Finite Group Databasebeta

← #29712838

Subgroups — #29712838

Order 15616.

1314 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 187 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#21459463C211yesmaximal, characteristic
2#21455750C212yesmaximal, complemented
3#21447036C214yesmaximal, complemented
4#21446262C212yesmaximal, complemented
5#21446028C212yesmaximal, complemented
6#21445772C212yesmaximal, complemented
7#21445747C211yesmaximal, characteristic, complemented
8C61×#2297C211yesmaximal, characteristic, complemented
9C2×#14519822C2211yescharacteristic
10#16840411—24no—
11#16840400—24no—
12#16840336C2212yes—
13#16840321—22no—
14#16840273C2212yes—
15C2×#14519760—22no—
16#16840237C2211yescharacteristic
17#16840103—24no—
18#16840086—24no—
19#16839802C2214yescomplemented
20C2×#14519705C2214yes—
21#16839761C2214yes—
22C2×#14519696C2212yescomplemented
23C2×#14519694C2212yes—
24#16839726C2212yescomplemented
25#16839522C2214yescomplemented
26C22×#41×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic, complemented
28C22×C42×C61C2211yescharacteristic, complemented
29C61×#391C2212yescomplemented
30C61×#390C2212yes—
31C61×#386—22no—
32C4×C22⋊C4×C61—22no—
33#14519822—24no—
34#14519822—42no—
35C2×#11758222C2312yes—
36C2×#11758224—22no—
37C22×C22⋊D61—22no—
38C2×D4×D61—22no—
39C2×#11758194C2314yes—
40#14519775—22no—
41#14519766—22no—
42#14519760—28no—
43#14519760—22no—
44#14519760—24no—
45#14519758—24no—
46#14519756D414yes—
47#14519753—24no—
48#14519751D414yes—
49#14519749—28no—
50#14519746—22no—
51#14519745—24no—
52D4×Dic61—24no—
53C4×C22⋊D61—24no—
54#14519736C2312yes—
55C2×#11758196—24no—
56C2×C22⋊Dic61—22no—
57C2×#11758198—24no—
58#14519710—24no—
59#14519708—24no—
60#14519705—44no—
61#14519700—24no—
62#14519697—24no—
63#14519696—42no—
64#14519694—24no—
65C22⋊#92418—24no—
66C2×#41×C61—24no—
67C2×#41×C61—24no—
68C2×#41×C61—42no—
69C22×Q8×C61C2311yescharacteristic
70C22×D4×C61D412yes—
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61—21no—
73C23×C4×C61D412yes—
74C23×C4×C61C2311yescharacteristic
75#117×C61—24no—
76C2×C4⋊C4×C61C2312yes—
77C2×C22⋊C4×C61—24no—
78C2×C22⋊C4×C61—21no—
79C2×C42×C61—21no—
80C2×C42×C61—21no—
81C2×C42×C61—42no—
82C2×C42×C61C2312yes—
83#95×C61—24no—
84C24×C61C2×D411yescharacteristic
85C24×C61C2×D412yes—
86C23×D61—22no—
87#41×C61—416no—
88C2×Q8×C61—22no—
89C2×Q8×C61—22no—
90C2×Q8×C61—22no—
91C2×D4×C61—24no—
92C2×D4×C61—24no—
93C2×D4×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—42no—
96C2×D4×C61#4114yes—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—24no—
100C22×C4×C61—22no—
101C22×C4×C61—24no—
102C22×C4×C61—22no—
103C22×C4×C61—42no—
104C22×C4×C61—44no—
105C22×C4×C61—42no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D414yes—
109C22×C4×C61C2411yescharacteristic
110C2×C22⋊D61—24no—
111C2×C22⋊D61—22no—
112C2×C22⋊D61—24no—
113C2×C22⋊D61—44no—
114C2×C22⋊D61—44no—
115C22×Dic61—22no—
116C22×Dic61—24no—
117#11758224—48no—
118D4×D61—48no—
119#11758222—24no—
120#11758222—24no—
121C2×D244C2×D412yes—
122C2×C4×D61—22no—
123C2×Dic122C2×D412yes—
124C4⋊C4×C61—24no—
125C22⋊C4×C61—48no—
126C22⋊C4×C61—42no—
