Finite Group Databasebeta

← #30600097

Subgroups — #30600097

Order 15616.

492 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 86 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#21462547C211yesmaximal, characteristic
2#21461993C211yesmaximal, characteristic, complemented
3#21461317C211yesmaximal, characteristic
4#21461147C211yesmaximal, characteristic, complemented
5#21460710C211yesmaximal, characteristic, complemented
6#21460658C211yesmaximal, characteristic, complemented
7#21455845C211yesmaximal, characteristic
8C2×#14519864—21no—
9#16840794—21no—
10#16840793—21no—
11#16840792—21no—
12#16840791—21no—
13#16840779C2211yescharacteristic
14#16840765C2211yescharacteristic
15C2×#14519844C2211yescharacteristic
16#16840735—21no—
17C2×#14519832C2211yescharacteristic
18#16840628—21no—
19#16840628—21no—
20#16840621—21no—
21#16840620—21no—
22Q8⋊#92420—21no—
23Q8⋊#92420—21no—
24#16840307C2211yescharacteristic
25C2×Q8×Dic61C2211yescharacteristic
26#16840081C411yescharacteristic
27#16840078C411yescharacteristic
28#16840045—21no—
29#16839960—21no—
30#16839901C411yescharacteristic
31SD16×Dic61—21no—
32#16839847—21no—
33#16839812C2211yescharacteristic
34C61×#448C411yescharacteristic
35#14519864—42no—
36#14519864—42no—
37C2×#11758213—21no—
38C2×#11758217—21no—
39C2×#11758217—21no—
40C2×D61⋊C8—21no—
41C22×#92420—21no—
42C22×#92420—21no—
43C22×#92420D411yescharacteristic
44C22×#92420D411yescharacteristic
45#14519844—21no—
46#14519844—21no—
47#14519844—21no—
48#14519844—21no—
49C4⋊#92420D411yescharacteristic
50C4⋊#92420D411yescharacteristic
51#14519834—21no—
52#14519834—21no—
53C22⋊#92420—21no—
54C22⋊#92420—21no—
55C22⋊#92420—21no—
56#14519832—21no—
57#14519832—21no—
58C4×#92420—21no—
59C2×#11758224—21no—
60C2×#11758201—21no—
61#14519776D411yescharacteristic
62#14519773D411yescharacteristic
63#14519763—21no—
64#14519751C2×C411yescharacteristic
65#14519749—21no—
66Q8×Dic61—21no—
67Q8×Dic61—21no—
68Q8×Dic61—21no—
69Q8×Dic61—21no—
70D4×Dic61D411yescharacteristic
71#14519739D411yescharacteristic
72#14519737—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2311yescharacteristic
77#14519709—21no—
78#14519708—21no—
79#14519706—21no—
80#14519701—21no—
81#14519699—21no—
82#14519698—21no—
83C22⋊#92418C2×C411yescharacteristic
84C8×Dic61—21no—
85C22×Q8×C61C2×C411yescharacteristic
86C2×SD16×C61—21no—
87#121×C61C2×C411yescharacteristic
88#103×C61—21no—
89C22⋊C8×C61C2×C411yescharacteristic
90C2×Q8×C61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—42no—
93C2×D4×C61—21no—
94C2×D4×C61C22⋊C411yescharacteristic
95C22×C4×C61—21no—
96C22×C4×C61C22×C411yescharacteristic
97C2×C22⋊D61—21no—
98C2×C22⋊D61—21no—
99C2×C22⋊D61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61C2×D411yescharacteristic
103C22×Dic61#4111yescharacteristic
104#11758224—41no—
105#11758224—41no—
106#11758224—41no—
107#11758224—41no—
108C2×D244C22⋊C411yescharacteristic
109C2×C4×D61—21no—
110C2×Dic122—21no—
111#11758217—42no—
112#11758217—41no—
113#11758217—41no—
114C2×#92420—21no—
115C2×#92420—21no—
116C2×#92420—21no—
117C2×#92420—21no—
118C2×#92420—21no—
119C2×#92420—21no—
120C2×#92420—21no—
121C2×#92420—42no—
122C2×#92420—41no—
123C2×#92420—41no—
124C2×#92420—41no—
125C2×#92420—41no—
126C2×#92420SD1612yescomplemented
127#11758213—42no—
128D61⋊C8—42no—
129SD16×C61—42no—
130C2×C8×C61—21no—
