Finite Group Databasebeta

← #29596845

Subgroups — #29596845

Order 15616.

492 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 89 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#21448930C211yesmaximal, characteristic
2C2×#16839558C211yesmaximal, characteristic
3#21446222C211yesmaximal, characteristic
4#21446034C211yesmaximal, characteristic, complemented
5#21443946C211yesmaximal, characteristic, complemented
6#21443945C211yesmaximal, characteristic, complemented
7C61×#1857C211yesmaximal, characteristic, complemented
8C2×#14519766C2211yescharacteristic
9#16839805—21no—
10#16839766—21no—
11C2×#14519696C2211yescharacteristic
12C2×#14519696C2211yescharacteristic
13#16839669C2211yescharacteristic
14#16839654—21no—
15#16839652—21no—
16#16839640—21no—
17C4×#11758198—21no—
18#16839609—21no—
19#16839607—21no—
20#16839558C414yescomplemented
21#16839521—21no—
22#16839521—21no—
23C61×#519C2211yescharacteristic
24C61×#428—21no—
25C2×#101×C61C2211yescharacteristic
26C61×#391C2211yescharacteristic
27C61×#339—21no—
28C22×D244D411yescharacteristic
29C22×C4×D61—21no—
30C2×#11758194—21no—
31C2×#11758194—21no—
32#14519766—21no—
33#14519766—21no—
34#14519766C2×C412yes—
35#14519766Q812yes—
36C2×C4⋊Dic61D411yescharacteristic
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46#14519710—41no—
47#14519708—41no—
48#14519697—41no—
49#14519697—41no—
50#14519696—21no—
51#14519696—21no—
52#14519696—21no—
53#14519696C2×C412yescomplemented
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61C2311yescharacteristic
59C2×M16×C61—21no—
60C4×Q8×C61—41no—
61C4×Q8×C61—41no—
62#117×C61—21no—
63#117×C61—21no—
64#117×C61—21no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C42×C61D411yescharacteristic
67#107×C61—41no—
68#106×C61—41no—
69#101×C61—21no—
70#101×C61C2×C412yescomplemented
71#95×C61—21no—
72#95×C61—21no—
73C23×D61—21no—
74C2×Q8×C61—41no—
75C2×Q8×C61—41no—
76C2×Q8×C61C22×C411yescharacteristic
77C2×Q8×C61C2×Q811yescharacteristic
78C2×Q8×C61#4111yescharacteristic
79C2×Q8×C61#4111yescharacteristic
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C2×D244—21no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244C4⋊C412yes—
92C2×C4×D61—42no—
93C2×C4×D61—42no—
94C2×C4×D61—81no—
95M16×C61—22no—
96M16×C61—41no—
97C2×C8×C61—41no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C42×C61—41no—
107C42×C61—41no—
108C42×C61—41no—
109C42×C61—41no—
110#11758198—22no—
111#11758198—22no—
112#11758198—22no—
113#11758198—22no—
114#11758198—42no—
115#11758198—42no—
116#11758198—81no—
117C4⋊Dic61—41no—
118C4×Dic61—21no—
119C4×Dic61—42no—
120C4×Dic61—42no—
121C4×Dic61—41no—
122C4×Dic61C4⋊C412yes—
123#11758194—22no—
124#11758194—41no—
125#11758194—41no—
126#11758194—41no—
127C2×#92418—41no—
128C2×#92418—41no—
129C23×C61#12011yescharacteristic
130C22×D61—22no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137Q8×C61—41no—
138Q8×C61—41no—
139Q8×C61—41no—
140Q8×C61—41no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61C2×C4⋊C411yescharacteristic
156C2×C4×C61C4×D411yescharacteristic
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61C22⋊Q811yescharacteristic
159C2×C4×C61C22⋊Q811yescharacteristic
160C2×C4×C61#12311yescharacteristic
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170D244—42no—
171D244—81no—
172C4×D61—84no—
173C8×C61—42no—
174#92418—42no—
175#92418—81no—
176#12972—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61#38511yescharacteristic
180C22×C61B(4,𝔽2)11yescharacteristic
181C22×C61#45811yescharacteristic
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—81no—
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—42no—
196C4×C61—81no—
197C4×C61—81no—
198C4×C61—81no—
199Dic61—42no—
200Dic61—42no—
201Dic61—81no—
202#2372—611no—
203C2×#355—611no—
204#1888—611no—
205#1857—611no—
206#1857D6111yescharacteristic, complemented
207#1394—611no—
208#1394—611no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61#186711yescharacteristic