127C42×C61—24no—
128C42×C61—42no—
129C42×C61—42no—
130C22⋊Dic61—24no—
131C22⋊Dic61—24no—
132C22⋊Dic61—22no—
133#11758198—28no—
134#11758198—24no—
135C4⋊Dic61C2×D414yes—
136#11758196—28no—
137#11758196—24no—
138C4×Dic61—22no—
139#11758194—24no—
140#11758194—24no—
141C2×#92418—24no—
142C23×C61—22no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—22no—
146C23×C61—42no—
147C23×C61—44no—
148C23×C61#12012yes—
149C23×C61#12314yes—
150C23×C61C22×D411yescharacteristic
151C22×D61—22no—
152C22×D61—44no—
153C22×D61—84no—
154Q8×C61—44no—
155D4×C61—28no—
156D4×C61—48no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—28no—
160C2×C4×C61—24no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—48no—
165C2×C4×C61—42no—
166C2×C4×C61—48no—
167C2×C4×C61—42no—
168C2×C4×C61—44no—
169C2×C4×C61—42no—
170C2×C4×C61#12012yes—
171C2×C4×C61C22×D412yes—
172C2×C4×C61C2×#4112yes—
173C2×C4×C61H(5,𝔽2)12yes—
174C22⋊D61—48no—
175C22⋊D61—42no—
176C22⋊D61—42no—
177C22⋊D61—84no—
178C2×Dic61—24no—
179C2×Dic61—22no—
180C2×Dic61—44no—
181C2×Dic61—84no—
182D244—24no—
183C4×D61—44no—
184Dic122—24no—
185#92418—44no—
186#24021—611nomaximal, Sylow
187C22×C61—22no—
188C22×C61—24no—
189C22×C61—22no—
190C22×C61—24no—
191C22×C61—44no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—44no—
195C22×C61—82no—
196C22×C61C2×#12011yescharacteristic
197C22×C61C2×#12311yescharacteristic
198C22×C61#53411yescharacteristic
199C2×D61—42no—
200C2×D61—44no—
201C2×D61—164no—
202C4×C61—22no—
203C4×C61—44no—
204C4×C61—42no—
205C4×C61—82no—
206C4×C61#54614yes—
207Dic61—42no—
208Dic61—44no—
209#3485—611no—
210#3093—612no—
211#2297D6111yescharacteristic, complemented
212#2038—614no—
213#1896—612no—
214#1856—612no—
215#1792—612no—
216#1785—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—44no—
220C2×C61—42no—
221C2×C61#240511yescharacteristic
222C2×C61#301812yes—
223D61—82no—
224C2×H(5,𝔽2)—611no—
225C22×#41C2×D6111yescharacteristic
226#547—1224no—
227#546—1224no—
228#536—612no—
229#534—1222no—
230#525—612no—
231C2×#123—1222no—
232#518—611no—
233C2×#117C2×D6111yescharacteristic, complemented
234C22×C42C2×D6111yescharacteristic, complemented
235#485—1224no—
236#482—1224no—
237#428—614no—
238C2×#104—614no—
239#417—614no—
240C2×#101—612no—
241C2×#100—612no—
242#407—612no—
243#391C2×D6112yescomplemented
244#390C2×D6112yes—
245#386—22no—
246C4×C22⋊C4—22no—
247#339—614no—
248C61#2402111yescharacteristic, complemented, Sylow
249H(5,𝔽2)—1224no—
250H(5,𝔽2)—2442no—
251C2×#41—24no—
252C2×#41—24no—
253C2×#41—42no—
254C2×#41—612no—
255C2×#41—1222no—
256C22×Q8C22×D6111yescharacteristic
257C22×D4—1222no—
258C22×D4—1222no—
259C22×D4C22⋊D6112yes—
260C22×D4C22×D6111yescharacteristic
261C23×C4—21no—
262C23×C4C22⋊D6112yes—
263C23×C4C22×D6111yescharacteristic
264C2×M16—614no—
265C4⋊Q8—1222no—
266#124—1222no—
267#123—1228no—
268#123—1224no—
269#123—1222no—
270#123—1224no—
271C22⋊Q8—614no—
272C22⋊Q8—1224no—
273#121—614no—
274#121—1228no—
275#120—1224no—
276#120—1222no—
277C4×D4—1224no—
278C4×D4—1224no—
279#117—24no—
280#117—612no—
281C2×C4⋊C4—1224no—
282C2×C4⋊C4C22×D6112yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—1224no—
