Finite Group Databasebeta

← #29580690

Subgroups — #29580690

Order 15616.

750 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 89 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#21456616C211yesmaximal, characteristic
2C2×#16839901C211yesmaximal, characteristic, complemented
3#21445743C211yesmaximal, characteristic, complemented
4#21444179C211yesmaximal, characteristic, complemented
5#21444178C211yesmaximal, characteristic, complemented
6#21443707C211yesmaximal, characteristic, complemented
7C2×C61×#448C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519751C2211yescharacteristic
10#16840080—21no—
11#16840078—21no—
12#16840013—22no—
13#16839901—22no—
14#16839901—22no—
15#16839896—21no—
16#16839895—21no—
17C2×#14519701—21no—
18C2×#14519696—22no—
19#16839726C2211yescharacteristic
20#16839726C2211yescharacteristic
21C2×C22⋊#92418C2211yescharacteristic
22#16839486—22no—
23#16839485—22no—
24C23×Q8×C61C2211yescharacteristic
25C22×SD16×C61—21no—
26C2×#121×C61C2211yescharacteristic
27C61×#448—22no—
28C61×#448—22no—
29C2×#103×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C2×#11758201—24no—
34C2×#11758201—44no—
35C2×#11758199—22no—
36C2×#11758194—21no—
37C2×#11758194D412yes—
38C22×#92418—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42#14519751—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×#11758198—21no—
45#14519708—22no—
46#14519701—24no—
47#14519698—22no—
48#14519697—22no—
49#14519696—44no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418D412yes—
56C22×Q8×C61—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—44no—
62C22×Q8×C61D412yes—
63C22×D4×C61—21no—
64C22×D4×C61D411yescharacteristic
65C23×C4×C61—21no—
66C23×C4×C61C2311yescharacteristic
67C2×SD16×C61—24no—
68C2×SD16×C61—44no—
69C22×C8×C61—21no—
70#121×C61—21no—
71#121×C61—21no—
72#121×C61—21no—
73#121×C61—21no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76#103×C61—24no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61D412yes—
79C24×C61—21no—
80C24×C61#4111yescharacteristic
81C23×D61—21no—
82C2×Q8×C61—24no—
83C2×Q8×C61—24no—
84C2×Q8×C61—24no—
85C2×Q8×C61—22no—
86C2×Q8×C61—22no—
87C2×Q8×C61—44no—
88C2×Q8×C61—42no—
89C2×Q8×C61—42no—
90C2×Q8×C61—42no—
91C2×Q8×C61—44no—
92C2×Q8×C61—44no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C2×D4×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—42no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—82no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—42no—
104C22×C4×C61—44no—
105C22×C4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—41no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—82no—
116C22×Dic61—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—42no—
120SD16×C61—48no—
121C2×C8×C61—22no—
122C2×C8×C61—42no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127#11758201—48no—
128#11758199—82no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758194—42no—
134#11758194—42no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—42no—
138C2×#92418—41no—
139C23×C61—22no—
140C23×C61—22no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—42no—
144C23×C61—81no—
145C23×C61—81no—
146C23×C61#12111yescharacteristic
147C23×C61#12111yescharacteristic
148C23×C61#12111yescharacteristic
149C22×D61—41no—
150C22×D61—42no—
151C22×D61—81no—
152C22×D61—81no—
153Q8×C61—24no—
154Q8×C61—24no—
155Q8×C61—42no—
156Q8×C61—44no—
157Q8×C61—42no—
158Q8×C61—44no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—82no—
162D4×C61—82no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—24no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—42no—
171C2×C4×C61—44no—
172C2×C4×C61—44no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—81no—
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12711yescharacteristic
180C22⋊D61—82no—
181C22⋊D61—82no—
182C2×Dic61—41no—
183C2×Dic61—81no—
184D244—42no—
185D244—42no—
186C8×C61—42no—
187#92418—42no—
188#92418—42no—
189#92418—81no—
190#9988—611nomaximal, Sylow
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—44no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61—162no—
203C22×C61#39211yescharacteristic
204C22×C61#47112yes—
205C22×C61#47112yes—
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C2×D61—162no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—44no—
