Finite Group Databasebeta

← #30600826

Subgroups — #30600826

Order 15616.

496 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462737C211yesmaximal, characteristic
2#21461985C211yesmaximal, characteristic, complemented
3#21461694C211yesmaximal, characteristic
4#21461335C211yesmaximal, characteristic, complemented
5#21460715C211yesmaximal, characteristic, complemented
6#21460688C211yesmaximal, characteristic, complemented
7#21455633C211yesmaximal, characteristic
8Q8×#92420—21no—
9#16840790—21no—
10#16840790—22no—
11#16840780—21no—
12#16840780C2211yescharacteristic
13#16840767C2211yescharacteristic
14D61⋊(C4⋊C8)—21no—
15D61⋊(C4⋊C8)C2211yescharacteristic
16C2×#14519842—21no—
17C2×#14519840—22no—
18C4×C4⋊#6923C2211yescharacteristic
19#16840632—21no—
20#16840621—21no—
21#16840621—21no—
22C4⋊Q8×D61C2211yescharacteristic
23C4×#11758226C2211yescharacteristic
24#16840057C411yescharacteristic, complemented
25#16840053C411yescharacteristic, complemented
26#16840049C411yescharacteristic, complemented
27#16840045—21no—
28#16840041—21no—
29#16839803—21no—
30#16839784C2211yescharacteristic
31#16839762—21no—
32C61×#477C411yescharacteristic, complemented
33C2×#11758213—21no—
34C2×D61⋊C8—22no—
35C2×D61⋊C8D412yes—
36C2×C4⋊#6923—21no—
37C2×C4×#6923—21no—
38#14519844—21no—
39#14519844—21no—
40#14519844—21no—
41#14519844D411yescharacteristic
42#14519844D411yescharacteristic
43C4⋊#92420—22no—
44C4⋊#92420—21no—
45#14519842—42no—
46#14519840—44no—
47#14519834—22no—
48#14519834—21no—
49C4×#92420—22no—
50C4×#92420—21no—
51(C2×#6923)⋊C4—21no—
52C2×#11758226D411yescharacteristic
53C2×Q8×D61—21no—
54C2×#11758189—21no—
55#14519777C2×C411yescharacteristic
56#14519776—21no—
57#14519776—21no—
58#14519775—21no—
59Q8×Dic61D411yescharacteristic
60C4×D244—21no—
61C4×D244C2×C411yescharacteristic
62#14519739—21no—
63#14519739D411yescharacteristic
64#14519735—21no—
65#14519735D411yescharacteristic
66C4⋊C4×D61—21no—
67C4⋊C4×D61—21no—
68C42×D61—21no—
69C42×D61C2311yescharacteristic
70#14519710—21no—
71#14519709—21no—
72#14519708—21no—
73C4⋊#92418C2×C411yescharacteristic
74#14519706—21no—
75#14519701—21no—
76#14519699—21no—
77#14519697—21no—
78C4⋊Q8×C61C2×C411yescharacteristic
79C4×Q8×C61C2×C411yescharacteristic
80#107×C61—21no—
81C4⋊C8×C61C2×C411yescharacteristic
82#103×C61—21no—
83C22×#6923—41no—
84C2×Q8×C61—21no—
85C2×Q8×C61C22⋊C411yescharacteristic
86#11758226—21no—
87#11758226—21no—
88#11758226SD1612yes—
89#11758226Q1612yes—
90Q8×D61—42no—
91Q8×D61—42no—
92C2×D244—21no—
93C2×D244C22⋊C411yescharacteristic
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—41no—
98C2×C4×D61C2×D411yescharacteristic
99C2×C4×D61C2×D411yescharacteristic
100C2×C4×D61#4111yescharacteristic
101C2×Dic122—21no—
102C2×Dic122—41no—
103C2×#92420—21no—
104C2×#92420—22no—
105C2×#92420—21no—
106C2×#92420—41no—
107C4⋊#6923—22no—
108C4⋊#6923—41no—
109C4×#6923—41no—
110C4×#6923—41no—
111#11758213—41no—
112#11758213—41no—
113D61⋊C8—42no—
114D61⋊C8—82no—
115C2×C8×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61C22⋊C411yescharacteristic
120C42×C61—21no—
121C42×C61C22×C411yescharacteristic
122#11758198—21no—
123#11758198—21no—
124#11758198—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61C22⋊C411yescharacteristic
