Finite Group Databasebeta

← #30606286

Subgroups — #30606286

Order 15616.

750 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 142 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
1D61⋊#413C212yesmaximal
2#21460911C212yesmaximal, complemented
3#21460910C212yesmaximal, complemented
4#408×D61C211yesmaximal, characteristic
5C2×D61⋊M16—22no—
6#16840749—22no—
7#16840749C414yes—
8#16840748—22no—
9#16840748C2214yes—
10D61⋊#101—22no—
11D61⋊#100C414yes—
12C2×#41×D61C2211yescharacteristic
13C2×M16×D61C411yescharacteristic
14C22×C8×D61C411yescharacteristic
15#16839730C2211yescharacteristic
16#16839729C411yescharacteristic
17#16839728C2211yescharacteristic
18#16839727—22no—
19C22⋊C8×D61—22no—
20C61×#408C411yescharacteristic
21C2×#11758213—42no—
22D61⋊M16—24no—
23D61⋊M16—24no—
24D61⋊M16—42no—
25D61⋊M16—42no—
26C2×#11758217—24no—
27C2×D61⋊C8—42no—
28#14519838—24no—
29#14519838—24no—
30#14519837—24no—
31#14519837C2×C418yes—
32C2×#11758226Q811yescharacteristic
33C2×#11758222D412yes—
34C2×#11758222C2311yescharacteristic
35C2×#11758224C2×C412yes—
36C2×#11758224D411yescharacteristic
37#41×D61—41no—
38#41×D61—41no—
39#41×D61—41no—
40#41×D61—41no—
41C2×Q8×D61D411yescharacteristic
42C2×D4×D61C2×C412yes—
43C2×D4×D61Q811yescharacteristic
44C22×C4×D61C2×C411yescharacteristic
45C22×C4×D61D412yes—
46C2×#11758194C2×C411yescharacteristic
47C2×#11758189—21no—
48M16×D61—41no—
49M16×D61—41no—
50C22×#92418C2×C411yescharacteristic
51C2×C8×D61—21no—
52C2×C8×D61—21no—
53C2×C8×D61—21no—
54C2×C8×D61—41no—
55C2×C8×D61—41no—
56C22⋊#92418—22no—
57#14519689—22no—
58#14519689—22no—
59C2×#41×C61C2×C411yescharacteristic
60C2×M16×C61C2×C411yescharacteristic
61C22×C8×C61C2×C411yescharacteristic
62C22⋊C8×C61—22no—
63C23×D61C4211yescharacteristic
64C23×D61C4⋊C412yes—
65#41×C61—41no—
66#41×C61—41no—
67C2×Q8×C61C4⋊C411yescharacteristic
68C2×D4×C61C4212yes—
69C2×D4×C61C4⋊C411yescharacteristic
70C22×C4×C61C22⋊C412yes—
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61C22⋊C412yes—
73C2×C22⋊D61C4⋊C412yes—
74C2×C22⋊D61C22×C412yes—
75C22×Dic61C4211yescharacteristic
76C22×Dic61C22⋊C412yes—
77#11758226—41no—
78#11758226—41no—
79Q8×D61—41no—
80Q8×D61—41no—
81#11758224—42no—
82#11758224—42no—
83#11758224—41no—
84#11758224—41no—
85D4×D61—42no—
86D4×D61—42no—
87D4×D61—41no—
88D4×D61—41no—
89#11758222—42no—
90#11758222—42no—
91#11758222—41no—
92#11758222—41no—
93C2×D244C4⋊C412yes—
94C2×D244C2×Q811yescharacteristic
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—21no—
98C2×C4×D61—21no—
99C2×C4×D61—22no—
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—22no—
103C2×C4×D61C22⋊C411yescharacteristic
104C2×C4×D61C22⋊C412yes—
105C2×C4×D61C22⋊C411yescharacteristic
106C2×C4×D61C4⋊C412yes—
107C2×C4×D61C4⋊C411yescharacteristic
108C2×C4×D61C4⋊C411yescharacteristic
109C2×C4×D61C2×D412yes—
110C2×Dic122C22⋊C412yes—
111C2×Dic122C2×D411yescharacteristic
112#11758217—28no—
113#11758217—44no—
114C2×#92420—44no—
115#11758213—44no—
116#11758213—44no—
117D61⋊C8—44no—
118D61⋊C8—44no—
119M16×C61—41no—
120C2×C8×C61—21no—
121C2×C8×C61—21no—
122C2×C8×C61—41no—
123#11758194—41no—
124C2×#92418—21no—
