C2×C411yescharacteristic
83#99×C61C2×C411yescharacteristic
84C24×C61C4211yescharacteristic
85C2×D4×C61—21no—
86C2×D4×C61C22×C411yescharacteristic
87C2×C22⋊D61—21no—
88C2×C22⋊D61C22×C411yescharacteristic
89C2×C22⋊D61#4111yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—41no—
93C22×Dic61—41no—
94C22×Dic61C4211yescharacteristic
95C22×Dic61C22×C411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97#11758224—41no—
98#11758224—41no—
99C2×C4×D61C2×D411yescharacteristic
100C2×Dic122#4111yescharacteristic
101C22⋊#6923—22no—
102#11758217—22no—
103#11758217—41no—
104#11758217—41no—
105#11758217—42no—
106#11758217—41no—
107#11758217—41no—
108#11758217—41no—
109#11758217—41no—
110C2×#92420—21no—
111C2×#92420—21no—
112C2×#92420—21no—
113C2×#92420—21no—
114C2×#92420—21no—
115C2×#92420—21no—
116C2×#92420—41no—
117C2×#92420—41no—
118C2×#92420C22⋊C411yescharacteristic
119C2×#92420C22⋊C411yescharacteristic
120C2×#92420C22⋊C411yescharacteristic
121C2×#92420C22⋊C411yescharacteristic
122C4⋊#6923—41no—
123C4×#6923—41no—
124#11758213—41no—
125D61⋊C8—21no—
126D61⋊C8—21no—
127M16×C61—21no—
128C22⋊C4×C61—21no—
129C22⋊C4×C61—21no—
130C22⋊C4×C61C4211yescharacteristic
131C22⋊Dic61—21no—
132C22⋊Dic61—21no—
133C22⋊Dic61—21no—
134C22⋊Dic61C4211yescharacteristic
135#11758198—21no—
136C4⋊Dic61—21no—
137#11758196—21no—
138#11758196—21no—
139#11758196—41no—
140C4×Dic61—21no—
141C4×Dic61—41no—
142#11758194—21no—
143#11758189—41no—
144C8×D61—41no—
145C23×C61—21no—
146C23×C61—41no—
147C23×C61—81no—
148C23×C61C2×C4211yescharacteristic
149C23×C61C4×D411yescharacteristic
150C22×D61C4×D411yescharacteristic
151C2×#6923—42no—
152D4×C61—41no—
153D4×C61—81no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61C4×D411yescharacteristic
157C22⋊D61—41no—
158C22⋊D61—41no—
159C22⋊D61—81no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61C2×C22⋊C411yescharacteristic
172C2×Dic61#11711yescharacteristic
173C2×Dic61C4×D411yescharacteristic
174C4×D61—21no—
175C4×D61—21no—
176Dic122—41no—
177#92420—21no—
178#92420—21no—
179#92420—21no—
180#92420—42no—
181#92420—41no—
182#92420—41no—
183#92420#9911yescharacteristic
184#92420#9911yescharacteristic
185C8×C61—41no—
186#92418—41no—
187#12624—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61—41no—
194C22×C61—41no—
195C22×C61—82no—
196C22×C61C4×C22⋊C411yescharacteristic
197C2×D61—21no—
198C2×D61—41no—
199#6923—82no—
200C4×C61—21no—
201C4×C61—41no—
202C4×C61—81no—
203Dic61—21no—
204Dic61—21no—
205Dic61—41no—
206Dic61—41no—
207Dic61—81no—
208Dic61C2×#9911yescharacteristic
209Dic61#41011yescharacteristic
210#2930—611no—
211#2120—611no—
212#2120—611no—
213#1757—611no—
214#1757—611no—
215#1392—611no—
216#1391—611no—
217C2×C61—21no—
218C2×C61—41no—
219C2×C61—41no—
220C2×C61—81no—
221C2×C61C4×#9911yescharacteristic
222D61—41no—
223C2×#131—611no—
224C22×M16—611no—
225#525—611no—
226#436—1221no—
227#436—1221no—
228#435—1221no—
229#435—1221no—
230#424—611no—
231#413—611no—
232#413—611no—
233#410—611no—
234#410—611no—
235#408—611no—
236#407—611no—
237#405—611no—
238#405—1221no—
239#405—1221no—
240#352—612no—
241#352—611no—
242#352—611no—
243AΣL(1,𝔽16)—612no—
244AΣL(1,𝔽16)—611no—
245AΣL(1,𝔽16)#692311yescharacteristic, complemented
246#344—1221no—
247#344—1221no—
248C61#1262411yescharacteristic, complemented, Sylow
249C2×#41—611no—
250C23×C4—611no—
251#131—2441no—
252#131—2441no—
253C2×M16—611no—
254C2×M16—611no—
255C2×M16—611no—
256C2×M16—611no—
257C2×M16—1221no—
258C2×M16—1221no—
259C2×M16—1221no—
260C2×M16—1221no—
261C2×M16—2441no—
262C2×M16—2441no—
