Finite Group Databasebeta

← C2×#21447832 #29834056

Subgroups — C2×#21447832 #29834056

Order 15616.

1136 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1136 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×#16839951C212yesmaximal
2C2×#16839939C211yesmaximal, characteristic
3#21447832C218yesmaximal, direct product factor
4C2×#16839557C211yesmaximal, characteristic, complemented
5C2×#16839552C212yesmaximal, complemented
6C2×C61×#563C211yesmaximal, characteristic, complemented
7C2×#14519822C2211yescharacteristic
8C2×#14519808—21no—
9C2×#14519760—22no—
10#16839951—24no—
11#16839951C2218yes—
12#16839939—22no—
13#16839939C2214yes—
14C2×#14519705—21no—
15C2×#14519694C2211yescharacteristic
16C2×#14519692C2212yes—
17C2×#14519691—22no—
18#16839557C2214yescomplemented
19#16839552C2218yescomplemented
20C61×#563—22no—
21C61×#563—24no—
22C61×#563C2214yescomplemented
23C2×C4⋊Q8×C61—21no—
24C2×#126×C61—22no—
25C2×#125×C61—21no—
26C2×#124×C61C2211yescharacteristic
27C2×#123×C61C2212yes—
28C2×C22⋊Q8×C61—22no—
29#14519822—21no—
30#14519822—22no—
31#14519822—42no—
32#14519822C2312yes—
33C2×#11758222—21no—
34C2×#11758224—41no—
35C22×C22⋊D61—22no—
36C2×D4×D61—41no—
37#14519808—24no—
38#14519808—42no—
39C2×#11758202—41no—
40C2×#11758201—41no—
41C2×#11758194—21no—
42#14519760—28no—
43#14519760—44no—
44C2×#11758196—42no—
45C2×C22⋊Dic61—22no—
46C2×#11758198—42no—
47#14519705—24no—
48#14519694—21no—
49#14519694C2312yes—
50#14519692—22no—
51#14519692C2314yes—
52#14519691—28no—
53C22×Q8×C61—21no—
54C22×Q8×C61D411yescharacteristic
55C22×D4×C61C2311yescharacteristic
56C23×C4×C61D412yes—
57C4⋊Q8×C61—24no—
58C4⋊Q8×C61—24no—
59#126×C61—28no—
60#126×C61—28no—
61#125×C61—24no—
62#125×C61—24no—
63#124×C61—22no—
64#124×C61D412yes—
65#124×C61C2312yes—
66#123×C61—22no—
67#123×C61—22no—
68#123×C61D414yes—
69#123×C61C2314yes—
70C22⋊Q8×C61—28no—
71C22⋊Q8×C61—28no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61D412yes—
80C2×C42×C61—21no—
81C2×C42×C61D411yescharacteristic
82C24×C61C2×D412yes—
83C23×D61—22no—
84C2×Q8×C61—24no—
85C2×Q8×C61—21no—
86C2×Q8×C61—41no—
87C2×Q8×C61—41no—
88C2×Q8×C61—81no—
89C2×Q8×C61—81no—
90C2×Q8×C61C2×D412yes—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61C2×D412yes—
94C2×D4×C61C2×D411yescharacteristic
95C2×D4×C61C2411yescharacteristic
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—42no—
102C22×C4×C61—42no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D414yes—
105C2×C22⋊D61—24no—
106C2×C22⋊D61—24no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—42no—
109C2×C22⋊D61—41no—
110C2×C22⋊D61—42no—
111C22×Dic61—22no—
112C22×Dic61—42no—
113#11758224—42no—
114#11758224—42no—
115#11758224—44no—
116D4×D61—42no—
117D4×D61—42no—
118D4×D61—44no—
119#11758222—22no—
120#11758222—22no—
121#11758222—41no—
122#11758222—41no—
123C2×D244—21no—
124C2×C4×D61—41no—
125C2×Dic122—21no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61—28no—
128C4⋊C4×C61—28no—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—24no—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61—24no—
