Finite Group Databasebeta

← #29580709

Subgroups — #29580709

Order 15616.

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

normal subgroups only

This group has 698 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#21456640C211yesmaximal, characteristic
2C2×#16839920C211yesmaximal, characteristic
3C2×#16839917C211yesmaximal, characteristic
4#21445742C211yesmaximal, characteristic
5#21444184C211yesmaximal, characteristic
6#21443707C211yesmaximal, characteristic
7C61×#1472C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C22×Dic244—21no—
10C22×#11758202—21no—
11C2×#14519756C2211yescharacteristic
12#16840092—21no—
13#16840090—21no—
14#16840018—22no—
15#16839920—22no—
16#16839920—22no—
17#16839918—21no—
18#16839917—22no—
19#16839917—21no—
20#16839917—22no—
21C2×#14519702—21no—
22C2×#14519700—21no—
23C2×#14519695—22no—
24#16839726C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26#16839487—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×Dic122D411yescharacteristic
38C22×Dic122D412yes—
39C23×Dic61—21no—
40C2×Dic244—24no—
41C2×Dic244—44no—
42C2×#11758202—24no—
43C2×#11758202—22no—
44C2×#11758202—44no—
45C2×#11758194D412yes—
46C22×#92418—21no—
47#14519756—21no—
48#14519756—21no—
49#14519756—21no—
50#14519756—21no—
51C2×C4⋊Dic61D411yescharacteristic
52C2×#11758196—21no—
53C2×C22⋊Dic61—21no—
54#14519708—22no—
55#14519702—24no—
56#14519701—22no—
57#14519700—24no—
58#14519700—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
80C2×Q8×C61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—42no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—42no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×Dic61—22no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—44no—
96C22×Dic61—41no—
97C22×Dic61—41no—
98C2×Dic122—24no—
99C2×Dic122—24no—
100C2×Dic122—22no—
101C2×Dic122—22no—
102C2×Dic122—24no—
103C2×Dic122—21no—
104C2×Dic122—21no—
105C2×Dic122—44no—
106C2×Dic122—44no—
107C2×Dic122—44no—
108C2×Dic122—42no—
109C2×Dic122—42no—
110C2×Dic122—42no—
111C2×Dic122—41no—
112C2×Dic122—41no—
113C2×Dic122—42no—
114M16×C61—42no—
115C2×C8×C61—22no—
116C2×C8×C61—42no—
117C2×C8×C61—41no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—41no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126#11758202—48no—
127#11758202—82no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130#11758196—41no—
131#11758196—41no—
132#11758194—42no—
133C2×#92418—22no—
134C2×#92418—22no—
135C2×#92418—42no—
136Dic244—48no—
137C23×C61—22no—
138C23×C61—22no—
139C23×C61—41no—
140C23×C61#12111yescharacteristic
141C23×C61#12111yescharacteristic
142C23×C61#12111yescharacteristic
143Q8×C61—42no—
144Q8×C61—42no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
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—42no—
154C2×C4×C61—81no—
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12711yescharacteristic
159C2×Dic61—24no—
160C2×Dic61—44no—
161C2×Dic61—44no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—81no—
168Dic122—24no—
169Dic122—24no—
170Dic122—42no—
171Dic122—42no—
172Dic122—44no—
173Dic122—44no—
174Dic122—42no—
175Dic122—42no—
176C8×C61—42no—
177C8×C61—81no—
178#92418—42no—
179#92418—42no—
180#9991—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—44no—
189C22×C61#39211yescharacteristic
190C22×C61#47312yes—
191C22×C61#47312yes—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61B(4,𝔽2)12yes—
198Dic61—44no—
199Dic61—81no—
200Dic61—81no—
201#3195—611no—
202C2×#451—611no—
203C2×#451—611no—
204#1784—611no—
205#1472—611no—
206#1472D6111yescharacteristic
207#1298—611no—
208C2×C61—22no—
209C2×C61—22no—
210C2×C61—41no—
211C2×C61#202011yescharacteristic
212C2×C61#202011yescharacteristic
213C2×C61#202111yescharacteristic
214C23×Q8—611no—
215C22×Q16—1221no—
