Finite Group Databasebeta

← C41×#9119 #23785320

Subgroups — C41×#9119 #23785320

Order 10496.

696 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#97×C41C211yesmaximal, characteristic, complemented
2C41×#2850C211yesmaximal, characteristic
3C41×#1749C211yesmaximal, characteristic, complemented
4C2×C41×#329C212yesmaximal, complemented
5C41×#1295C212yesmaximal, complemented
6C23×C8×C41—21no—
7C2×#124×C41C2211yescharacteristic
8C2×C22⋊Q8×C41C411yescharacteristic
9C2×#121×C41C411yescharacteristic
10C22×C42×C41C2211yescharacteristic
11C2×C22⋊C8×C41—21no—
12C2×#97×C41—21no—
13C2×#97×C41—24no—
14C2×#97×C41C2212yes—
15C2×#97×C41C2212yes—
16C41×#391C411yescharacteristic
17C41×#390C411yescharacteristic
18C41×#388—21no—
19C41×#388—21no—
20C41×#386—21no—
21C41×#386—21no—
22C41×#382C2211yescharacteristic
23C41×#336—21no—
24C41×#329—24no—
25C22×Q8×C41C2×C411yescharacteristic
26C22×D4×C41—21no—
27C22×D4×C41C2×C411yescharacteristic
28C23×C4×C41C2×C411yescharacteristic
29C23×C4×C41D411yescharacteristic
30C23×C4×C41D411yescharacteristic
31C22×C8×C41—24no—
32C22×C8×C41—24no—
33C22×C8×C41—24no—
34C22×C8×C41—22no—
35C22×C8×C41—22no—
36#124×C41—22no—
37#124×C41—22no—
38C22⋊Q8×C41—42no—
39#121×C41—42no—
40C2×C4⋊C4×C41—21no—
41C2×C4⋊C4×C41C2×C411yescharacteristic
42C2×C4⋊C4×C41C2×C411yescharacteristic
43C2×C22⋊C4×C41—21no—
44C2×C22⋊C4×C41—21no—
45C2×C22⋊C4×C41—21no—
46C2×C22⋊C4×C41—21no—
47C2×C22⋊C4×C41—21no—
48C2×C22⋊C4×C41—21no—
49C2×C42×C41—22no—
50C2×C42×C41—22no—
51C2×C42×C41C2×C411yescharacteristic
52C2×C42×C41D411yescharacteristic
53C2×C42×C41D411yescharacteristic
54C2×C42×C41C2311yescharacteristic
55C22⋊C8×C41—24no—
56#97×C41—24no—
57#97×C41—22no—
58#97×C41—22no—
59#97×C41—22no—
60#97×C41D414yescomplemented
61#95×C41—21no—
62#95×C41—21no—
63#95×C41—21no—
64#95×C41—21no—
65C24×C41—21no—
66C24×C41C22⋊C411yescharacteristic
67C2×Q8×C41—22no—
68C2×Q8×C41—41no—
69C2×Q8×C41—41no—
70C2×D4×C41—22no—
71C2×D4×C41—41no—
72C2×D4×C41—41no—
73C2×D4×C41—42no—
74C2×D4×C41—42no—
75C2×D4×C41—42no—
76C22×C4×C41—22no—
77C22×C4×C41—22no—
78C22×C4×C41—21no—
79C22×C4×C41—21no—
80C22×C4×C41—22no—
81C22×C4×C41—22no—
82C22×C4×C41—22no—
83C22×C4×C41—22no—
84C22×C4×C41—21no—
85C22×C4×C41—21no—
86C22×C4×C41—21no—
87C22×C4×C41—41no—
88C22×C4×C41—41no—
89C22×C4×C41C22⋊C411yescharacteristic
90C22×C4×C41C22⋊C411yescharacteristic
91C22×C4×C41C22⋊C411yescharacteristic
92C22×C4×C41C22×C411yescharacteristic
93C22×C4×C41C2×D411yescharacteristic
94C22×C4×C41C2×D411yescharacteristic
95C2×C8×C41—216no—
96C2×C8×C41—24no—
97C2×C8×C41—24no—
98C2×C8×C41—24no—
99C2×C8×C41—24no—
100C2×C8×C41—42no—
101C2×C8×C41—42no—
102C4⋊C4×C41—41no—
103C4⋊C4×C41—41no—
104C4⋊C4×C41—42no—
105C22⋊C4×C41—42no—
106C22⋊C4×C41—42no—
107C22⋊C4×C41—42no—
108C22⋊C4×C41—42no—
109C22⋊C4×C41—41no—
110C22⋊C4×C41—42no—
111C22⋊C4×C41—41no—
112C42×C41—22no—
113C42×C41—21no—
114C42×C41—22no—
115C42×C41—21no—
116C42×C41C2×D412yes—
117C42×C41#4112yes—
118C23×C41—21no—