137C22⋊C4×C61C2×D414yes—
138C42×C61—24no—
139C42×C61—21no—
140C42×C61C2×D412yes—
141C22⋊Dic61—24no—
142C22⋊Dic61—42no—
143#11758202—44no—
144#11758201—44no—
145#11758198—48no—
146#11758196—48no—
147#11758194—22no—
148#11758194—41no—
149C2×#92418—41no—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61—44no—
153C23×C61#12011yescharacteristic
154C23×C61#12012yes—
155C23×C61C22×D412yes—
156C22×D61—24no—
157C22×D61—41no—
158C22×D61—42no—
159C22×D61—84no—
160Q8×C61—42no—
161Q8×C61—81no—
162Q8×C61—84no—
163D4×C61—41no—
164D4×C61—42no—
165D4×C61—41no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—24no—
170C2×C4×C61—22no—
171C2×C4×C61—24no—
172C2×C4×C61—44no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—44no—
179C2×C4×C61—82no—
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—44no—
183C22⋊D61—44no—
184C22⋊D61—42no—
185C22⋊D61—42no—
186C2×Dic61—24no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—44no—
190C2×Dic61—84no—
191D244—22no—
192D244—41no—
193C4×D61—42no—
194C4×D61—42no—
195Dic122—22no—
196Dic122—41no—
197#92418—42no—
198C2×#2201—611nomaximal, Sylow
199C22×C61—22no—
200C22×C61—22no—
201C22×C61—22no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—44no—
205C22×C61—82no—
206C22×C61B(4,𝔽2)11yescharacteristic
207C22×C61B(4,𝔽2)11yescharacteristic
208C22×C61C2×#12011yescharacteristic
209C2×D61—42no—
210C2×D61—82no—
211C2×D61—84no—
212C2×D61—162no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—44no—
216C4×C61—42no—
217C4×C61—44no—
218C4×C61—42no—
219Dic61—42no—
220Dic61—84no—
221C2×#563D6111yescharacteristic, complemented
222C2×#458—612no—
223C2×#455—611no—
224#2201—618no—
225C2×#354—611no—
226C2×#352—612no—
227C2×C61—21no—
228C2×C61—21no—
229C2×C61—42no—
230C2×C61—42no—
231C2×C61#220112yescomplemented
232C2×C61C2×B(4,𝔽2)11yescharacteristic
233D61—82no—
234C2×H(5,𝔽2)—611no—
235C2×#137—1221no—
236#563—22no—
237#563—24no—
238#563C2×D6114yescomplemented
239C2×C4⋊Q8—21no—
240C2×#126—22no—
241C2×#125—21no—
242C2×#124C2×D6111yescharacteristic
243C2×#123—1222no—
244C2×#123C2×D6112yes—
245C2×C22⋊Q8—22no—
246#458—618no—
247#458—1224no—
248#455—614no—
249#455—1222no—
250C2×#104—1221no—
251C2×#100—611no—
252C2×#99—612no—
253C2×#99—1222no—
254#354—614no—
255#352—618no—
256C61C2×#220111yescharacteristic, complemented, Sylow
257H(5,𝔽2)—612no—
258H(5,𝔽2)—1222no—
259H(5,𝔽2)—1221no—
260H(5,𝔽2)—2442no—
261C2×#41—1221no—
262C2×#41—2441no—
263C22×Q8—21no—
264C22×Q8C22⋊D6111yescharacteristic
265C22×D4—1222no—
266C22×D4—2441no—
267C22×D4C22×D6111yescharacteristic
268C23×C4C22⋊D6112yes—
269#137—1224no—
270#137—2442no—
271C2×Q16—2441no—
272C2×SD16—2441no—
273C2×M16—1221no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276#126—28no—
277#126—28no—
278#125—24no—
279#125—24no—
280#124—22no—
281#124C22⋊D6112yes—
282#124C22×D6112yes—
283#123—22no—
284#123—22no—
285#123—1228no—
286#123—2444no—
287#123C22⋊D6114yes—
288#123C22×D6114yes—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—2442no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—1222no—
300C2×C22⋊C4—2442no—
301C2×C22⋊C4C22⋊D6112yes—
