Finite Group Databasebeta

← #29596785

Subgroups — #29596785

Order 15616.

504 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 95 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#21448967C211yesmaximal, characteristic
2C2×#16839552C211yesmaximal, characteristic
3#21446100C211yesmaximal, characteristic
4#21446029C211yesmaximal, characteristic
5#21443943C211yesmaximal, characteristic
6#21443936C211yesmaximal, characteristic
7C61×#1890C211yesmaximal, characteristic
8#16839805—21no—
9C2×#14519694C2211yescharacteristic
10C2×#14519692C2211yescharacteristic
11#16839665C2211yescharacteristic
12#16839661—21no—
13#16839655—21no—
14#16839655—21no—
15#16839648—21no—
16#16839642C2211yescharacteristic
17#16839637—21no—
18#16839634—21no—
19#16839622—21no—
20#16839552C414yescomplemented
21#16839529—21no—
22#16839521—21no—
23C2×#123×C61C2211yescharacteristic
24C61×#518C2211yescharacteristic
25C2×#99×C61C2211yescharacteristic
26C61×#381—21no—
27C61×#342—21no—
28C23×Dic61D411yescharacteristic
29C2×#11758194—21no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61D411yescharacteristic
34C2×#11758196—41no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61D411yescharacteristic
40C2×C4×Dic61—21no—
41C2×C4×Dic61D411yescharacteristic
42#14519710—41no—
43#14519708—41no—
44#14519694—21no—
45#14519694C2×C412yes—
46#14519692—21no—
47#14519692C2×C412yes—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—41no—
54C22×D4×C61C2311yescharacteristic
55C23×C4×C61D411yescharacteristic
56#123×C61—21no—
57#123×C61—21no—
58#123×C61C2×C412yes—
59#123×C61Q812yes—
60C4×D4×C61—41no—
61C4×D4×C61—41no—
62#117×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61D411yescharacteristic
69#99×C61—21no—
70#99×C61—21no—
71#95×C61—41no—
72#95×C61—41no—
73C24×C61C2×D411yescharacteristic
74C24×C61C2×D411yescharacteristic
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61C22×C411yescharacteristic
78C2×D4×C61C2×Q811yescharacteristic
79C2×D4×C61#4111yescharacteristic
80C2×D4×C61#4111yescharacteristic
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61C4⋊C412yescomplemented
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—41no—
98C22×Dic61—41no—
99C22×Dic61—81no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C22⋊C4×C61—22no—
105C22⋊C4×C61—22no—
106C22⋊C4×C61—21no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61C4⋊C412yescomplemented
113C42×C61—41no—
114C22⋊Dic61—22no—
115C22⋊Dic61—41no—
116C22⋊Dic61—41no—
117C22⋊Dic61—81no—
118C22⋊Dic61—81no—
119C22⋊Dic61—81no—
120C4⋊Dic61—41no—
121C4⋊Dic61—81no—
122C4⋊Dic61—81no—
123C4⋊Dic61—81no—
124#11758196—82no—
125C4×Dic61—41no—
126C4×Dic61—81no—
127#11758194—22no—
128#11758194—41no—
129C2×#92418—41no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61C2×C4⋊C411yescharacteristic
139C23×C61C4×D411yescharacteristic
140C23×C61#12011yescharacteristic
141C23×C61C22⋊Q811yescharacteristic
142C23×C61#12311yescharacteristic
143D4×C61—41no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—41no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61—81no—
163C2×C4×C61—81no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C2×C4×C61C4×D411yescharacteristic
167C2×C4×C61C22⋊Q811yescharacteristic
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—81no—
183C2×Dic61—81no—
184C2×Dic61—81no—
185#92418—42no—
186#12961—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—41no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61#38511yescharacteristic
204C22×C61B(4,𝔽2)11yescharacteristic
205C22×C61#45811yescharacteristic
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—42no—
209C4×C61—42no—
210C4×C61—81no—
211C4×C61—81no—
212C4×C61—161no—
213Dic61—81no—
214Dic61—81no—
215Dic61—82no—
216Dic61—81no—
217Dic61—81no—
218#2376—611no—
219C2×#352—611no—
220#1890D6111yescharacteristic
221#1867—611no—
222#1856—611no—
223#1393—611no—
224#1390—611no—
225C2×C61—21no—
226C2×C61—21no—
227C2×C61—41no—
228C2×C61—41no—
