Finite Group Databasebeta

← #29653125

Subgroups — #29653125

Order 15616.

1222 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 203 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#21455815C212yesmaximal, complemented
2#21450724C211yesmaximal, characteristic, complemented
3#21444822C214yesmaximal, complemented
4#21444754C214yesmaximal, complemented
5#21444379C212yesmaximal, complemented
6#21444143C211yesmaximal, characteristic, complemented
7C61×#3497C211yesmaximal, characteristic, complemented
8C2×#14519753—21no—
9C2×C4×D244C2211yescharacteristic
10#16840073C2214yes—
11#16840039—22no—
12#16839954—22no—
13#16839895C2214yes—
14C4×#11758199—24no—
15#16839815C2212yes—
16#16839692C2212yes—
17#16839679C2212yes—
18#16839653—22no—
19#16839628C2211yescharacteristic
20#16839619—21no—
21#16839496C2218yes—
22#16839488C2214yes—
23C2×2−1+4×C61—21no—
24C61×#550—22no—
25D4×Q8×C61—22no—
26C61×#548—24no—
27C61×#537—21no—
28C61×#537C2214yes—
29C61×#532—22no—
30C61×#532C2212yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C4×D61—21no—
34C2×#11758199—24no—
35#14519753—44no—
36C4×D244—22no—
37C4×D244—22no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61C2311yescharacteristic
40C2×#11758196—22no—
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44#14519710—24no—
45C4⋊#92418—22no—
46#14519698—24no—
47#14519697—24no—
48C22⋊#92418C2314yes—
49C4×#92418—22no—
50#14519684—22no—
51#14519684—21no—
52#14519684C2312yes—
532−1+4×C61—44no—
542−1+4×C61—44no—
55C2×#41×C61—24no—
56C2×#41×C61—24no—
57C2×#41×C61—22no—
58C22×Q8×C61—24no—
59C22×Q8×C61D411yescharacteristic
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61D412yes—
62#127×C61—22no—
63#127×C61D412yes—
64#124×C61—24no—
65#124×C61—22no—
66C22⋊Q8×C61—24no—
67C22⋊Q8×C61—28no—
68#121×C61—24no—
69#121×C61—28no—
70#121×C61—22no—
71#121×C61—24no—
72#121×C61C2314yes—
73C4×Q8×C61—22no—
74C4×Q8×C61—22no—
75C4×D4×C61—24no—
76C4×D4×C61—24no—
77C4×D4×C61—22no—
78C4×D4×C61D414yes—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C4⋊C4×C61C2312yes—
81C2×C42×C61D412yes—
82C2×C42×C61C2311yescharacteristic
83C23×D61—21no—
84#41×C61—48no—
85#41×C61—416no—
86#41×C61—48no—
87#41×C61—44no—
88#41×C61—44no—
89C2×Q8×C61—24no—
90C2×Q8×C61—22no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61—416no—
94C2×Q8×C61D814yescomplemented
95C2×D4×C61—24no—
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61C2×D414yes—
101C22×C4×C61—28no—
102C22×C4×C61—24no—
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2411yescharacteristic
108C22×Dic61—21no—
109C22×Dic61—22no—
110C2×D244—22no—
111C2×D244—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61C2×D414yes—
120C22⋊C4×C61—28no—
121C22⋊C4×C61—24no—
122C22⋊C4×C61—24no—
123C42×C61—22no—
124C42×C61—22no—
125C42×C61C2×D412yes—
126C42×C61C2×D412yes—
127#11758199—48no—
128#11758198—44no—
129#11758198—41no—
130#11758198—41no—
131#11758198—44no—
132C4⋊Dic61—22no—
133C4⋊Dic61—42no—
134#11758196—44no—
135C2×#92418—24no—
136C23×C61—24no—
137C23×C61C22×D411yescharacteristic
138C22×D61—22no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145Q8×C61—28no—
146Q8×C61—48no—
147Q8×C61—44no—
148D4×C61—44no—
149D4×C61—44no—
150D4×C61—48no—
151D4×C61—44no—
152D4×C61—44no—
153D4×C61—44no—
154D4×C61—84no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—24no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—416no—
162C2×C4×C61—48no—
163C2×C4×C61—42no—
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61#12012yes—
167C2×C4×C61C2×D814yes—
168C2×C4×C61C2×D812yes—
169C2×C4×C61AGL(1,ℤ/8)12yes—
170C2×C4×C61C22×D412yes—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177D244—24no—
178D244—42no—
179C4×D61—84no—
180#92418—44no—
181#18218—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—44no—
185C22×C61—44no—
186C22×C61—44no—
187C22×C61C2×#12011yescharacteristic
188C22×C61C22×D811yescharacteristic
189C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
190C2×D61—42no—
191C2×D61—42no—
192C2×D61—42no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—24no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—22no—
201C4×C61—44no—
202C4×C61—42no—
203C4×C61#44714yes—
204Dic61—82no—
205Dic61—81no—
206#3497D6111yescharacteristic, complemented
207#3100—612no—
208#2536—611no—
209#1610—614no—
210#1598—614no—
211#1517—612no—