214C4×C61—81no—
215C4×C61B(4,𝔽2)12yes—
216Dic61—81no—
217#3192—611no—
218C2×#448—611no—
219C2×#448D6111yescharacteristic, complemented
220#1784—611no—
221#1471—611no—
222#1471—611no—
223#1298—611no—
224C2×C61—22no—
225C2×C61—22no—
226C2×C61—41no—
227C2×C61—161no—
228C2×C61#201911yescharacteristic
229C2×C61#201911yescharacteristic
230C2×C61#202111yescharacteristic
231D61—161no—
232C23×Q8C2×D6111yescharacteristic
233C22×SD16—21no—
234C22×SD16—1221no—
235C2×#121—611no—
236C2×#121C2×D6111yescharacteristic
237#481—1221no—
238#481—1221no—
239#469—1222no—
240#448—22no—
241#448—22no—
242#448—1222no—
243#448—1222no—
244#447—1221no—
245#447—1221no—
246C2×#103—21no—
247C2×#103—1221no—
248C2×#101—1222no—
249#407—611no—
250#407—611no—
251C2×C22⋊C8—611no—
252C2×C22⋊C8C2×D6111yescharacteristic
253#323—1222no—
254#323—1222no—
255C61#998811yescharacteristic, complemented, Sylow
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—44no—
262C22×Q8C22⋊D6112yes—
263C22×D4—21no—
264C22×D4—611no—
265C22×D4—1221no—
266C22×D4C22⋊D6111yescharacteristic
267C23×C4—21no—
268C23×C4C22×D6111yescharacteristic
269C2×SD16—24no—
270C2×SD16—44no—
271C2×SD16—1224no—
272C2×SD16—2444no—
273C2×D8—1222no—
274C2×M16—612no—
275C2×M16—1221no—
276C22×C8—21no—
277C22×C8—1221no—
278#121—21no—
279#121—21no—
280#121—21no—
281#121—21no—
282#121—1221no—
283#121—1221no—
284#121—1221no—
285#121—1221no—
286C2×C4⋊C4—611no—
287C2×C4⋊C4C22⋊D6111yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—1221no—
290#106—1222no—
291#103—24no—
292#103—1224no—
293#102—1222no—
294#102—1222no—
295#101—2444no—
296C22⋊C8—21no—
297C22⋊C8—612no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301C22⋊C8—1221no—
302C22⋊C8—1221no—
303C22⋊C8C22⋊D6112yes—
304C24—21no—
305C24—1221no—
306C24#1175822411yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—44no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—82no—
326C2×D4—1221no—
327C2×D4—1221no—
328C2×D4—4882no—
329C2×D4—2442no—
330C2×D4—2441no—
331C2×D4—2441no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—1221no—
339C22×C4D4×D6111yescharacteristic
340C22×C4D4×D6111yescharacteristic
341C22×C4D4×D6111yescharacteristic
342C22×C4C2×C22⋊D6111yescharacteristic
343C22×C4C2×C22⋊D6111yescharacteristic
344C22×C4C2×C22⋊D6111yescharacteristic
345SD16—48no—
346SD16—2448no—
347D8—4882no—
348M16—2442no—
349M16—2442no—
350C2×C8—22no—
351C2×C8—42no—
352C2×C8—1222no—
353C2×C8—1222no—
354C2×C8—2441no—
355C2×C8—2442no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—2441no—
363C22⋊C4—2441no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—4881no—
372C23—4881no—
373C23—2442no—
374C23—2441no—
375C23#1451974811yescharacteristic
376C23#1451974911yescharacteristic
377C23#1451974911yescharacteristic
378Q8—24no—
379Q8—24no—
380Q8—42no—
381Q8—44no—
382Q8—42no—
383Q8—44no—
384D4—42no—
385D4—42no—
386D4—82no—
387D4—82no—
388D4—4882no—
389D4—4882no—
390D4—2442no—
391D4—2442no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—4881no—
406C2×C4—2441no—
407C2×C4#1451974511yescharacteristic
408C2×C4#1451974511yescharacteristic
409C2×C4#1451974611yescharacteristic
410C2×C4#1451977311yescharacteristic
411C8—42no—
412C8—4881no—
413C8—2442no—
414C8—2442no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—44no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—162no—
427C22—9762no—
428C22—4881no—
429C22—4881no—
430C22—4881no—
431C22#1683967011yescharacteristic
432C22#1684002212yes—
433C22#1684002712yes—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—44no—
438C4—81no—
439C4—4881no—
440C4#1683994812yes—
441C2—22no—
442C2—22no—
443C2—41no—
444C2—161no—
445C2—9761no—
446C2#2144693011yescharacteristic
447C2#2144693111yescharacteristic
448C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#448C2
socleC22×C61#392
prime radical#21456616C2
hypercenterC2×#448D61
hypocenterC61#9988
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C2×#16839901C2semidirect
3. #21445743C2semidirect
4. #21444179C2semidirect
5. #21444178C2semidirect
6. #21443707C2semidirect
7. C2×C61×#448C2semidirect
219. C2×#448D61semidirect
255. C61#9988semidirect

Maximal subgroups

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

subgroupindex
#214566162
C2×#168399012
#214457432
#214441792
#214441782
#214437072
C2×C61×#4482
#998861