Finite Group Databasebeta

← C2×#21447820 #29834042

Subgroups — C2×#21447820 #29834042

Order 15616.

1392 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1392 conjugacy classes of subgroups — too many to draw. Only its 171 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×#16839948C212yesmaximal, complemented
2C2×#16839939C211yesmaximal, characteristic, complemented
3#21447820C218yesmaximal, direct product factor
4C2×#16839557C211yesmaximal, characteristic, complemented
5C2×#16839549C212yesmaximal, complemented
6C2×C61×H(3,𝔽4)C211yesmaximal, characteristic, complemented
7C2×#14519822C2211yescharacteristic
8C2×#14519808—21no—
9C2×#14519746—22no—
10#16839948—24no—
11#16839948C2218yescomplemented
12#16839939—22no—
13#16839939C2214yescomplemented
14C2×#14519705—21no—
15C2×#14519694C2211yescharacteristic
16C2×#14519692C2212yes—
17C2×#14519691—22no—
18#16839557C2214yescomplemented
19#16839549C2218yescomplemented
20C61×H(3,𝔽4)—22no—
21C61×H(3,𝔽4)—24no—
22C61×H(3,𝔽4)C2214yescomplemented
23C2×#124×C61—21no—
24C2×#124×C61—22no—
25C2×#124×C61—21no—
26C2×#124×C61C2211yescharacteristic
27C2×#120×C61—22no—
28C2×#120×C61C2212yes—
29#14519822—21no—
30#14519822—22no—
31#14519822—42no—
32#14519822C2312yes—
33C2×#11758222—21no—
34C2×#11758224—41no—
35C22×C22⋊D61—22no—
36C22×C22⋊D61—42no—
37C2×D4×D61—41no—
38#14519808—24no—
39#14519808—42no—
40C2×#11758202—41no—
41C2×#11758201—41no—
42C2×#11758194—21no—
43#14519746—28no—
44#14519746—44no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—42no—
47#14519705—24no—
48#14519694—21no—
49#14519694C2312yes—
50#14519692—22no—
51#14519692C2314yes—
52#14519691—28no—
53C25×C61D412yescomplemented
54C22×Q8×C61—21no—
55C22×Q8×C61D411yescharacteristic, complemented
56C22×D4×C61—21no—
57C22×D4×C61—21no—
58C22×D4×C61—22no—
59C22×D4×C61C2311yescharacteristic
60#124×C61—28no—
61#124×C61—24no—
62#124×C61—24no—
63#124×C61—24no—
64#124×C61—22no—
65#124×C61—28no—
66#124×C61—24no—
67#124×C61D412yescomplemented
68#124×C61C2312yes—
69#120×C61—28no—
70#120×C61—28no—
71#120×C61—22no—
72#120×C61—22no—
73#120×C61D414yescomplemented
74#120×C61C2314yes—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61D412yescomplemented
80C2×C42×C61—21no—
81C2×C42×C61D411yescharacteristic, complemented
82C24×C61—22no—
83C24×C61—22no—
84C24×C61—42no—
85C24×C61—42no—
86C24×C61—84no—
87C24×C61C2×D412yes—
88C24×C61C2×D414yescomplemented
89C23×D61—22no—
90C2×Q8×C61—24no—
91C2×Q8×C61—21no—
92C2×Q8×C61—41no—
93C2×Q8×C61—41no—
94C2×Q8×C61—82no—
95C2×Q8×C61C2×D412yescomplemented
96C2×D4×C61—24no—
97C2×D4×C61—24no—
98C2×D4×C61—24no—
99C2×D4×C61—24no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61—84no—
103C2×D4×C61—82no—
104C2×D4×C61—81no—
105C2×D4×C61—81no—
106C2×D4×C61C2×D412yes—
107C2×D4×C61C2×D411yescharacteristic
108C2×D4×C61C2411yescharacteristic
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—24no—
114C2×C22⋊D61—24no—
115C2×C22⋊D61—42no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—41no—
118C2×C22⋊D61—42no—
119C2×C22⋊D61—48no—
120C2×C22⋊D61—88no—
121C22×Dic61—22no—
122C22×Dic61—42no—
123#11758224—42no—
124#11758224—42no—
125#11758224—44no—
126D4×D61—42no—
127D4×D61—42no—
128D4×D61—44no—
129#11758222—22no—
130#11758222—22no—
131#11758222—41no—
132#11758222—41no—
133C2×D244—21no—
134C2×C4×D61—41no—
135C2×Dic122—21no—
136C22⋊C4×C61—28no—
137C22⋊C4×C61—28no—
138C22⋊C4×C61—24no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—24no—
142C22⋊C4×C61—24no—
143C22⋊C4×C61C2×D414yescomplemented
144C42×C61—24no—
145C42×C61—21no—
146C42×C61C2×D412yescomplemented
147C22⋊Dic61—24no—
148C22⋊Dic61—48no—
149C22⋊Dic61—42no—
150#11758202—44no—
151#11758201—44no—
