Finite Group Databasebeta

← #30600390

Subgroups — #30600390

Order 15616.

612 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×#16840767C211yesmaximal, characteristic
2#21461676C211yesmaximal, characteristic
3C2×#16840622C212yesmaximal, complemented
4#21460697C212yesmaximal, complemented
5#21454168C211yesmaximal, characteristic
6C22×#11758213—21no—
7#16840767—24no—
8C2×#14519844—21no—
9C2×#14519844C2212yes—
10C2×#14519844C2212yes—
11C2×#14519834—21no—
12#16840628—21no—
13#16840622—24no—
14C2×#14519775C2211yescharacteristic
15C2×#14519757C411yescharacteristic
16C2×#14519753C411yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839668C411yescharacteristic
19#16839659—21no—
20#16839659—21no—
21#16839631C2211yescharacteristic
22#16839621—21no—
23#16839621—21no—
24C61×#391C411yescharacteristic
25C2×#11758213—24no—
26C2×#11758213—42no—
27C2×#11758213—42no—
28C22×#92420—22no—
29C22×#92420—22no—
30#14519844—24no—
31#14519844—22no—
32#14519844—22no—
33#14519844—22no—
34#14519844D414yes—
35#14519834—42no—
36C22×Dic122C2×C411yescharacteristic
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40#14519775—22no—
41#14519775—22no—
42#14519757—42no—
43#14519753—42no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×#11758198—21no—
52C2×#11758198—21no—
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61C2311yescharacteristic
56C42×D61—22no—
57C42×D61—22no—
58#14519684—21no—
59#14519684—21no—
60C22×Q8×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C42×C61C2×C411yescharacteristic
63#95×C61—21no—
64#95×C61—21no—
65C23×D61C22⋊C411yescharacteristic
66C2×Q8×C61—22no—
67C2×Q8×C61—41no—
68C2×Q8×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—22no—
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—21no—
87C2×C4×D61—22no—
88C2×C4×D61—21no—
89C2×C4×D61—41no—
90C2×C4×D61SD1612yes—
91C2×C4×D61Q1612yes—
92C2×Dic122—22no—
93C2×Dic122—41no—
94C2×Dic122—41no—
95C2×#92420—24no—
96C2×#92420—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99C2×#92420—42no—
100C2×#92420—42no—
101#11758213—48no—
102C4⋊C4×C61—41no—
103C42×C61—22no—
104#11758198—42no—
105#11758198—42no—
106C4⋊Dic61—42no—
107C4⋊Dic61—41no—
108#11758196—42no—
109#11758196—41no—
110#11758196—42no—
111#11758196—41no—
112#11758196—41no—
113#11758196—41no—
114#11758196—41no—
115#11758196—41no—
116C4×Dic61—21no—
117C4×Dic61—22no—
118C4×Dic61—21no—
119C4×Dic61C2×D412yes—
120C4×Dic61#4112yes—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61#10312yes—
129Q8×C61—82no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61#10312yes—
144C2×C4×C61C4×D412yes—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61#10012yes—
154C2×Dic61#12011yescharacteristic
155C2×Dic61#12112yes—
156C2×Dic61#12311yescharacteristic
157C2×Dic61C2×SD1611yescharacteristic
158C2×Dic61C2×Q1611yescharacteristic
159C2×Dic61AGL(1,ℤ/8)11yescharacteristic
160C2×Dic61AGL(1,ℤ/8)11yescharacteristic
161C2×Dic61#13711yescharacteristic
162C2×Dic61#13712yes—
163C2×Dic61#13711yescharacteristic
164C4×D61—22no—
165C4×D61—22no—
166C4×D61—22no—
167C4×D61—22no—
168C4×D61—44no—
169C4×D61—42no—
170C4×D61—42no—
171Dic122—82no—
172#92420—44no—
173#92420—82no—
174#9135—611nomaximal, Sylow
175C22×C61#38611yescharacteristic
176C22×C61C2×#10011yescharacteristic
177C22×C61#41311yescharacteristic
178C22×C61C2×#10311yescharacteristic
179C22×C61#41711yescharacteristic
180C22×C61#41711yescharacteristic
181C22×C61#41911yescharacteristic
182C2×D61—22no—
183C2×D61—22no—
184C2×D61—22no—
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—41no—
189C4×C61—22no—
190C4×C61—22no—
191C4×C61—41no—
192C4×C61—81no—
193C4×C61—81no—
194Dic61—22no—
195Dic61—81no—
196Dic61—81no—
197Dic61#45011yescharacteristic
198Dic61#45111yescharacteristic
199Dic61#46812yes—
200Dic61#47012yes—
201Dic61#48311yescharacteristic
202Dic61#48411yescharacteristic
203C2×#423—611no—
204#2857—611no—
205#1797—611no—
206C2×#333—612no—
207#1296—612no—