212C2×C61#220411yescharacteristic
213C2×C61#220411yescharacteristic
214D61—82no—
215C2×#124—611no—
216#519C2×D6111yescharacteristic
217#428—21no—
218#428—1221no—
219#418—1221no—
220C2×#101—611no—
221C2×#101—611no—
222C2×#101C2×D6111yescharacteristic
223#391—611no—
224#391C2×D6111yescharacteristic
225#388—1221no—
226#388—1221no—
227#386—1221no—
228C4×C22⋊C4—1221no—
229C4×C22⋊C4—1221no—
230#376—1221no—
231#355—614no—
232#339—21no—
233#339—1221no—
234#339—1221no—
235C61#1297211yescharacteristic, complemented, Sylow
236C22×Q8C22×D6111yescharacteristic
237C22×D4—611no—
238C23×C4—1221no—
239C2×M16—21no—
240C2×M16—1221no—
241C2×M16—1221no—
242#124—612no—
243#124—612no—
244#124—1221no—
245#124—1221no—
246C4×Q8—41no—
247C4×Q8—41no—
248#117—21no—
249#117—21no—
250#117—21no—
251C2×C4⋊C4—611no—
252C2×C4⋊C4C22⋊D6111yescharacteristic
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C42—611no—
259C2×C42—611no—
260C2×C42—1221no—
261C2×C42—1221no—
262C2×C42C22⋊D6111yescharacteristic
263#107—41no—
264#107—2441no—
265#106—41no—
266#106—2441no—
267#102—2441no—
268#102—2441no—
269#101—21no—
270#101—612no—
271#101—1221no—
272#101—1221no—
273#101—1221no—
274#101C4×D6112yescomplemented
275#95—21no—
276#95—21no—
277#95—1221no—
278#95—1221no—
279#95—1221no—
280#95—1221no—
281C24—1221no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8C2×C4×D6111yescharacteristic
285C2×Q8#1175822211yescharacteristic
286C2×Q8Q8×D6111yescharacteristic
287C2×Q8#1175822611yescharacteristic
288C2×D4—612no—
289C2×D4—1221no—
290C2×D4—2441no—
291C2×D4—2441no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—4881no—
298C22×C4—2442no—
299C22×C4—2442no—
300C22×C4D4×D6111yescharacteristic
301C22×C4D4×D6111yescharacteristic
302C22×C4C2×C22⋊D6111yescharacteristic
303M16—22no—
304M16—41no—
305M16—1222no—
306M16—2441no—
307M16—2441no—
308M16—2441no—
309C2×C8—41no—
310C2×C8—2441no—
311C2×C8—2441no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—2441no—
320C22⋊C4—41no—
321C22⋊C4—1222no—
322C22⋊C4—1222no—
323C22⋊C4—1222no—
324C22⋊C4—1222no—
325C22⋊C4—4881no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—612no—
333C42—1221no—
334C42—2441no—
335C42—2442no—
336C42—2442no—
337C23—1222no—
338C23—2441no—
339C23—2441no—
340C23—2441no—
341C23—2441no—
342C23—2441no—
343C23—2441no—
344C23#1451974511yescharacteristic
345Q8—41no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349D4—4881no—
350D4—2442no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—1222no—
366C2×C4—1222no—
367C2×C4—4884no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4C4⋊C4×D6111yescharacteristic
376C2×C4C4×C22⋊D6111yescharacteristic
377C2×C4#1451973911yescharacteristic
378C2×C4#1451975311yescharacteristic
379C2×C4#1451975511yescharacteristic
380C2×C4#1451976211yescharacteristic
381C8—42no—
382C8—4881no—
383C8—2442no—
384C22—21no—
385C22—21no—
386C22—4881no—
387C22—4881no—
388C22—4881no—
389C22—4881no—
390C22—2442no—
391C22—2442no—
392C22—2442no—
393C22#1683963311yescharacteristic
394C22#1683994811yescharacteristic
395C22#1683995111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—4881no—
407C4—2442no—
408C4—2442no—
409C2—21no—
410C2—21no—
411C2—4882no—
412C2#2144609811yescharacteristic
413C2#2144784311yescharacteristic
414C2#2144784411yescharacteristic
subgroupG/H
centerC22#16839948
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1857C2
socleC22×C61B(4,𝔽2)
prime radicalC22×D244D4
cosocleC22×Q8×C61C23
hypercenter#1857D61
hypocenterC61#12972
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
4. #21446034C2semidirect
5. #21443946C2semidirect
6. #21443945C2semidirect
7. C61×#1857C2semidirect
20. #16839558C4semidirect
53. #14519696C2×C4semidirect
70. #101×C61C2×C4semidirect
206. #1857D61semidirect
235. C61#12972semidirect
274. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214489302
C2×#168395582
#214462222
#214460342
#214439462
#214439452
C61×#18572
#1297261