119C23×C41—21no—
120C23×C41—21no—
121C23×C41—21no—
122C23×C41—41no—
123C23×C41—41no—
124C23×C41—41no—
125C23×C41—41no—
126C23×C41—41no—
127C23×C41—41no—
128C23×C41—41no—
129C23×C41#10012yes—
130C23×C41#10412yes—
131C23×C41#10412yes—
132C23×C41C2×C22⋊C411yescharacteristic
133Q8×C41—82no—
134D4×C41—82no—
135D4×C41—84no—
136C2×C4×C41—24no—
137C2×C4×C41—24no—
138C2×C4×C41—21no—
139C2×C4×C41—21no—
140C2×C4×C41—21no—
141C2×C4×C41—21no—
142C2×C4×C41—21no—
143C2×C4×C41—21no—
144C2×C4×C41—21no—
145C2×C4×C41—21no—
146C2×C4×C41—21no—
147C2×C4×C41—21no—
148C2×C4×C41—21no—
149C2×C4×C41—21no—
150C2×C4×C41—21no—
151C2×C4×C41—21no—
152C2×C4×C41—44no—
153C2×C4×C41—44no—
154C2×C4×C41—41no—
155C2×C4×C41—41no—
156C2×C4×C41—41no—
157C2×C4×C41—41no—
158C2×C4×C41—41no—
159C2×C4×C41—41no—
160C2×C4×C41—41no—
161C2×C4×C41—41no—
162C2×C4×C41—41no—
163C2×C4×C41#10412yes—
164C2×C4×C41C4×D412yes—
165C2×C4×C41#12011yescharacteristic
166C2×C4×C41#12112yes—
167C2×C4×C41#12311yescharacteristic
168C8×C41—28no—
169C8×C41—44no—
170#9119C4111yesmaximal, characteristic, direct product factor, Sylow
171C22×C41—22no—
172C22×C41—22no—
173C22×C41—22no—
174C22×C41—22no—
175C22×C41—22no—
176C22×C41—22no—
177C22×C41—81no—
178C22×C41—81no—
179C22×C41—81no—
180C22×C41—81no—
181C22×C41—81no—
182C22×C41—81no—
183C22×C41—81no—
184C22×C41#32914yescomplemented
185C22×C41#38611yescharacteristic
186C22×C41C2×#10011yescharacteristic
187C22×C41#41311yescharacteristic
188C22×C41C2×#10411yescharacteristic
189C22×C41C2×#10411yescharacteristic
190C22×C41C2×#10411yescharacteristic
191C22×C41#42111yescharacteristic
192C4×C41—22no—
193C4×C41—21no—
194C4×C41—22no—
195C4×C41—21no—
196C4×C41—22no—
197C4×C41—41no—
198C4×C41—41no—
199C4×C41—81no—
200C4×C41—81no—
201C4×C41—81no—
202C22×#97C2×C4111yescharacteristic, complemented
203#2850C2×C4111yescharacteristic
204#1749C2×C4111yescharacteristic, complemented
205C2×#329C2×C4112yescomplemented
206#1295C2×C4112yescomplemented
207C2×C41—24no—
208C2×C41—161no—
209C2×C41C2×#32911yescharacteristic
210C2×C41#152411yescharacteristic
211C2×C41#153011yescharacteristic
212C2×C41#153211yescharacteristic
213C2×C41#187111yescharacteristic
214C2×C41#189611yescharacteristic
215C2×C41#190011yescharacteristic
216C23×C8—21no—
217C2×#124C22×C4111yescharacteristic
218C2×C22⋊Q8C4×C4111yescharacteristic
219C2×#121C4×C4111yescharacteristic
220C22×C42C22×C4111yescharacteristic
221C2×C22⋊C8—21no—
222C2×#97—21no—
223C2×#97—24no—
224C2×#97C22×C4112yes—
225C2×#97C22×C4112yes—
226#391C4×C4111yescharacteristic
227#390C4×C4111yescharacteristic
228#388—21no—
229#388—21no—
230#386—21no—
231#386—21no—
232#382C22×C4111yescharacteristic
233#336—21no—
234#329—24no—
235C41#911911yescharacteristic, direct product factor, Sylow
236C22×Q8C2×C4×C4111yescharacteristic
237C22×D4—21no—
238C22×D4C2×C4×C4111yescharacteristic
239C23×C4C2×C4×C4111yescharacteristic
240C23×C4D4×C4111yescharacteristic
241C23×C4D4×C4111yescharacteristic