152#11758194—22no—
153#11758194—41no—
154C2×#92418—41no—
155C23×C61—22no—
156C23×C61—22no—
157C23×C61—24no—
158C23×C61—44no—
159C23×C61—44no—
160C23×C61—44no—
161C23×C61—44no—
162C23×C61—48no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—82no—
166C23×C61—84no—
167C23×C61—84no—
168C23×C61—88no—
169C23×C61—84no—
170C23×C61#12011yescharacteristic
171C23×C61#12012yes—
172C23×C61C22×D412yes—
173C22×D61—24no—
174C22×D61—41no—
175C22×D61—42no—
176C22×D61—84no—
177Q8×C61—42no—
178Q8×C61—81no—
179Q8×C61—84no—
180D4×C61—41no—
181D4×C61—42no—
182D4×C61—41no—
183D4×C61—84no—
184D4×C61—84no—
185D4×C61—84no—
186D4×C61—84no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—44no—
199C22⋊D61—44no—
200C22⋊D61—42no—
201C22⋊D61—42no—
202C22⋊D61—816no—
203C2×Dic61—24no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—44no—
207C2×Dic61—84no—
208D244—22no—
209D244—41no—
210C4×D61—42no—
211C4×D61—42no—
212Dic122—22no—
213Dic122—41no—
214#92418—42no—
215C2×#2199—611nomaximal, Sylow
216C22×C61—22no—
217C22×C61—22no—
218C22×C61—22no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—44no—
222C22×C61—44no—
223C22×C61—44no—
224C22×C61—48no—
225C22×C61—44no—
226C22×C61—82no—
227C22×C61—84no—
228C22×C61—84no—
229C22×C61—88no—
230C22×C61—84no—
231C22×C61B(4,𝔽2)11yescharacteristic
232C22×C61B(4,𝔽2)11yescharacteristic
233C22×C61C2×#12011yescharacteristic
234C2×D61—42no—
235C2×D61—82no—
236C2×D61—84no—
237C2×D61—162no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—44no—
241C4×C61—42no—
242C4×C61—42no—
243Dic61—42no—
244Dic61—84no—
245C2×H(3,𝔽4)D6111yescharacteristic, complemented
246C2×B(4,𝔽2)—612no—
247C2×#455—611no—
248#2199—618no—
249C2×#354—611no—
250C2×AΣL(1,𝔽16)—612no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61—42no—
254C2×C61—42no—
255C2×C61—84no—
256C2×C61#219912yescomplemented
257C2×C61C2×B(4,𝔽2)11yescharacteristic
258D61—82no—
259C2×H(5,𝔽2)—611no—
260C2×#137—1221no—
261H(3,𝔽4)—22no—
262H(3,𝔽4)—24no—
263H(3,𝔽4)C2×D6114yescomplemented
264C2×#124—21no—
265C2×#124—22no—
266C2×#124—21no—
267C2×#124C2×D6111yescharacteristic
268C2×#120—22no—
269C2×#120—1222no—
270C2×#120C2×D6112yes—
271B(4,𝔽2)—618no—
272B(4,𝔽2)—1224no—
273#455—614no—
274#455—1222no—
275C2×#104—1221no—
276C2×#100—611no—
277C2×#99—612no—
278C2×#99—1222no—
279#354—614no—
280AΣL(1,𝔽16)—618no—
281C61C2×#219911yescharacteristic, complemented, Sylow
282C25C22⋊D6112yescomplemented
283H(5,𝔽2)—612no—
284H(5,𝔽2)—1222no—
285H(5,𝔽2)—1221no—
286H(5,𝔽2)—2442no—
287C2×#41—1221no—
288C2×#41—2441no—
289C22×Q8—21no—
290C22×Q8C22⋊D6111yescharacteristic, complemented
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—22no—
294C22×D4—1222no—
295C22×D4—2441no—
296C22×D4—2442no—
297C22×D4C22×D6111yescharacteristic
298#137—1224no—
299#137—2442no—
300C2×Q16—2441no—
301C2×SD16—2441no—
302C2×M16—1221no—
303#124—28no—
304#124—24no—
305#124—24no—
306#124—24no—
307#124—22no—
308#124—28no—
309#124—24no—
310#124C22⋊D6112yescomplemented
311#124C22×D6112yes—
312#120—28no—
313#120—28no—
314#120—22no—
315#120—22no—
316#120—1228no—
317#120—2444no—
318#120C22⋊D6114yescomplemented
319#120C22×D6114yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—2442no—
326C2×C22⋊C4C22⋊D6112yescomplemented
327C2×C42—21no—
328C2×C42C22⋊D6111yescharacteristic, complemented
329#104—1224no—
330#100—612no—
331#100—1221no—
332#99—614no—
333#99—1222no—
334#99—1228no—
335C24—22no—
336C24—22no—
337C24—42no—
338C24—42no—
339C24—84no—
340C24—1222no—
341C24C2×C22⋊D6112yes—
342C24C2×C22⋊D6114yescomplemented
343#41—1222no—
344#41—1222no—
345#41—2441no—
346#41—2442no—
347#41—2444no—
348#41—2442no—
349#41—2441no—
350C2×Q8—24no—
351C2×Q8—21no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—82no—