125C2×#92418—21no—
126C2×#92418—41no—
127#11758189—42no—
128C8×D61—42no—
129C8×D61—41no—
130C8×D61—41no—
131C8×D61—41no—
132C8×D61—41no—
133C23×C61#9512yes—
134C23×C61C2×C4211yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—22no—
138C22×D61—22no—
139C22×D61—42no—
140C22×D61—44no—
141C22×D61#9511yescharacteristic
142C22×D61#9511yescharacteristic
143C22×D61C2×C4⋊C412yes—
144Q8×C61—41no—
145D4×C61—42no—
146D4×C61—41no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61#9512yes—
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61C2×C4⋊C411yescharacteristic
154C22⋊D61—42no—
155C22⋊D61—42no—
156C22⋊D61—42no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61#9511yescharacteristic
162C2×Dic61#9511yescharacteristic
163C2×Dic61C2×C22⋊C412yes—
164D244—42no—
165D244—41no—
166C4×D61—22no—
167C4×D61—22no—
168C4×D61—21no—
169C4×D61—21no—
170C4×D61—21no—
171C4×D61—21no—
172C4×D61—22no—
173C4×D61—22no—
174C4×D61—22no—
175C4×D61—22no—
176C4×D61—21no—
177C4×D61—21no—
178C4×D61#9911yescharacteristic
179C4×D61#9911yescharacteristic
180C4×D61#9911yescharacteristic
181C4×D61#9911yescharacteristic
182Dic122—42no—
183Dic122—41no—
184#92420—48no—
185C8×C61—41no—
186C8×C61—41no—
187#92418—41no—
188#92418—41no—
189C2×#1392—611nomaximal, Sylow
190C22×C61—22no—
191C22×C61—21no—
192C22×C61—42no—
193C22×C61—41no—
194C22×C61C2×#9511yescharacteristic
195C2×D61—22no—
196C2×D61—21no—
197C2×D61—22no—
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—44no—
201C2×D61—41no—
202C2×D61—42no—
203C2×D61—82no—
204C2×D61#34211yescharacteristic
205C2×D61#34211yescharacteristic
206C4×C61—22no—
207C4×C61—21no—
208C4×C61C2×#9911yescharacteristic
209C4×C61C2×#9911yescharacteristic
210Dic61—22no—
211Dic61—21no—
212Dic61#34211yescharacteristic
213Dic61#34211yescharacteristic
214C2×#413—612no—
215C2×#408—611no—
216#1392—612no—
217#1392—612no—
218C2×C61—21no—
219C2×C61—41no—
220C2×C61—42no—
221C2×C61C2×#34211yescharacteristic
222D61—21no—
223D61—41no—
224D61—42no—
225D61#139212yescomplemented
226C22×#41—611no—
227C22×M16—611no—
228C22×M16—1222no—
229C23×C8—611no—
230#413—614no—
231#413—614no—
232#413—1222no—
233#413—1222no—
234C2×#101—1222no—
235C2×#100—614no—
236#408—611no—
237#408—611no—
238#408—611no—
239#408—1222no—
240#408#692311yescharacteristic
241C2×C22⋊C8—1222no—
242C61C2×#139211yescharacteristic, complemented, Sylow
243C2×#41—611no—
244C2×#41—612no—
245C2×#41—611no—
246C2×#41—612no—
247C2×#41—611no—
248C2×#41—2441no—
249C2×#41—2441no—
250C2×#41—2441no—
251C2×#41—2441no—
252C2×#41C2×#692311yescharacteristic
253C22×Q8—611no—
254C22×D4—612no—
255C22×D4—611no—
256C23×C4—612no—
257C23×C4—611no—
258C2×M16—611no—
259C2×M16—1221no—
260C2×M16—1224no—
261C2×M16—1224no—
262C2×M16—1224no—
263C2×M16—2442no—
264C2×M16—2442no—
265C2×M16—2442no—
266C2×M16—2441no—
267C2×M16—2441no—
268C2×M16C2×#692311yescharacteristic
269C22×C8—611no—
270C22×C8—1221no—
271C22×C8—1221no—
272C22×C8—1221no—
273C22×C8—2441no—
274C22×C8—2441no—
275C22×C8—2442no—
276C22×C8C2×#692311yescharacteristic
277#101—1224no—
278#101—1224no—
279#100—618no—
280#100—1224no—
281C22⋊C8—22no—
282C22⋊C8—1222no—
283C22⋊C8—1222no—