129#11758196—41no—
130#11758196—41no—
131C4×Dic61—21no—
132C4×Dic61—21no—
133C4×Dic61C2×D411yescharacteristic
134C4×Dic61C2×D411yescharacteristic
135C4×Dic61#4111yescharacteristic
136C2×#92418—21no—
137#11758189—41no—
138#11758189—41no—
139C22×D61—21no—
140C22×D61#12111yescharacteristic
141C2×#6923—42no—
142C2×#6923—81no—
143C2×#6923—81no—
144Q8×C61—21no—
145Q8×C61—42no—
146Q8×C61#10312yescomplemented
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61C2×C22⋊C411yescharacteristic
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61#12011yescharacteristic
156C2×Dic61#12111yescharacteristic
157C2×Dic61#12311yescharacteristic
158D244—22no—
159D244—22no—
160D244—21no—
161D244#10312yescomplemented
162C4×D61—21no—
163C4×D61—21no—
164C4×D61—22no—
165C4×D61—22no—
166C4×D61—41no—
167C4×D61—41no—
168C4×D61—41no—
169C4×D61—81no—
170C4×D61—82no—
171C4×D61C2×SD1611yescharacteristic
172C4×D61C2×Q1611yescharacteristic
173C4×D61#13511yescharacteristic
174C4×D61#13511yescharacteristic
175Dic122—42no—
176Dic122—81no—
177Dic122—81no—
178#92420—42no—
179#92420—41no—
180#92420—41no—
181#92420—81no—
182C8×C61—41no—
183#92418—41no—
184#9317—611nomaximal, Sylow
185C22×C61#38611yescharacteristic
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—22no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61#46511yescharacteristic
192C2×D61#46711yescharacteristic
193#6923—82no—
194C4×C61—21no—
195C4×C61—22no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—81no—
199C4×C61C2×#10311yescharacteristic
200C4×C61#41611yescharacteristic
201Dic61—21no—
202Dic61—41no—
203Dic61—81no—
204Dic61#45311yescharacteristic
205Dic61#45411yescharacteristic
206Dic61#46511yescharacteristic
207Dic61#46711yescharacteristic
208#3079—611no—
209#2988—611no—
210#1981—611no—
211#1806—611no—
212#1549—611no—
213#1303—611no—
214#1292—611no—
215C2×C61#188311yescharacteristic
216C2×C61#189211yescharacteristic
217C2×C61#190211yescharacteristic
218D61—21no—
219D61—21no—
220D61—42no—
221C2×C4⋊Q8—611no—
222C4×#41—611no—
223#477—611no—
224#477#692311yescharacteristic, complemented
225#476—611no—
226#476—611no—
227#475—1221no—
228#474—1221no—
229C8×Q8—1221no—
230C8×D4—1221no—
231C8×D4—1222no—
232#431—611no—
233#431—1221no—
234#428—1221no—
235#423—611no—
236#423—611no—
237C2×C4⋊C8—611no—
238C2×C4⋊C8—1221no—
239C2×#104—1221no—
240#417—1221no—
241C2×#103—1222no—
242C4×C4⋊C4—611no—
243#339—1221no—
244#332—1221no—
245#332—1221no—
246C61#931711yescharacteristic, complemented, Sylow
247C2×#41—611no—
248C22×Q8—1221no—
249C2×M16—1221no—
250C2×M16—1221no—
251C22×C8—612no—
252C22×C8—1222no—
253C4⋊Q8—611no—
254C4⋊Q8—1221no—
255C4⋊Q8—1221no—
256C4⋊Q8—1221no—
257C4⋊Q8C2×#692311yescharacteristic
258C4×Q8—611no—
259C4×Q8C2×#692311yescharacteristic
260C4×D4—611no—
261C4×D4—611no—
262C4×D4—1221no—
263C4×D4—1221no—
264#117—611no—
265#117—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—1221no—
269C2×C42—611no—
270C2×C42—1221no—
271C2×C42—1221no—
272#107—21no—
273#107—1221no—
274#106—1221no—
275#106—1221no—
276C4⋊C8—611no—
277C4⋊C8—611no—
278C4⋊C8—611no—
279C4⋊C8—1221no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282C4⋊C8—1221no—