208C2×C61#154811yescharacteristic
209C2×C61#155011yescharacteristic
210C2×C61#155911yescharacteristic
211C2×C61#156011yescharacteristic
212C2×C61#187211yescharacteristic
213C2×C61#189411yescharacteristic
214C2×C61#190011yescharacteristic
215D61—42no—
216C22×M16—1221no—
217C2×C4⋊Q8—611no—
218C2×C22⋊Q8—611no—
219C2×C22⋊Q8—611no—
220C22×C42—611no—
221#423—1224no—
222C2×C4⋊C8—612no—
223C2×C4⋊C8—612no—
224C2×C4⋊C8—1221no—
225C2×C22⋊C8—1221no—
226#391—611no—
227#391#692311yescharacteristic
228#389—1221no—
229#389—1221no—
230#384—611no—
231#380—1221no—
232#380—1221no—
233#336—1221no—
234#333—1224no—
235C61#913511yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×Q8C2×#692311yescharacteristic
238C23×C4—611no—
239C23×C4—611no—
240C23×C4—611no—
241C2×M16—1224no—
242C2×M16—2442no—
243C2×M16—2442no—
244C22×C8—1222no—
245C22×C8—1222no—
246C4⋊Q8—1222no—
247C4⋊Q8—1222no—
248C22⋊Q8—2442no—
249C22⋊Q8—2442no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4C2×#692311yescharacteristic
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C42—611no—
261C2×C42—611no—
262C2×C42—611no—
263C2×C42—1222no—
264C2×C42—1222no—
265C2×C42C2×#692311yescharacteristic
266C4⋊C8—614no—
267C4⋊C8—1222no—
268C4⋊C8—1222no—
269C4⋊C8—1222no—
270C4⋊C8—1224no—
271C22⋊C8—2442no—
272#95—21no—
273#95—21no—
274#95—1221no—
275#95—1221no—
276C24—611no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—1222no—
281C2×Q8—2441no—
282C2×Q8—2441no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—611no—
286C22×C4—611no—
287C22×C4—611no—
288C22×C4—612no—
289C22×C4—612no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1221no—
298C22×C4—1222no—
299C22×C4—1222no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—2441no—
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22⋊#692311yescharacteristic
305C22×C4C22×#692311yescharacteristic
306M16—2448no—
307C2×C8—1224no—
308C2×C8—2442no—
309C2×C8—2442no—
310C2×C8—2442no—
311C2×C8—2442no—
312C2×C8—2442no—
313C4⋊C4—41no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C4⋊C4—2442no—
320C4⋊C4—2441no—
321C4⋊C4—2442no—
322C4⋊C4—2441no—
323C4⋊C4—2442no—
324C22⋊C4—2442no—
325C22⋊C4—2442no—
326C42—22no—
327C42—612no—
328C42—612no—
329C42—1221no—
330C42—1222no—
331C42—1221no—
332C23—612no—
333C23—1221no—
334C23—1221no—
335C23—1221no—
336C23—1221no—
337C23—1221no—
338C23—1221no—
339C23C2×C22⋊#692311yescharacteristic
340Q8—82no—
341Q8—4882no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—611no—
356C2×C4—612no—
357C2×C4—611no—
358C2×C4—611no—
359C2×C4—611no—
360C2×C4—611no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—612no—
365C2×C4—612no—
366C2×C4—1222no—
367C2×C4—1222no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—2444no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4#1451984012yes—
382C2×C4D4×#692312yes—
383C8—4882no—
384C8—2444no—
385C22—1222no—
386C22—1222no—
387C22—1222no—
388C22—2441no—
389C22—2441no—
390C22—2441no—
391C22—2441no—
392C22#1684071011yescharacteristic
393C22D61⋊#10011yescharacteristic
394C22#1684074811yescharacteristic
395C22C2×#1451984011yescharacteristic
396C22#1684075311yescharacteristic
397C22#1684075311yescharacteristic
398C22#1684075511yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—81no—
403C4—81no—
404C4—611no—
405C4—611no—
406C4—611no—
407C4—611no—
408C4—612no—
409C4—612no—
410C4—1222no—
411C4—4881no—
412C4—4881no—
413C2—2442no—
414C2#2146133311yescharacteristic
415C2#2146133611yescharacteristic
416C2#2146135411yescharacteristic
417C2#2146135711yescharacteristic
418C2#2146180411yescharacteristic
419C2#2146183411yescharacteristic
420C2#2146184511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#9135
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C2×#16840622C2semidirect
4. #21460697C2semidirect
235. C61#9135semidirect

Maximal subgroups

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

subgroupindex
C2×#168407672
#214616762
C2×#168406222
#214606972
#214541682
#913561