212#1461—611no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61—84no—
216C2×C61C2×#44711yescharacteristic
217C2×C61#302011yescharacteristic
218C2×C61#302511yescharacteristic
219D61—82no—
220C2×2−1+4—21no—
221#550—22no—
222D4×Q8—22no—
223#548—24no—
224#537—21no—
225#537C2×D6114yes—
226#532—22no—
227#532C2×D6112yes—
228C2×C22⋊Q8—1221no—
229C2×C4×Q8C2×D6111yescharacteristic
230C2×C4×D4—611no—
231#480—614no—
232#474—1222no—
233#459—1222no—
234#447—614no—
235C4×D8—1224no—
236#431—612no—
237#398—612no—
238#394—612no—
239#388—1222no—
240#382—611no—
241#380—1221no—
242#327—618no—
243#324—614no—
244C61#1821811yescharacteristic, complemented, Sylow
2452−1+4—44no—
2462−1+4—44no—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—22no—
250C22×Q8—24no—
251C22×Q8C22⋊D6111yescharacteristic
252C22×D4—611no—
253C23×C4—1221no—
254C2×D8—1224no—
255C4⋊Q8—22no—
256C4⋊Q8C22⋊D6112yes—
257#127—22no—
258#127C22⋊D6112yes—
259#124—24no—
260#124—22no—
261C22⋊Q8—24no—
262C22⋊Q8—28no—
263C22⋊Q8—2444no—
264#121—24no—
265#121—28no—
266#121—22no—
267#121—24no—
268#121C22×D6114yes—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—1222no—
275C4×D4—1222no—
276C4×D4C22⋊D6114yes—
277C2×C4⋊C4—611no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4—1222no—
280C2×C4⋊C4C22⋊D6111yescharacteristic
281C2×C4⋊C4C22×D6112yes—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1222no—
284C2×C22⋊C4—1222no—
285C2×C42C22⋊D6112yes—
286C2×C42C22×D6111yescharacteristic
287#107—1224no—
288C4⋊C8—1222no—
289#102—1224no—
290#102—1224no—
291C22⋊C8—614no—
292C4×C8—1222no—
293#95—612no—
294#95—1222no—
295#95—1221no—
296C24—1221no—
297#41—48no—
298#41—416no—
299#41—48no—
300#41—44no—
301#41—44no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—416no—
307C2×Q8#1175819914yescomplemented
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—1222no—
314C2×D4—2442no—
315C2×D4C2×C22⋊D6114yes—
316C22×C4—28no—
317C22×C4—24no—
318C22×C4—1222no—
319C22×C4—1221no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4C2×C22⋊D6112yes—
324C22×C4C2×C22⋊D6112yes—
325C22×C4C2×C22⋊D6111yescharacteristic
326C22×C4C2×C22⋊D6111yescharacteristic
327C22×C4C23×D6111yescharacteristic
328D8—2448no—
329C2×C8—1224no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—1222no—
335C4⋊C4—2444no—
336C4⋊C4—2442no—
337C4⋊C4C2×C22⋊D6114yes—
338C22⋊C4—28no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—2444no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C22⋊C4—2444no—
345C42—22no—
346C42—22no—
347C42C2×C22⋊D6112yes—
348C42C2×C22⋊D6112yes—
349C23—24no—
350C23—1222no—
351C23—2441no—
352C23—2441no—
353C23—2441no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23C22×C22⋊D6111yescharacteristic
358Q8—28no—
359Q8—48no—
360Q8—44no—
361D4—44no—
362D4—44no—
363D4—48no—
364D4—44no—
365D4—44no—
366D4—44no—
367D4—84no—
368D4—1224no—
369D4—2442no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—416no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—4884no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4—2442no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4#1451974611yescharacteristic
387C2×C4#1451974611yescharacteristic
388C2×C4#1451974612yes—
389C2×C4C2×#1175819914yes—
390C2×C4C2×#1175819912yes—
391C2×C4#1451980312yes—
392C2×C4C22×C22⋊D6112yes—
393C8—2444no—
394C22—21no—
395C22—21no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—4881no—
400C22—4881no—
401C22—4881no—
402C22—4881no—
403C22—2442no—
404C22—2442no—
405C22—2442no—
406C22C2×#1451974611yescharacteristic
407C22C22×#1175819911yescharacteristic
408C22C2×#1451980311yescharacteristic
409C4—21no—
410C4—21no—
411C4—22no—
412C4—24no—
413C4—42no—
414C4—44no—
415C4—4882no—
416C4—4881no—
417C4#1683989614yes—
418C2—21no—
419C2—21no—
420C2—84no—
421C2—4882no—
422C2C2×#1683989611yescharacteristic
423C2#2145518311yescharacteristic
424C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18218
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21455815C2semidirect
2. #21450724C2semidirect
3. #21444822C2semidirect
4. #21444754C2semidirect
5. #21444379C2semidirect
6. #21444143C2semidirect
7. C61×#3497C2semidirect
94. C2×Q8×C61D8semidirect
206. #3497D61semidirect
244. C61#18218semidirect
307. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214558152
#214507242
#214448222
#214447542
#214443792
#214441432
C61×#34972
#1821861