302C2×C42—21no—
303C2×C42C22⋊D6111yescharacteristic
304#104—1224no—
305#100—612no—
306#100—1221no—
307#99—614no—
308#99—1222no—
309#99—1228no—
310C24—1222no—
311C24C2×C22⋊D6112yes—
312#41—1222no—
313#41—1222no—
314#41—2441no—
315#41—2442no—
316#41—2444no—
317#41—2442no—
318#41—2441no—
319C2×Q8—24no—
320C2×Q8—21no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—81no—
324C2×Q8—81no—
325C2×Q8—1221no—
326C2×Q8C2×C22⋊D6112yes—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—1221no—
330C2×D4—1224no—
331C2×D4—1224no—
332C2×D4—2442no—
333C2×D4—2442no—
334C2×D4—2444no—
335C2×D4—2442no—
336C2×D4—2442no—
337C2×D4—2442no—
338C2×D4—2441no—
339C2×D4C2×C22⋊D6111yescharacteristic
340C2×D4C2×C22⋊D6112yes—
341C2×D4C23×D6111yescharacteristic
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—1222no—
350C22×C4—2441no—
351C22×C4—2442no—
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C2×C22⋊D6114yes—
354Q16—2444no—
355SD16—2444no—
356M16—1222no—
357M16—2441no—
358C2×C8—2441no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—2448no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—1224no—
372C22⋊C4—2448no—
373C22⋊C4—2442no—
374C22⋊C4C2×C22⋊D6114yes—
375C42—24no—
376C42—21no—
377C42C2×C22⋊D6112yes—
378C23—22no—
379C23—22no—
380C23—44no—
381C23—1224no—
382C23—4884no—
383C23—2442no—
384C23—2441no—
385C23#1451974612yes—
386C23#1451974611yescharacteristic
387C23C22×C22⋊D6112yes—
388Q8—42no—
389Q8—81no—
390Q8—84no—
391Q8—1222no—
392Q8—2441no—
393D4—41no—
394D4—42no—
395D4—41no—
396D4—1222no—
397D4—2441no—
398D4—2444no—
399D4—2444no—
400D4—2442no—
401D4—2442no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—82no—
416C2×C4—1224no—
417C2×C4—4884no—
418C2×C4—2442no—
419C2×C4—2442no—
420C2×C4—2444no—
421C2×C4—2442no—
422C2×C4—2441no—
423C2×C4#1451974611yescharacteristic
424C2×C4C22×C22⋊D6111yescharacteristic
425C8—2442no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—42no—
430C22—42no—
431C22—44no—
432C22—82no—
433C22—9762no—
434C22—4884no—
435C22—4882no—
436C22—2442no—
437C22#1683994911yescharacteristic
438C22#1683994911yescharacteristic
439C22C2×#1451974611yescharacteristic
440C4—21no—
441C4—21no—
442C4—44no—
443C4—42no—
444C4—44no—
445C4—42no—
446C4—4884no—
447C4—2442no—
448C2—21no—
449C2—21no—
450C2—42no—
451C2—42no—
452C2—4882no—
453C2#2144783212yesdirect product factor
454C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×D4×C61C24
FrattiniC2×D4C23×D61
FittingC2×C61×#563C2
socleC22×C61B(4,𝔽2)
prime radicalC2×#14519822C22
hypercenterC2×#563D61
hypocenterC61C2×#2201
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
3. #21447832C2direct (N × H)
4. C2×#16839557C2semidirect
5. C2×#16839552C2semidirect
6. C2×C61×#563C2semidirect
18. #16839557C22semidirect
19. #16839552C22semidirect
22. C61×#563C22semidirect
221. C2×#563D61semidirect
231. C2×C61#2201semidirect
238. #563C2×D61semidirect
256. C61C2×#2201semidirect
453. C2#21447832direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168399512
C2×#168399392
#214478322
C2×#168395572
C2×#168395522
C2×C61×#5632
C2×#220161