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—22no—
246C22×C8—22no—
247#124—22no—
248#124—22no—
249C22⋊Q8—42no—
250#121—42no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C2×C4×C4111yescharacteristic
253C2×C4⋊C4C2×C4×C4111yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42C2×C4×C4111yescharacteristic
263C2×C42D4×C4111yescharacteristic
264C2×C42D4×C4111yescharacteristic
265C2×C42C23×C4111yescharacteristic
266C22⋊C8—24no—
267#97—24no—
268#97—22no—
269#97—22no—
270#97—22no—
271#97D4×C4114yescomplemented
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C24C22⋊C4×C4111yescharacteristic
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C22⋊C4×C4111yescharacteristic
301C22×C4C22⋊C4×C4111yescharacteristic
302C22×C4C22⋊C4×C4111yescharacteristic
303C22×C4C22×C4×C4111yescharacteristic
304C22×C4C2×D4×C4111yescharacteristic
305C22×C4C2×D4×C4111yescharacteristic
306C2×C8—216no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C42—22no—
324C42—21no—
325C42—22no—
326C42—21no—
327C42C2×D4×C4112yes—
328C42#41×C4112yes—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23#100×C4112yes—
341C23#104×C4112yes—
342C23#104×C4112yes—
343C23C2×C22⋊C4×C4111yescharacteristic
344Q8—82no—
345D4—82no—
346D4—84no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4#104×C4112yes—
375C2×C4C4×D4×C4112yes—
376C2×C4#120×C4111yescharacteristic
377C2×C4#121×C4112yes—
378C2×C4#123×C4111yescharacteristic
379C8—28no—
380C8—44no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22C41×#32914yescomplemented
395C22C41×#38611yescharacteristic
396C22C2×#100×C4111yescharacteristic
397C22C41×#41311yescharacteristic
398C22C2×#104×C4111yescharacteristic
399C22C2×#104×C4111yescharacteristic
400C22C2×#104×C4111yescharacteristic
401C22C41×#42111yescharacteristic
402C4—22no—
403C4—21no—
404C4—22no—
405C4—21no—
406C4—22no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—81no—
412C2—24no—
413C2—161no—
414C2C2×C41×#32911yescharacteristic
415C2C41×#152411yescharacteristic
416C2C41×#153011yescharacteristic
417C2C41×#153211yescharacteristic
418C2C41×#187111yescharacteristic
419C2C41×#189611yescharacteristic
420C2C41×#190011yescharacteristic
subgroupG/H
center, socleC23×C41C2×C22⋊C4
commutatorC22×C4C22×C4×C41
Frattini, cosocleC2×C42C23×C41
prime radicalC2×#121×C41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C41 ◁ C22×C42×C41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#97×C41C2semidirect
3. C41×#1749C2semidirect
4. C2×C41×#329C2semidirect
5. C41×#1295C2semidirect
60. #97×C41D4semidirect
170. #9119C41direct (N × H)
184. C22×C41#329semidirect
202. C22×#97C2×C41semidirect
204. #1749C2×C41semidirect
205. C2×#329C2×C41semidirect
206. #1295C2×C41semidirect
235. C41#9119direct (N × H)
271. #97D4×C41semidirect
394. C22C41×#329semidirect

Maximal subgroups

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

subgroupindex
C22×#97×C412
C41×#28502
C41×#17492
C2×C41×#3292
C41×#12952
#911941