263C22×C8—611no—
264C22×C8—611no—
265C22×C8—611no—
266C22×C8—611no—
267#123—611no—
268#123—611no—
269C22⋊Q8—1221no—
270#121—1221no—
271#120—611no—
272#120C2×#692311yescharacteristic
273C4×D4—1221no—
274C4×D4—1221no—
275#117—611no—
276#117—1221no—
277#117—1221no—
278C4⋊C8—1221no—
279C4⋊C8—1221no—
280#101—1221no—
281#101—1221no—
282#100—612no—
283#100—611no—
284#100C2×#692311yescharacteristic
285#99—612no—
286#99—611no—
287#99—1221no—
288#99—1221no—
289#99C2×#692311yescharacteristic
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292C22⋊C8—1221no—
293C22⋊C8—1221no—
294#97—1221no—
295#97—2441no—
296#97—2441no—
297C4×C8—1221no—
298C4×C8—2441no—
299C4×C8—2441no—
300C24C4×#692311yescharacteristic
301#41—2441no—
302#41—2441no—
303C2×Q8—611no—
304C2×D4—21no—
305C2×D4—611no—
306C2×D4—611no—
307C2×D4—1221no—
308C2×D4C22×#692311yescharacteristic
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—611no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—2441no—
316C22×C4—2441no—
317M16—21no—
318M16—1222no—
319M16—1221no—
320M16—2441no—
321M16—2441no—
322M16—2441no—
323M16—2441no—
324M16—2441no—
325M16—2441no—
326M16—2442no—
327M16—2441no—
328M16—2441no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—611no—
332C2×C8—611no—
333C2×C8—1221no—
334C2×C8—1221no—
335C2×C8—1221no—
336C2×C8—1221no—
337C2×C8—1221no—
338C2×C8—1221no—
339C2×C8—1221no—
340C2×C8—1221no—
341C2×C8—2441no—
342C2×C8—2441no—
343C2×C8—2441no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—2441no—
348C4⋊C4—2441no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—611no—
352C22⋊C4—1221no—
353C22⋊C4—1221no—
354C22⋊C4—1221no—
355C22⋊C4—1222no—
356C22⋊C4—1221no—
357C22⋊C4C4×#692311yescharacteristic
358C42—1221no—
359C42—2441no—
360C42—2441no—
361C23—21no—
362C23—41no—
363C23—81no—
364C23—611no—
365C23C2×C4×#692311yescharacteristic
366C23D4×#692311yescharacteristic
367Q8—2441no—
368D4—41no—
369D4—81no—
370D4—4881no—
371D4—2441no—
372D4—2441no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—611no—
376C2×C4—611no—
377C2×C4—611no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4D4×#692311yescharacteristic
393C8—41no—
394C8—611no—
395C8—611no—
396C8—1221no—
397C8—1221no—
398C8—1221no—
399C8—2441no—
400C8—2441no—
401C8—2441no—
402C8—2442no—
403C22—21no—
404C22—21no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—82no—
411C22—1221no—
412C22—2441no—
413C22C22⋊C4×#692311yescharacteristic
414C4—21no—
415C4—41no—
416C4—81no—
417C4—611no—
418C4—611no—
419C4—1221no—
420C4—1221no—
421C4—4881no—
422C4—4882no—
423C4—2441no—
424C4—2441no—
425C2—21no—
426C2—41no—
427C2—41no—
428C2—81no—
429C2—2441no—
430C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC23×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC61×AΣL(1,𝔽16)C4
socleC2×C61C4×#99
prime radical#16840268C22
cosocleC2×#11758224C23
hypercenterAΣL(1,𝔽16)#6923
hypocenterC61#12624
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
3. #21461577C2semidirect
4. #21461576C2semidirect
5. #21460909C2semidirect
6. #21460906C2semidirect
9. C22×#11758217C4semidirect
14. #16840768C4semidirect
18. #16840732C4semidirect
21. #16840726C4semidirect
22. #16840665C4semidirect
23. #16840662C4semidirect
29. #16839553C4semidirect
30. #16839551C4semidirect
31. #16839549C4semidirect
32. C61×AΣL(1,𝔽16)C4semidirect
245. AΣL(1,𝔽16)#6923semidirect
248. C61#12624semidirect

Maximal subgroups

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

subgroupindex
#214625382
#214620712
#214615772
#214615762
#214609092
#214609062
#214474572
#1262461