286C2×C22⋊C4—1222no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—42no—
290C2×C42C22×D6112yes—
291#107—1224no—
292#106—1224no—
293#104—2444no—
294#103—1224no—
295#102—1224no—
296#101—2442no—
297#100—1224no—
298C22⋊C8—1224no—
299#95—24no—
300C24—1222no—
301C24C2×C22⋊D6111yescharacteristic
302C24C2×C22⋊D6112yes—
303#41—416no—
304#41—1224no—
305#41—1224no—
306#41—2448no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—612no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—42no—
316C2×D4—612no—
317C2×D4—1224no—
318C2×D4—1224no—
319C2×D4—1222no—
320C2×D4—2448no—
321C2×D4—2444no—
322C2×D4—2444no—
323C2×D4#1175822214yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—1224no—
334C22×C4—1222no—
335C22×C4—1222no—
336C22×C4D4×D6114yes—
337C22×C4C2×C22⋊D6112yes—
338C22×C4C2×C22⋊D6111yescharacteristic
339C22×C4C23×D6111yescharacteristic
340M16—1224no—
341M16—1224no—
342C2×C8—1224no—
343C4⋊C4—24no—
344C4⋊C4—614no—
345C4⋊C4—1228no—
346C4⋊C4—1224no—
347C22⋊C4—48no—
348C22⋊C4—42no—
349C22⋊C4—1228no—
350C22⋊C4—1224no—
351C22⋊C4—1222no—
352C22⋊C4—1224no—
353C22⋊C4—1224no—
354C42—24no—
355C42—42no—
356C42—42no—
357C42—1222no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—42no—
363C23—44no—
364C23—1222no—
365C23—4884no—
366C23—2444no—
367C23#1451974512yes—
368C23#1451976014yes—
369C23C22×C22⋊D6111yescharacteristic
370Q8—44no—
371Q8—1224no—
372D4—28no—
373D4—48no—
374D4—1224no—
375D4—4884no—
376D4—2448no—
377D4—2442no—
378D4—2442no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—1224no—
393C2×C4—1222no—
394C2×C4—4884no—
395C2×C4—2444no—
396C2×C4—2444no—
397C2×C4#1451974512yes—
398C2×C4C2×D4×D6112yes—
399C2×C4C2×#1175822212yes—
400C2×C4#1451982212yes—
401C8—2444no—
402C22—22no—
403C22—24no—
404C22—22no—
405C22—24no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—82no—
411C22—9764no—
412C22—2444no—
413C22—2442no—
414C22C2×#1451974511yescharacteristic
415C22C2×#1451976011yescharacteristic
416C22#1684032111yescharacteristic
417C4—22no—
418C4—44no—
419C4—42no—
420C4—82no—
421C4—2444no—
422C4—2442no—
423C4#1684040214yes—
424C2—21no—
425C2—21no—
426C2—42no—
427C2—44no—
428C2—4882no—
429C2#2144914611yescharacteristic
430C2#2145516712yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2297C2
socleC22×C61C2×#120
prime radical#21459463C2
hypercenter#2297D61
hypocenterC61#24021
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #21455750C2semidirect
3. #21447036C2semidirect
4. #21446262C2semidirect
5. #21446028C2semidirect
6. #21445772C2semidirect
7. #21445747C2semidirect
8. C61×#2297C2semidirect
19. #16839802C22semidirect
22. C2×#14519696C22semidirect
24. #16839726C22semidirect
25. #16839522C22semidirect
27. C2×#117×C61C22semidirect
28. C22×C42×C61C22semidirect
29. C61×#391C22semidirect
211. #2297D61semidirect
233. C2×#117C2×D61semidirect
234. C22×C42C2×D61semidirect
243. #391C2×D61semidirect
248. C61#24021semidirect

Maximal subgroups

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

subgroupindex
#214594632
#214557502
#214470362
#214462622
#214460282
#214457722
#214457472
C61×#22972
#2402161