131C4⋊C4×C61C22⋊C411yescharacteristic
132C22⋊C4×C61—21no—
133C22⋊Dic61—21no—
134#11758201—42no—
135#11758198—21no—
136#11758198—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—41no—
140C4⋊Dic61—41no—
141C4⋊Dic61C22⋊C411yescharacteristic
142#11758196—21no—
143C4×Dic61—41no—
144C4×Dic61—41no—
145C4×Dic61C2×D411yescharacteristic
146C4×Dic61C2×D411yescharacteristic
147C4×Dic61C2×D411yescharacteristic
148C4×Dic61#4111yescharacteristic
149C2×#92418—21no—
150C23×C61—21no—
151C23×C61C4×D411yescharacteristic
152C22×D61—21no—
153Q8×C61—22no—
154Q8×C61—22no—
155Q8×C61—41no—
156Q8×C61—41no—
157D4×C61—41no—
158D4×C61—41no—
159D4×C61—41no—
160C2×C4×C61—22no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61C2×C22⋊C411yescharacteristic
166C2×C4×C61C4×D411yescharacteristic
167C22⋊D61—41no—
168C22⋊D61—41no—
169C22⋊D61—41no—
170C22⋊D61—41no—
171C22⋊D61—81no—
172C2×Dic61—21no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—42no—
177C2×Dic61—41no—
178C2×Dic61#12011yescharacteristic
179C2×Dic61#12111yescharacteristic
180C2×Dic61#12111yescharacteristic
181C2×Dic61#12311yescharacteristic
182C2×Dic61C2×SD1611yescharacteristic
183C2×Dic61#13511yescharacteristic
184C2×Dic61AGL(1,ℤ/8)11yescharacteristic
185C2×Dic61#13711yescharacteristic
186D244—41no—
187C4×D61—41no—
188C4×D61—41no—
189Dic122—41no—
190Dic122—41no—
191#92420—22no—
192#92420—41no—
193#92420—41no—
194#92420—41no—
195#92420—81no—
196#92420—81no—
197C8×C61—41no—
198#92418—41no—
199#9047—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—41no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61#38611yescharacteristic
206C22×C61C4×SD1611yescharacteristic
207C22×C61#44011yescharacteristic
208C2×D61—41no—
209C2×D61—41no—
210C2×D61—41no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—41no—
214C4×C61—42no—
215Dic61—21no—
216Dic61—21no—
217Dic61—41no—
218Dic61—81no—
219Dic61#45011yescharacteristic
220Dic61#45211yescharacteristic
221Dic61#46111yescharacteristic
222Dic61#46311yescharacteristic
223#3103—611no—
224#2932—611no—
225#1985—611no—
226#1540—611no—
227#1487—611no—
228#1301—611no—
229#1281—611no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61—81no—
233C2×C61#187711yescharacteristic
234C2×C61#188111yescharacteristic
235C2×C61#189111yescharacteristic
236D61—81no—
237C2×#131—1221no—
238#532—611no—
239C2×C4×Q8—611no—
240#481—611no—
241#481—611no—
242#475—1221no—
243#460—1221no—
244#448—611no—
245#448#692311yescharacteristic
246C4×SD16—1221no—
247C4×SD16—1221no—
248#435—1221no—
249#435—1221no—
250#435—1221no—
251#435—1221no—
252#431—611no—
253#431—611no—
254#423—611no—
255C2×C4⋊C8—611no—
256#408—1221no—
257C2×#97—611no—
258#336—1221no—
259#336—1221no—
260#332—1221no—
261#331—1221no—
262Q8⋊C8—1221no—
263Q8⋊C8—1221no—
264C61#904711yescharacteristic, complemented, Sylow
265C2×#41—1221no—
266C22×Q8C2×#692311yescharacteristic
267C2×SD16—21no—
268C2×SD16—1221no—
269#131—2442no—
270#131—2442no—
271C2×M16—1221no—
272C2×M16—1221no—
273C2×M16—1221no—
274C22×C8—611no—
275C22×C8—611no—
276C22×C8—1221no—
277C22×C8—1221no—
278C22×C8—1221no—
279C4⋊Q8—611no—
280#127—611no—
281#124—1221no—
282#121—611no—
283#121—1221no—
284#121C2×#692311yescharacteristic
285C4×Q8—1221no—
286C4×Q8—1221no—
287C4×Q8—1221no—
288C4×Q8—1221no—
289C4×D4—611no—
290C4×D4—611no—
291C4×D4—1221no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1221no—