216C22×Q16—1221no—
217C2×C22⋊Q8—611no—
218C2×C22⋊Q8C2×D6111yescharacteristic
219#483—1221no—
220#483—1221no—
221#470—1222no—
222#451—1222no—
223#451—1222no—
224#451—1221no—
225#451—1222no—
226#451—1221no—
227#451—1222no—
228C2×#103—1221no—
229C2×#103—1221no—
230C2×#101—1222no—
231#407—611no—
232#407C2×D6111yescharacteristic
233C2×C22⋊C8—611no—
234C2×C22⋊C8C2×D6111yescharacteristic
235#324—22no—
236#324—1222no—
237C61#999111yescharacteristic, complemented, Sylow
238C22×Q8—611no—
239C22×Q8—612no—
240C22×Q8—1221no—
241C22×Q8—1221no—
242C22×Q8—1221no—
243C22×Q8—1221no—
244C22×Q8—1221no—
245C22×Q8—2444no—
246C22×Q8D24411yescharacteristic
247C23×C4—1221no—
248C23×C4C22×D6111yescharacteristic
249C2×Q16—1224no—
250C2×Q16—1222no—
251C2×Q16—1224no—
252C2×Q16—2444no—
253C2×Q16—2444no—
254C2×M16—21no—
255C2×M16—612no—
256C22×C8—21no—
257C22×C8—1221no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C22⋊Q8—21no—
262C22⋊Q8—1221no—
263C22⋊Q8—1221no—
264C22⋊Q8—1221no—
265C22⋊Q8—1221no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4D24411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1221no—
272#106—1222no—
273#103—1224no—
274#103—1222no—
275#103—1224no—
276#103—1222no—
277#101—2444no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—612no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284C22⋊C8—1221no—
285C22⋊C8C22⋊D6112yes—
286C24#1175822211yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C2×Q8—1224no—
291C2×Q8—1224no—
292C2×Q8—1224no—
293C2×Q8—1222no—
294C2×Q8—1221no—
295C2×Q8—1222no—
296C2×Q8—1221no—
297C2×Q8—2442no—
298C2×Q8—2442no—
299C2×Q8—2442no—
300C2×Q8—2442no—
301C2×Q8—2444no—
302C2×Q8—2444no—
303C2×Q8—2444no—
304C2×Q8—2441no—
305C2×Q8—2441no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—1222no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—2444no—
313C22×C4—2441no—
314C22×C4—2441no—
315C22×C4C2×D24411yescharacteristic
316C22×C4D4×D6111yescharacteristic
317C22×C4D4×D6111yescharacteristic
318C22×C4D4×D6111yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321Q16—4882no—
322Q16—2448no—
323Q16—2448no—
324M16—42no—
325M16—2442no—
326C2×C8—22no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—2442no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—2441no—
339C4⋊C4—2441no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23#1451975011yescharacteristic
348C23#1451975011yescharacteristic
349C23#1451975111yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—1224no—
353Q8—1224no—
354Q8—2442no—
355Q8—2442no—
356Q8—2444no—
357Q8—2442no—
358Q8—2442no—
359Q8—2444no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—1224no—
371C2×C4—4881no—
372C2×C4—2442no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2442no—
376C2×C4—2444no—
377C2×C4—2444no—
378C2×C4—2442no—
379C2×C4#1451974511yescharacteristic
380C2×C4#1451974711yescharacteristic
381C2×C4#1451974711yescharacteristic
382C2×C4#1451977311yescharacteristic
383C8—42no—
384C8—81no—
385C8—2442no—
386C8—2442no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—44no—
395C22#1683967111yescharacteristic
396C22#1684003612yes—
397C22#1684003712yes—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—4881no—
404C4—4881no—
405C4—2444no—
406C4#1683995012yes—
407C2—22no—
408C2—22no—
409C2—41no—
410C2#2144693911yescharacteristic
411C2#2144694011yescharacteristic
412C2#2144694411yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1472C2
socleC22×C61#392
prime radicalC24×C61#41
hypercenter#1472D61
hypocenterC61#9991
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
237. C61#9991semidirect

Maximal subgroups

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

subgroupindex
#214566402
C2×#168399202
C2×#168399172
#214457422
#214441842
#214437072
C61×#14722
#999161