Finite Group Databasebeta

← #29580697

Subgroups — #29580697

Order 15616.

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

normal subgroups only

This group has 724 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#21456621C211yesmaximal, characteristic
2C2×#16839917C211yesmaximal, characteristic, complemented
3C2×#16839903C211yesmaximal, characteristic
4#21445742C211yesmaximal, characteristic, complemented
5#21444180C211yesmaximal, characteristic
6#21443707C211yesmaximal, characteristic, complemented
7C61×#1472C211yesmaximal, characteristic, complemented
8C23×Dic122C2211yescharacteristic
9C22×#11758202—21no—
10C22×#11758190—21no—
11C2×#14519751C2211yescharacteristic
12#16840095—21no—
13#16840075—21no—
14#16840010—22no—
15#16839917—22no—
16#16839917—22no—
17#16839912—21no—
18#16839903—22no—
19#16839903—22no—
20#16839901—21no—
21C2×#14519702—21no—
22C2×#14519700—21no—
23C2×#14519695—22no—
24#16839726C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26#16839485—22no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#324—22no—
31C22×Dic122—21no—
32C22×Dic122—21no—
33C22×Dic122—21no—
34C22×Dic122—21no—
35C22×Dic122—21no—
36C22×Dic122—44no—
37C22×Dic122D412yes—
38C22×D244D411yescharacteristic
39C22×C22⋊D61—21no—
40C23×Dic61—21no—
41C2×#11758202—24no—
42C2×#11758202—44no—
43C2×#11758190—24no—
44C2×#11758190—44no—
45C2×#11758201—22no—
46C2×#11758194D412yes—
47C22×#92418—21no—
48#14519751—21no—
49#14519751—21no—
50#14519751—21no—
51#14519751—21no—
52C2×C4⋊Dic61D411yescharacteristic
53C2×#11758198—21no—
54#14519710—22no—
55#14519702—24no—
56#14519701—22no—
57#14519700—24no—
58#14519697—22no—
59#14519695—44no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63C22⋊#92418D412yes—
64C22×Q8×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C2×M16×C61—21no—
67C22×C8×C61—21no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61D412yes—
79C24×C61#4111yescharacteristic
80C23×D61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—42no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—42no—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95C2×C22⋊D61—82no—
96C22×Dic61—22no—
97C22×Dic61—21no—
98C22×Dic61—44no—
99C22×Dic61—41no—
100C22×Dic61—41no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244—42no—
104C2×Dic122—24no—
105C2×Dic122—24no—
106C2×Dic122—22no—
107C2×Dic122—22no—
108C2×Dic122—24no—
109C2×Dic122—44no—
110C2×Dic122—44no—
111C2×Dic122—44no—
112C2×Dic122—42no—
113C2×Dic122—42no—
114C2×Dic122—42no—
115C2×Dic122—41no—
116C2×Dic122—41no—
117M16×C61—42no—
118C2×C8×C61—22no—
119C2×C8×C61—42no—
120C2×C8×C61—41no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127#11758202—48no—
128#11758201—82no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418—42no—
137#11758190—48no—
138C23×C61—22no—
139C23×C61—22no—
140C23×C61—41no—
141C23×C61#12111yescharacteristic
142C23×C61#12111yescharacteristic
143C23×C61#12111yescharacteristic
144C22×D61—41no—
145C22×D61—42no—
146C22×D61—81no—
147C22×D61—81no—
148Q8×C61—42no—
149Q8×C61—42no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—81no—
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12011yescharacteristic
163C2×C4×C61#12711yescharacteristic
164C22⋊D61—82no—
165C22⋊D61—82no—
166C2×Dic61—24no—
167C2×Dic61—44no—
168C2×Dic61—44no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—81no—
173D244—42no—
174D244—42no—
175Dic122—24no—
176Dic122—24no—
177Dic122—42no—
178Dic122—42no—
179Dic122—44no—
180Dic122—44no—
181C8×C61—42no—
182C8×C61—81no—
183#92418—42no—
184#92418—42no—
185#9989—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—44no—
194C22×C61#39211yescharacteristic
195C22×C61#47112yes—
196C22×C61#47312yes—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—162no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—81no—
205C4×C61—81no—
206C4×C61B(4,𝔽2)12yes—
207Dic61—44no—
208Dic61—81no—
209#3193—611no—
210C2×#451—611no—
211C2×#448—611no—
212#1784—611no—
213#1472D6111yescharacteristic, complemented
214#1471—611no—
215#1298—611no—
216C2×C61—22no—
217C2×C61—22no—
218C2×C61—41no—
219C2×C61#201911yescharacteristic
220C2×C61#202011yescharacteristic
221C2×C61#202111yescharacteristic
