Finite Group Databasebeta

← #29653124

Subgroups — #29653124

Order 15616.

1222 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 187 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#21459590C211yesmaximal, characteristic, complemented
2#21455819C212yesmaximal, complemented
3#21444821C214yesmaximal, complemented
4#21444753C214yesmaximal, complemented
5#21444378C212yesmaximal, complemented
6#21444143C211yesmaximal, characteristic, complemented
7C61×#2536C211yesmaximal, characteristic, complemented
8C2×#14519825—21no—
9#16840434—22no—
10Q8×D244—22no—
11#16840424—24no—
12#16840343—21no—
13#16840343C2214yes—
14#16840312—22no—
15#16840312C2212yes—
16#16840075C2214yes—
17#16840042—22no—
18#16839957—22no—
19#16839896C2214yes—
20C4×#11758199—24no—
21#16839815C2212yes—
22#16839495C2218yes—
23#16839488C2214yes—
24C2×C22⋊Q8×C61—21no—
25C2×C4×Q8×C61C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#398C2212yes—
28C61×#394C2212yes—
29C61×#388—22no—
30C61×#382C2211yescharacteristic
31C61×#380—21no—
32#14519825—44no—
33#14519825—44no—
34C2×#11758226—24no—
35C2×#11758222—24no—
36C2×#11758222—22no—
37C2×Q8×D61—24no—
38C2×#11758199—24no—
39#14519777—22no—
40#14519777D412yes—
41#14519774—22no—
42#14519774D412yes—
43#14519765—24no—
44#14519765—22no—
45#14519757—24no—
46#14519757—28no—
47#14519752—24no—
48#14519752—28no—
49#14519751—22no—
50#14519751—24no—
51#14519751C2314yes—
52C4×D244—24no—
53C4×D244—24no—
54C4×D244—22no—
55C4×D244D414yes—
56#14519710—24no—
57C4⋊#92418—22no—
58#14519698—24no—
59#14519697—24no—
60C22⋊#92418C2314yes—
61C4×#92418—22no—
62C22×Q8×C61D411yescharacteristic
63C22×D4×C61C2311yescharacteristic
64C23×C4×C61—21no—
65C22⋊Q8×C61—44no—
66C4×Q8×C61—22no—
67C4×Q8×C61—22no—
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C4⋊C4×C61C2311yescharacteristic
74C2×C4⋊C4×C61C2312yes—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—22no—
78C2×C42×C61D412yes—
79C2×C42×C61C2311yescharacteristic
80#95×C61—22no—
81#95×C61—21no—
82#95×C61C2312yes—
83C24×C61—21no—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61D814yescomplemented
87C2×D4×C61—22no—
88C2×D4×C61—42no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61—42no—
92C22×C4×C61—42no—
93C22×C4×C61—42no—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2411yescharacteristic
99C2×C22⋊D61—24no—
100#11758226—416no—
101Q8×D61—416no—
102#11758222—48no—
103#11758222—48no—
104#11758222—44no—
105#11758222—44no—
106C2×D244—24no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—24no—
110C2×D244C2×D414yes—
111C2×C4×D61—28no—
112C2×C4×D61—24no—
113C2×Dic122—24no—
114C2×Dic122—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—24no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—42no—
119C4⋊C4×C61—44no—
120C22⋊C4×C61—44no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—44no—
124C42×C61—22no—
125C42×C61—22no—
126C42×C61C2×D412yes—
127C42×C61C2×D412yes—
128#11758199—48no—
129#11758198—28no—
130#11758198—24no—
131#11758198—24no—
132C4⋊Dic61—24no—
133C4⋊Dic61—22no—
134C4⋊Dic61C2×D414yes—
135C2×#92418—24no—
136C23×C61—22no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61C22×D411yescharacteristic
144C22×D61—24no—
145Q8×C61—28no—
146D4×C61—24no—
147D4×C61—42no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—84no—
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12011yescharacteristic
163C2×C4×C61#12012yes—
164C2×C4×C61C2×D814yes—
165C2×C4×C61C2×D812yes—
166C2×C4×C61AGL(1,ℤ/8)12yes—
167C2×C4×C61C22×D412yes—
168C22⋊D61—44no—
169C22⋊D61—44no—
170C2×Dic61—24no—
171D244—44no—
172D244—44no—
173D244—48no—
174D244—44no—
175D244—84no—
176C4×D61—416no—
177C4×D61—48no—
178Dic122—48no—
179Dic122—44no—
180#92418—44no—
181#18218—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—42no—
185C22×C61—42no—
186C22×C61—42no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61C2×#12011yescharacteristic
192C22×C61C22×D811yescharacteristic
193C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
194C2×D61—44no—
195C2×D61—44no—
196C2×D61—44no—
197C4×C61—24no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—22no—
201C4×C61—42no—