283C4⋊C8—1221no—
284C4⋊C8—1222no—
285C4⋊C8C2×#692311yescharacteristic
286#104—2442no—
287#103—21no—
288#103—1221no—
289#103—2444no—
290#102—1221no—
291#102—1221no—
292C22⋊C8—1221no—
293C22⋊C8—1222no—
294C4×C8—1221no—
295C4×C8—1222no—
296#95—1221no—
297#41—612no—
298#41—612no—
299#41—1221no—
300#41—1221no—
301C2×Q8—21no—
302C2×Q8—1221no—
303C2×Q8—2441no—
304C2×Q8—2442no—
305C2×Q8—2442no—
306C2×Q8C22⋊#692311yescharacteristic
307C2×D4—611no—
308C2×D4—1221no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—2441no—
316C22×C4—2441no—
317M16—2441no—
318M16—2441no—
319M16—2441no—
320M16—2441no—
321C2×C8—21no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—1222no—
325C2×C8—1221no—
326C2×C8—4882no—
327C2×C8—2441no—
328C2×C8—2442no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—611no—
333C4⋊C4—1221no—
334C4⋊C4—1222no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—2441no—
338C4⋊C4—2441no—
339C4⋊C4—2441no—
340C4⋊C4C22⋊#692311yescharacteristic
341C22⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C42—21no—
345C42—611no—
346C42—611no—
347C42—611no—
348C42—1221no—
349C42—1221no—
350C42—2441no—
351C42—2441no—
352C42C22×#692311yescharacteristic
353C23—611no—
354C23—1221no—
355Q8—21no—
356Q8—42no—
357Q8—4881no—
358Q8—4881no—
359Q8—2442no—
360Q8#1451984012yescomplemented
361D4—612no—
362D4—1222no—
363D4—1222no—
364D4—1221no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—611no—
369C2×C4—611no—
370C2×C4—611no—
371C2×C4—611no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—611no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1222no—
378C2×C4—1222no—
379C2×C4—1221no—
380C2×C4—4881no—
381C2×C4—4882no—
382C2×C4—4881no—
383C2×C4—4881no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2442no—
388C2×C4—2441no—
389C2×C4C2×C22⋊#692311yescharacteristic
390C2×C4D4×#692311yescharacteristic
391C2×C4D4×#692311yescharacteristic
392C8—41no—
393C8—4881no—
394C8—2441no—
395C8—2441no—
396C8—2442no—
397C8—2441no—
398C22—611no—
399C22—611no—
400C22—1221no—
401C22—1221no—
402C22—1222no—
403C22—2441no—
404C22—2441no—
405C22#1684071011yescharacteristic
406C4—21no—
407C4—22no—
408C4—21no—
409C4—41no—
410C4—81no—
411C4—611no—
412C4—611no—
413C4—611no—
414C4—611no—
415C4—1221no—
416C4—4882no—
417C4—4881no—
418C4—2441no—
419C4C2×#1451984011yescharacteristic
420C4#1684075211yescharacteristic
421C2—1221no—
422C2—1221no—
423C2—2442no—
424C2#2146181911yescharacteristic
425C2#2146183211yescharacteristic
426C2#2146184911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#477C4
socleC22×C61#386
prime radicalC2×#11758226D4
cosocleC42×D61C23
hypercenter#477#6923
hypocenterC61#9317
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #21461985C2semidirect
4. #21461335C2semidirect
5. #21460715C2semidirect
6. #21460688C2semidirect
24. #16840057C4semidirect
25. #16840053C4semidirect
26. #16840049C4semidirect
32. C61×#477C4semidirect
146. Q8×C61#103semidirect
161. D244#103semidirect
224. #477#6923semidirect
246. C61#9317semidirect
360. Q8#14519840semidirect

Maximal subgroups

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

subgroupindex
#214627372
#214619852
#214616942
#214613352
#214607152
#214606882
#214556332
#931761