284C22⋊C8—1222no—
285C24—612no—
286C24—611no—
287#41—41no—
288#41—41no—
289#41—2441no—
290#41—2442no—
291#41—2441no—
292#41—2442no—
293#41—2441no—
294#41—2442no—
295#41—2441no—
296#41—2442no—
297#41—2441no—
298#41—2441no—
299C2×Q8—612no—
300C2×Q8—611no—
301C2×Q8—2441no—
302C2×Q8—2441no—
303C2×Q8C4⋊#692311yescharacteristic
304C2×D4—612no—
305C2×D4—612no—
306C2×D4—612no—
307C2×D4—612no—
308C2×D4—611no—
309C2×D4—2442no—
310C2×D4—2442no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4C4×#692312yes—
314C2×D4C4⋊#692311yescharacteristic
315C22×C4—611no—
316C22×C4—612no—
317C22×C4—611no—
318C22×C4—612no—
319C22×C4—612no—
320C22×C4—612no—
321C22×C4—611no—
322C22×C4—611no—
323C22×C4—611no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1222no—
330C22×C4—1222no—
331C22×C4—1222no—
332C22×C4C22⋊#692312yes—
333C22×C4C22×#692311yescharacteristic
334M16—41no—
335M16—1228no—
336M16—2442no—
337M16—2444no—
338M16—2444no—
339M16—2444no—
340M16—2441no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—1221no—
345C2×C8—1221no—
346C2×C8—2441no—
347C2×C8—2441no—
348C2×C8—2441no—
349C2×C8—2441no—
350C2×C8—2442no—
351C2×C8—2444no—
352C2×C8—2444no—
353C2×C8—2444no—
354C2×C8—2441no—
355C23—611no—
356C23—611no—
357C23—612no—
358C23—1222no—
359C23—1222no—
360C23—1221no—
361C23—1221no—
362C23—2444no—
363C23—2442no—
364C23(C2×#6923)⋊C412yes—
365C23C2×C4×#692311yescharacteristic
366Q8—41no—
367Q8—2442no—
368Q8—2441no—
369D4—42no—
370D4—41no—
371D4—2442no—
372D4—2441no—
373D4—2442no—
374D4—2442no—
375D4—2442no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—611no—
381C2×C4—611no—
382C2×C4—611no—
383C2×C4—611no—
384C2×C4—611no—
385C2×C4—611no—
386C2×C4—612no—
387C2×C4—1222no—
388C2×C4—1222no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—1222no—
392C2×C4—1222no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1222no—
396C2×C4—1222no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1222no—
400C2×C4—1221no—
401C2×C4—1222no—
402C2×C4—1221no—
403C2×C4(C2×#6923)⋊C412yes—
404C2×C4C2×C22⋊#692311yescharacteristic
405C2×C4C2×C4⋊#692311yescharacteristic
406C8—41no—
407C8—41no—
408C8—2448no—
409C8—2441no—
410C8—2441no—
411C22—22no—
412C22—21no—
413C22—42no—
414C22—41no—
415C22—611no—
416C22—611no—
417C22—1222no—
418C22—1222no—
419C22—1221no—
420C22—4882no—
421C22—2442no—
422C22—2441no—
423C22—2444no—
424C22—2442no—
425C22—2442no—
426C22C2×(C2×#6923)⋊C411yescharacteristic
427C4—22no—
428C4—21no—
429C4—611no—
430C4—611no—
431C4—1222no—
432C4—1221no—
433C4D61⋊#9911yescharacteristic
434C4D61⋊#9911yescharacteristic
435C2—21no—
436C2—42no—
437C2—41no—
438C2—612no—
439C2—1221no—
440C2—2442no—
441C2—2441no—
442C2#2146151611yescharacteristic
subgroupG/H
centerC4D61⋊#99
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#408C4
socleC2×C61C2×#342
prime radicalC2×#41×D61C22
cosocleC2×#11758222C23
hypercenter#408#6923
hypocenterC61C2×#1392
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #21460911C2semidirect
3. #21460910C2semidirect
225. D61#1392semidirect
242. C61C2×#1392semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4132
#214609112
#214609102
#408×D612
C2×#139261