202C4×C61—82no—
203C4×C61—81no—
204C4×C61#44714yes—
205Dic61—44no—
206#3497—611no—
207#3100—612no—
208#2536D6111yescharacteristic, complemented
209#1610—614no—
210#1598—614no—
211#1517—612no—
212#1461—611no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61—82no—
216C2×C61C2×#44711yescharacteristic
217C2×C61#302011yescharacteristic
218C2×C61#302511yescharacteristic
219D61—84no—
220C2×2−1+4—1221no—
221#550—1222no—
222D4×Q8—1222no—
223#548—1224no—
224#537—614no—
225#537—1221no—
226#532—612no—
227#532—1222no—
228C2×C22⋊Q8—21no—
229C2×C4×Q8C2×D6111yescharacteristic
230C2×C4×D4C2×D6111yescharacteristic
231#480—614no—
232#474—1222no—
233#459—1222no—
234#447—614no—
235C4×D8—1224no—
236#431—612no—
237#398C2×D6112yes—
238#394C2×D6112yes—
239#388—22no—
240#382C2×D6111yescharacteristic
241#380—21no—
242#327—618no—
243#324—614no—
244C61#1821811yescharacteristic, complemented, Sylow
2452−1+4—2444no—
2462−1+4—2444no—
247C2×#41—1224no—
248C2×#41—1222no—
249C2×#41—1224no—
250C22×Q8—1224no—
251C22×Q8C22⋊D6111yescharacteristic
252C22×D4C22×D6111yescharacteristic
253C23×C4—21no—
254C2×D8—1224no—
255C4⋊Q8—612no—
256C4⋊Q8—1222no—
257#127—612no—
258#127—1222no—
259#124—1224no—
260#124—1222no—
261C22⋊Q8—44no—
262C22⋊Q8—1228no—
263C22⋊Q8—1224no—
264#121—614no—
265#121—1228no—
266#121—1224no—
267#121—1222no—
268#121—1224no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—614no—
274C4×D4—1224no—
275C4×D4—1224no—
276C4×D4—1222no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4C22⋊D6111yescharacteristic
280C2×C4⋊C4C22×D6111yescharacteristic
281C2×C4⋊C4C22×D6112yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
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—22no—
294#95—21no—
295#95C22×D6112yes—
296C24—21no—
297#41—2448no—
298#41—24416no—
299#41—2448no—
300#41—2444no—
301#41—2444no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—1224no—
305C2×Q8—1222no—
306C2×Q8—24416no—
307C2×Q8#1175819914yescomplemented
308C2×D4—22no—
309C2×D4—42no—
310C2×D4—614no—
311C2×D4—1224no—
312C2×D4—1224no—
313C2×D4—1224no—
314C2×D4—1222no—
315C2×D4—1222no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—1228no—
322C22×C4—1224no—
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—22no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—614no—
336C4⋊C4—1224no—
337C4⋊C4—1222no—
338C22⋊C4—44no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—44no—
342C22⋊C4—1228no—
343C22⋊C4—1224no—
344C22⋊C4—1224no—
345C42—22no—
346C42—22no—
347C42D4×D6112yes—
348C42D4×D6112yes—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—1224no—
357C23C22×C22⋊D6111yescharacteristic
358Q8—28no—
359Q8—2448no—
360Q8—2444no—
361D4—24no—
362D4—42no—
363D4—4884no—
364D4—2444no—
365D4—2444no—
366D4—2448no—
367D4—2444no—
368D4—2444no—
369D4—2444no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—84no—
383C2×C4—1224no—
384C2×C4—24416no—
385C2×C4—2448no—
386C2×C4#1451974511yescharacteristic
387C2×C4#1451974511yescharacteristic
388C2×C4#1451974512yes—
389C2×C4C2×#1175819914yes—
390C2×C4C2×#1175819912yes—
391C2×C4#1451980212yes—
392C2×C4C2×D4×D6112yes—
393C8—2444no—
394C22—21no—
395C22—21no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—2444no—
404C22—2444no—
405C22—2444no—
406C22C2×#1451974511yescharacteristic
407C22C22×#1175819911yescharacteristic
408C22C2×#1451980211yescharacteristic
409C4—21no—
410C4—21no—
411C4—22no—
412C4—24no—
413C4—42no—
414C4—82no—
415C4—81no—
416C4—2444no—
417C4#1683989514yes—
418C2—21no—
419C2—21no—
420C2—82no—
421C2—4884no—
422C2C2×#1683989511yescharacteristic
423C2#2145518211yescharacteristic
424C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2536C2
socleC22×C61C2×#120
hypercenter#2536D61
hypocenterC61#18218
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #21459590C2semidirect
2. #21455819C2semidirect
3. #21444821C2semidirect
4. #21444753C2semidirect
5. #21444378C2semidirect
6. #21444143C2semidirect
7. C61×#2536C2semidirect
86. C2×Q8×C61D8semidirect
208. #2536D61semidirect
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
#214595902
#214558192
#214448212
#214447532
#214443782
#214441432
C61×#25362
#1821861