Finite Group Databasebeta

← #29580509

Subgroups — #29580509

Order 15616.

744 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 107 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#21456591C211yesmaximal, characteristic
2C2×#16839917C211yesmaximal, characteristic
3#21449463C211yesmaximal, characteristic
4#21445753C211yesmaximal, characteristic
5#21444390C211yesmaximal, characteristic
6#21443667C211yesmaximal, characteristic
7C61×#1996C211yesmaximal, characteristic
8C23×Dic122C2211yescharacteristic
9C22×#11758202—21no—
10C2×#14519777—21no—
11C2×#14519756C2211yescharacteristic
12#16840097—21no—
13#16840095—21no—
14#16840001—22no—
15#16839918—21no—
16#16839917—22no—
17#16839917—22no—
18#16839917—21no—
19C2×#14519700—22no—
20C2×#14519700—21no—
21C2×C22⋊#92418C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23#16839683—22no—
24#16839668—21no—
25#16839637—22no—
26#16839498—22no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C61×#380C2211yescharacteristic
29C4×C22⋊C4×C61C2211yescharacteristic
30C61×#342—22no—
31C22×Dic122—21no—
32C22×Dic122—21no—
33C22×Dic122—21no—
34C22×Dic122—21no—
35C22×Dic122—21no—
36C22×Dic122—44no—
37C22×Dic122D412yes—
38C22×Dic122D411yescharacteristic
39C23×Dic61—21no—
40C2×#11758202—24no—
41C2×#11758202—22no—
42C2×#11758202—44no—
43C22×#92418—21no—
44C22×#92418D412yes—
45#14519777—44no—
46#14519756—21no—
47#14519756—21no—
48#14519756—21no—
49#14519756—21no—
50C2×C4⋊Dic61—24no—
51C2×C4⋊Dic61D411yescharacteristic
52C2×#11758196—21no—
53C2×C22⋊Dic61—21no—
54#14519710—22no—
55#14519701—22no—
56#14519700—22no—
57#14519700—24no—
58#14519700—44no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62C22⋊#92418D412yes—
63#14519684—24no—
64C22×Q8×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—21no—
68C22⋊Q8×C61—21no—
69C22⋊Q8×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61D412yes—
76C2×C42×C61—21no—
77#95×C61—21no—
78#95×C61—21no—
79C24×C61C2×D411yescharacteristic
80C2×Q8×C61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61—42no—
83C22×C4×C61—22no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—42no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C22×Dic61—22no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—44no—
98C22×Dic61—41no—
99C22×Dic61—41no—
100C2×Dic122—24no—
101C2×Dic122—24no—
102C2×Dic122—22no—
103C2×Dic122—22no—
104C2×Dic122—24no—
105C2×Dic122—21no—
106C2×Dic122—21no—
107C2×Dic122—44no—
108C2×Dic122—44no—
109C2×Dic122—44no—
110C2×Dic122—42no—
111C2×Dic122—42no—
112C2×Dic122—42no—
113C2×Dic122—41no—
114C2×Dic122—41no—
115C2×Dic122—42no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—42no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—41no—
121C22⋊C4×C61—22no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—41no—
124C22⋊C4×C61—42no—
125C22⋊C4×C61Q1614yescomplemented
126C42×C61—42no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129#11758202—48no—
130#11758202—82no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—48no—
134#11758196—41no—
135#11758196—41no—
136C2×#92418—22no—
137C2×#92418—22no—
138C2×#92418—42no—
139C2×#92418—42no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61#12011yescharacteristic
143C23×C61#12111yescharacteristic
144C23×C61#12711yescharacteristic
145C23×C61C2×Q1612yes—
146C23×C61#13712yes—
147Q8×C61—42no—
148Q8×C61—42no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—24no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61—81no—
163C2×C4×C61—81no—
164C2×C4×C61—81no—
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×Dic61—24no—
170C2×Dic61—44no—
171C2×Dic61—44no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178Dic122—24no—
179Dic122—24no—
180Dic122—42no—
181Dic122—42no—
182Dic122—44no—
183Dic122—44no—
184Dic122—42no—
185Dic122—81no—
186#92418—42no—
187#92418—42no—
188#9957—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61#39211yescharacteristic
198C22×C61#45112yes—
199C22×C61#46212yes—
200C22×C61#49411yescharacteristic
201C22×C61#49711yescharacteristic
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—44no—
206C4×C61—81no—
207C4×C61—81no—
208C4×C61—161no—
209C4×C61B(4,𝔽2)12yes—
210Dic61—44no—
211Dic61—81no—
212Dic61—81no—
213#3189—611no—
214C2×#451—611no—
215#2434—611no—