355C2×Q8—1221no—
356C2×Q8C2×C22⋊D6112yescomplemented
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—84no—
364C2×D4—82no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4—1221no—
368C2×D4—1224no—
369C2×D4—1224no—
370C2×D4—4888no—
371C2×D4—2442no—
372C2×D4—2442no—
373C2×D4—2444no—
374C2×D4—2448no—
375C2×D4—2442no—
376C2×D4—2442no—
377C2×D4—2442no—
378C2×D4—2441no—
379C2×D4C2×C22⋊D6111yescharacteristic
380C2×D4C2×C22⋊D6112yes—
381C2×D4C23×D6111yescharacteristic
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—1222no—
386C22×C4—2441no—
387C22×C4—2442no—
388C22×C4C2×C22⋊D6111yescharacteristic
389Q16—2444no—
390SD16—2444no—
391M16—1222no—
392M16—2441no—
393C2×C8—2441no—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—1224no—
402C22⋊C4—2448no—
403C22⋊C4—2442no—
404C22⋊C4C2×C22⋊D6114yescomplemented
405C42—24no—
406C42—21no—
407C42C2×C22⋊D6112yescomplemented
408C23—22no—
409C23—22no—
410C23—24no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—48no—
416C23—42no—
417C23—42no—
418C23—82no—
419C23—84no—
420C23—84no—
421C23—84no—
422C23—88no—
423C23—1224no—
424C23—4884no—
425C23—2442no—
426C23—2441no—
427C23#1451974612yes—
428C23#1451974611yescharacteristic
429C23C22×C22⋊D6112yes—
430Q8—42no—
431Q8—81no—
432Q8—84no—
433Q8—1222no—
434Q8—2441no—
435D4—41no—
436D4—42no—
437D4—41no—
438D4—84no—
439D4—84no—
440D4—84no—
441D4—84no—
442D4—1222no—
443D4—48816no—
444D4—2441no—
445D4—2444no—
446D4—2444no—
447D4—2442no—
448D4—2442no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—1224no—
459C2×C4—4884no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2444no—
463C2×C4—2442no—
464C2×C4—2441no—
465C2×C4#1451974611yescharacteristic
466C2×C4C22×C22⋊D6111yescharacteristic
467C8—2442no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—48no—
477C22—44no—
478C22—82no—
479C22—84no—
480C22—84no—
481C22—88no—
482C22—84no—
483C22—9762no—
484C22—4884no—
485C22—4882no—
486C22—2442no—
487C22#1683994911yescharacteristic
488C22#1683994911yescharacteristic
489C22C2×#1451974611yescharacteristic
490C4—21no—
491C4—21no—
492C4—44no—
493C4—42no—
494C4—42no—
495C4—4884no—
496C4—2442no—
497C2—21no—
498C2—21no—
499C2—42no—
500C2—42no—
501C2—84no—
502C2—4882no—
503C2#2144782012yesdirect product factor
504C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×D4×C61C24
FrattiniC2×D4C23×D61
FittingC2×C61×H(3,𝔽4)C2
socleC22×C61B(4,𝔽2)
hypercenterC2×H(3,𝔽4)D61
hypocenterC61C2×#2199
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C2×#16839948C2semidirect
2. C2×#16839939C2semidirect
3. #21447820C2direct (N × H)
4. C2×#16839557C2semidirect
5. C2×#16839549C2semidirect
6. C2×C61×H(3,𝔽4)C2semidirect
11. #16839948C22semidirect
13. #16839939C22semidirect
18. #16839557C22semidirect
19. #16839549C22semidirect
22. C61×H(3,𝔽4)C22semidirect
53. C25×C61D4semidirect
55. C22×Q8×C61D4semidirect
67. #124×C61D4semidirect
73. #120×C61D4semidirect
79. C2×C22⋊C4×C61D4semidirect
81. C2×C42×C61D4semidirect
88. C24×C61C2×D4semidirect
95. C2×Q8×C61C2×D4semidirect
143. C22⋊C4×C61C2×D4semidirect
146. C42×C61C2×D4semidirect
245. C2×H(3,𝔽4)D61semidirect
256. C2×C61#2199semidirect
263. H(3,𝔽4)C2×D61semidirect
281. C61C2×#2199semidirect
282. C25C22⋊D61semidirect
290. C22×Q8C22⋊D61semidirect
310. #124C22⋊D61semidirect
318. #120C22⋊D61semidirect
326. C2×C22⋊C4C22⋊D61semidirect
328. C2×C42C22⋊D61semidirect
342. C24C2×C22⋊D61semidirect
356. C2×Q8C2×C22⋊D61semidirect
404. C22⋊C4C2×C22⋊D61semidirect
407. C42C2×C22⋊D61semidirect
503. C2#21447820direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168399482
C2×#168399392
#214478202
C2×#168395572
C2×#168395492
C2×C61×H(3,𝔽4)2
C2×#219961