294C2×C42—611no—
295C2×C42—1221no—
296#106—1221no—
297#106—1221no—
298C4⋊C8—611no—
299C4⋊C8—611no—
300C4⋊C8—1221no—
301C4⋊C8—1221no—
302C4⋊C8—1221no—
303C4⋊C8—1221no—
304C4⋊C8—1221no—
305#103—21no—
306#103—1221no—
307#102—1221no—
308#102—1221no—
309C22⋊C8—611no—
310C22⋊C8—1221no—
311C22⋊C8—1221no—
312C22⋊C8—1221no—
313C22⋊C8—1221no—
314C22⋊C8—1221no—
315C22⋊C8C2×#692311yescharacteristic
316#97—1221no—
317#97—1221no—
318C4×C8—1221no—
319C4×C8—1221no—
320#41—2441no—
321#41—2441no—
322#41—2441no—
323#41—2441no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—42no—
327C2×Q8—1221no—
328C2×D4—21no—
329C2×D4—611no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4C22⋊#692311yescharacteristic
334C22×C4—21no—
335C22×C4—611no—
336C22×C4—611no—
337C22×C4—1221no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4C22×#692311yescharacteristic
341SD16—42no—
342SD16—2442no—
343M16—2442no—
344M16—2442no—
345M16—2441no—
346M16—2441no—
347C2×C8—21no—
348C2×C8—612no—
349C2×C8—1221no—
350C2×C8—1221no—
351C2×C8—1221no—
352C2×C8—1221no—
353C2×C8—1221no—
354C2×C8—1221no—
355C2×C8—1221no—
356C2×C8—1221no—
357C2×C8—2442no—
358C2×C8—2442no—
359C2×C8—2441no—
360C2×C8—2441no—
361C2×C8—2441no—
362C2×C8—2441no—
363C4⋊C4—611no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—2441no—
368C4⋊C4—2441no—
369C4⋊C4C22⋊#692311yescharacteristic
370C22⋊C4—21no—
371C22⋊C4—1221no—
372C22⋊C4—1221no—
373C22⋊C4—1221no—
374C42—611no—
375C42—611no—
376C42—611no—
377C42—611no—
378C42—2441no—
379C42—2441no—
380C23—21no—
381C23—1221no—
382C23D4×#692311yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—41no—
386Q8—41no—
387Q8—2441no—
388Q8—2441no—
389D4—41no—
390D4—41no—
391D4—41no—
392D4—4881no—
393D4—2441no—
394D4—2441no—
395D4—2441no—
396D4—2441no—
397D4—2441no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—611no—
404C2×C4—611no—
405C2×C4—611no—
406C2×C4—611no—
407C2×C4—611no—
408C2×C4—611no—
409C2×C4—611no—
410C2×C4—611no—
411C2×C4—1221no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2442no—
418C2×C4—2441no—
419C2×C4C2×C22⋊#692311yescharacteristic
420C2×C4D4×#692311yescharacteristic
421C8—41no—
422C8—1222no—
423C8—4881no—
424C8—4881no—
425C8—2441no—
426C8—2441no—
427C8—2441no—
428C8—2441no—
429C22—21no—
430C22—21no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—2441no—
435C22—2441no—
436C22—2441no—
437C22#1684071011yescharacteristic
438C22SD16×#692311yescharacteristic
439C22#1684080111yescharacteristic
440C4—21no—
441C4—21no—
442C4—41no—
443C4—42no—
444C4—611no—
445C4—611no—
446C4—611no—
447C4—611no—
448C4—1221no—
449C4—1221no—
450C4—4881no—
451C4—2441no—
452C2—21no—
453C2—21no—
454C2—81no—
455C2—4881no—
456C2#2146181211yescharacteristic
457C2#2146181611yescharacteristic
458C2#2146183011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#448C4
socleC22×C61#386
prime radical#16840307C22
cosocleC2×C4×Dic61C23
hypercenter#448#6923
hypocenterC61#9047
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #21461993C2semidirect
4. #21461147C2semidirect
5. #21460710C2semidirect
6. #21460658C2semidirect
126. C2×#92420SD16semidirect
264. C61#9047semidirect

Maximal subgroups

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

subgroupindex
#214625472
#214619932
#214613172
#214611472
#214607102
#214606582
#214558452
#904761