222D61—161no—
223C23×Q8—611no—
224C22×Q16—1221no—
225C22×SD16—1221no—
226C2×C22⋊Q8C2×D6111yescharacteristic
227C2×#121—611no—
228#484—1221no—
229#480—1221no—
230#468—1222no—
231#451—1222no—
232#451—1222no—
233#450—1221no—
234#448—1222no—
235#448—1221no—
236#448—1222no—
237C2×#103—1221no—
238C2×#103—1221no—
239C2×#101—1222no—
240#407—611no—
241#407C2×D6111yescharacteristic
242C2×C22⋊C8—611no—
243C2×C22⋊C8C2×D6111yescharacteristic
244#324—22no—
245#323—1222no—
246C61#998911yescharacteristic, complemented, Sylow
247C22×Q8—612no—
248C22×Q8—1221no—
249C22×Q8—1221no—
250C22×Q8—1221no—
251C22×Q8—1221no—
252C22×Q8—1221no—
253C22×Q8—2444no—
254C22×Q8D24411yescharacteristic
255C22×D4—611no—
256C22×D4—1221no—
257C23×C4—1221no—
258C23×C4C22×D6111yescharacteristic
259C2×Q16—1224no—
260C2×Q16—2444no—
261C2×SD16—1224no—
262C2×SD16—1222no—
263C2×SD16—2444no—
264C2×M16—21no—
265C2×M16—612no—
266C22×C8—21no—
267C22×C8—1221no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272#121—1221no—
273#121—1221no—
274#121—1221no—
275#121—1221no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—611no—
278C2×C4⋊C4D24411yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—1221no—
281#107—1222no—
282#103—1224no—
283#103—1224no—
284#103—1222no—
285#102—1222no—
286#101—2444no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—612no—
291C22⋊C8—1221no—
292C22⋊C8—1221no—
293C22⋊C8—1221no—
294C22⋊C8C22⋊D6112yes—
295C24—1221no—
296C24#1175822211yescharacteristic
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×Q8—1224no—
301C2×Q8—1224no—
302C2×Q8—1224no—
303C2×Q8—1222no—
304C2×Q8—1222no—
305C2×Q8—2442no—
306C2×Q8—2442no—
307C2×Q8—2442no—
308C2×Q8—2444no—
309C2×Q8—2444no—
310C2×Q8—2444no—
311C2×Q8—2441no—
312C2×Q8—2441no—
313C2×D4—1221no—
314C2×D4—1221no—
315C2×D4—4882no—
316C2×D4—2442no—
317C2×D4—2441no—
318C2×D4—2441no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—1222no—
323C22×C4—1221no—
324C22×C4—2444no—
325C22×C4—2441no—
326C22×C4—2441no—
327C22×C4C2×D24411yescharacteristic
328C22×C4D4×D6111yescharacteristic
329C22×C4D4×D6111yescharacteristic
330C22×C4D4×D6111yescharacteristic
331C22×C4D4×D6111yescharacteristic
332C22×C4C2×C22⋊D6111yescharacteristic
333Q16—2448no—
334SD16—4882no—
335SD16—2448no—
336M16—42no—
337M16—2442no—
338C2×C8—22no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—1222no—
342C2×C8—1222no—
343C2×C8—2442no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—2441no—
353C22⋊C4—2441no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—4881no—
358C23—4881no—
359C23—2442no—
360C23—2441no—
361C23#1451975011yescharacteristic
362C23#1451975011yescharacteristic
363C23#1451975111yescharacteristic
364Q8—42no—
365Q8—42no—
366Q8—1224no—
367Q8—1224no—
368Q8—2442no—
369Q8—2444no—
370Q8—2442no—
371Q8—2444no—
372D4—4882no—
373D4—4882no—
374D4—2442no—
375D4—2442no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—1224no—
387C2×C4—4881no—
388C2×C4—2441no—
389C2×C4—2442no—
390C2×C4—2444no—
391C2×C4—2444no—
392C2×C4—2442no—
393C2×C4#1451974511yescharacteristic
394C2×C4#1451974711yescharacteristic
395C2×C4#1451974711yescharacteristic
396C2×C4#1451977311yescharacteristic
397C8—42no—
398C8—81no—
399C8—2442no—
400C8—2442no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—44no—
409C22—9762no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22#1683967111yescharacteristic
414C22#1684002512yes—
415C22#1684003612yes—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—81no—
421C4—4881no—
422C4—2444no—
423C4#1683995012yes—
424C2—22no—
425C2—22no—
426C2—41no—
427C2—9761no—
428C2#2144693411yescharacteristic
429C2#2144693911yescharacteristic
430C2#2144694411yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1472C2
socleC22×C61#392
prime radicalC2×#14519751C22
hypercenter#1472D61
hypocenterC61#9989
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
2. C2×#16839917C2semidirect
4. #21445742C2semidirect
6. #21443707C2semidirect
7. C61×#1472C2semidirect
213. #1472D61semidirect
246. C61#9989semidirect

Maximal subgroups

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

subgroupindex
#214566212
C2×#168399172
C2×#168399032
#214457422
#214441802
#214437072
C61×#14722
#998961