216#1996D6111yescharacteristic
217#1787—611no—
218#1520—611no—
219#1282—611no—
220C2×C61—22no—
221C2×C61—21no—
222C2×C61—22no—
223C2×C61—21no—
224C2×C61#200911yescharacteristic
225C2×C61#201611yescharacteristic
226C2×C61#202111yescharacteristic
227C23×Q8—611no—
228C22×Q16—1221no—
229C2×C4⋊Q8—1221no—
230C2×C22⋊Q8—611no—
231C2×C22⋊Q8C2×D6111yescharacteristic
232#484—1221no—
233#484—1221no—
234#467—1222no—
235#451—1222no—
236#451—1221no—
237#451—1222no—
238#451—1221no—
239C2×#103—1221no—
240C2×#103—1222no—
241C2×C22⋊C8—611no—
242C2×C22⋊C8—611no—
243#395—1222no—
244#391—1221no—
245#385—1222no—
246#380C2×D6111yescharacteristic
247C4×C22⋊C4C2×D6111yescharacteristic
248#342—22no—
249#328—1222no—
250C61#995711yescharacteristic, complemented, Sylow
251C22×Q8—611no—
252C22×Q8—612no—
253C22×Q8—1221no—
254C22×Q8—1221no—
255C22×Q8—1221no—
256C22×Q8—1221no—
257C22×Q8—1221no—
258C22×Q8—2444no—
259C22×Q8D24411yescharacteristic
260C23×C4—1221no—
261C23×C4C22×D6111yescharacteristic
262C2×Q16—1222no—
263C2×Q16—1224no—
264C2×Q16—2444no—
265C22×C8—612no—
266C22×C8—1221no—
267C4⋊Q8—2444no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—1221no—
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275C22⋊Q8—1221no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—1224no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4D24411yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4C22⋊D6112yes—
286C2×C42—21no—
287#107—1222no—
288#103—1224no—
289#103—1222no—
290#103—1222no—
291#103—2444no—
292C22⋊C8—612no—
293C22⋊C8—1221no—
294C22⋊C8—1221no—
295C22⋊C8—1221no—
296#95—21no—
297#95—21no—
298#95—1224no—
299C24C2×C22⋊D6111yescharacteristic
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—42no—
303C2×Q8—1224no—
304C2×Q8—1224no—
305C2×Q8—1224no—
306C2×Q8—1222no—
307C2×Q8—1221no—
308C2×Q8—1222no—
309C2×Q8—1221no—
310C2×Q8—2442no—
311C2×Q8—2442no—
312C2×Q8—2442no—
313C2×Q8—2442no—
314C2×Q8—2444no—
315C2×Q8—2444no—
316C2×Q8—2444no—
317C2×Q8—2441no—
318C2×Q8—2441no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—1222no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—2444no—
328C22×C4—2441no—
329C22×C4—2441no—
330C22×C4C2×D24411yescharacteristic
331C22×C4#1175822211yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4D4×D6111yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336Q16—4882no—
337Q16—2448no—
338C2×C8—1222no—
339C2×C8—1222no—
340C2×C8—2442no—
341C2×C8—2442no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—2448no—
350C4⋊C4—2441no—
351C4⋊C4—2441no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—2441no—
357C22⋊C4—2441no—
358C22⋊C4#1175820214yescomplemented
359C42—42no—
360C23—41no—
361C23—41no—
362C23#1451974511yescharacteristic
363C23#1451975111yescharacteristic
364C23#1451977311yescharacteristic
365C23C2×#1175820212yes—
366C23#1451980712yes—
367Q8—42no—
368Q8—42no—
369Q8—1224no—
370Q8—1224no—
371Q8—4881no—
372Q8—2442no—
373Q8—2442no—
374Q8—2444no—
375Q8—2442no—
376Q8—2444no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—1224no—
394C2×C4—2442no—
395C2×C4—2442no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2442no—
399C2×C4—2444no—
400C2×C4—2444no—
401C2×C4—2442no—
402C2×C4#1451974711yescharacteristic
403C2×C4#1451974711yescharacteristic
404C2×C4#1451975011yescharacteristic
405C2×C4#1451975011yescharacteristic
406C8—2442no—
407C8—2442no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22#1683967111yescharacteristic
417C22#1683991712yes—
418C22#1683997212yes—
419C22#1684014111yescharacteristic
420C22#1684015711yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—44no—
425C4—81no—
426C4—81no—
427C4—161no—
428C4—4881no—
429C4—4881no—
430C4—2444no—
431C4#1683995012yes—
432C2—22no—
433C2—21no—
434C2—22no—
435C2—21no—
436C2#2144687811yescharacteristic
437C2#2144691811yescharacteristic
438C2#2144694411yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1996C2
socleC22×C61#392
prime radicalC24×C61C2×D4
hypercenter#1996D61
hypocenterC61#9957
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
125. C22⋊C4×C61Q16semidirect
250. C61#9957semidirect
358. C22⋊C4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214565912
C2×#168399172
#214494632
#214457532
#214443902
#214436672
C61×#19962
#995761