229C2×C61—41no—
230C2×C61—41no—
231C2×C61#186711yescharacteristic
232C2×C61#220011yescharacteristic
233C2×C61#220111yescharacteristic
234C2×#123C2×D6111yescharacteristic
235#518C2×D6111yescharacteristic
236#428—1221no—
237C2×#100—611no—
238C2×#99—611no—
239C2×#99C2×D6111yescharacteristic
240#390—611no—
241#389—1221no—
242#388—1221no—
243#388—1221no—
244#387—1221no—
245#386—611no—
246#385—1221no—
247#385—1221no—
248#381—21no—
249#380—1221no—
250#352—614no—
251#342—21no—
252#342—1221no—
253#339—1221no—
254C61#1296111yescharacteristic, complemented, Sylow
255C22×D4C22×D6111yescharacteristic
256C23×C4—611no—
257C23×C4C22⋊D6111yescharacteristic
258C2×M16—1221no—
259#123—21no—
260#123—21no—
261#123Dic12212yes—
262#123C4×D6112yes—
263C4×D4—41no—
264C4×D4—41no—
265#117—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1221no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—2441no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—611no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4C22⋊D6111yescharacteristic
282C2×C42—611no—
283C2×C42—1221no—
284#107—2441no—
285#106—2441no—
286#100—612no—
287#100—1221no—
288#99—21no—
289#99—21no—
290#99—612no—
291#99—1221no—
292#95—41no—
293#95—41no—
294#95—1221no—
295#95—1221no—
296#95—1221no—
297#95—1221no—
298#95—1221no—
299#95—2441no—
300C24D4×D6111yescharacteristic
301C24C2×C22⋊D6111yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C2×Dic12211yescharacteristic
305C2×D4C2×C4×D6111yescharacteristic
306C2×D4#1175822411yescharacteristic
307C2×D4#1175822411yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—4881no—
323C22×C4—2441no—
324C22×C4—2441no—
325C22×C4#1175819612yescomplemented
326C22×C4D4×D6111yescharacteristic
327M16—1222no—
328M16—2441no—
329C2×C8—2441no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—4881no—
335C4⋊C4—4881no—
336C4⋊C4—4881no—
337C4⋊C4—4882no—
338C4⋊C4—2441no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—21no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—1222no—
348C22⋊C4—4881no—
349C22⋊C4—4881no—
350C22⋊C4—4881no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C22⋊C4#1175819612yescomplemented
354C42—41no—
355C42—4881no—
356C42—2441no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×#1175819611yescharacteristic
366C23#1451973711yescharacteristic
367C23#1451974511yescharacteristic
368C23#1451975411yescharacteristic
369C23#1451975811yescharacteristic
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—4881no—
394C2×C4—4881no—
395C2×C4—4881no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2442no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2442no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4#1451973711yescharacteristic
411C2×C4#1451975411yescharacteristic
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—81no—
428C22—81no—
429C22#1683963411yescharacteristic
430C22#1683994811yescharacteristic
431C22#1683995111yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—81no—
437C4—81no—
438C4—161no—
439C4—4881no—
440C4—4881no—
441C4—4882no—
442C4—4881no—
443C4—4881no—
444C2—21no—
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—41no—
450C2#2144609911yescharacteristic
451C2#2144782311yescharacteristic
452C2#2144783011yescharacteristic
subgroupG/H
centerC22#16839948
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1890C2
socleC22×C61B(4,𝔽2)
prime radical, cosocleC22×D4×C61C23
hypercenter#1890D61
hypocenterC61#12961
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1890

Complements

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

NH ≅ G/Ntype
20. #16839552C4semidirect
91. C22×C4×C61C4⋊C4semidirect
112. C22⋊C4×C61C4⋊C4semidirect
254. C61#12961semidirect
325. C22×C4#11758196semidirect
353. C22⋊C4#11758196semidirect

Maximal subgroups

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

subgroupindex
#214489672
C2×#168395522
#214461002
#214460292
#214439432
#214439362
C61×#18902
#1296161