Finite Group Databasebeta

← C4⋊C4×#1204 #14327246

Subgroups — C4⋊C4×#1204 #14327246

Order 1920.

1136 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1204C212yesmaximal
2C2×C4×#1204C211yesmaximal, characteristic
3C4⋊C4×Dic15C211yesmaximal, characteristic
4C5×Dic3×C4⋊C4C211yesmaximal, characteristic
5C3×C4⋊C4×Dic5C211yesmaximal, characteristic
6#11748866C211yesmaximal, characteristic
7#11748861C211yesmaximal, characteristic
8#11748834C211yesmaximal, characteristic
9#11748804C212yesmaximal
10#11748803C212yesmaximal
11#11748802C212yesmaximal
12C4⋊C4×Dic10—31nomaximal, Hall
13C22×#1204C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic15C2211yescharacteristic
16C2×C4×Dic15C2212yes—
17C2×C4×Dic15C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C4×C5×Dic3C2212yes—
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic5C2211yescharacteristic
22C2×C3×C4×Dic5C2212yes—
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×Dic3⋊Dic5C2212yes—
25C2×Dic3⋊Dic5C2212yes—
26C2×Dic3⋊Dic5C2211yescharacteristic
27C2×#6669C2212yes—
28C2×#6669C2212yes—
29C2×#6669C2211yescharacteristic
30C2×#6668C2212yes—
31C2×#6668C2212yes—
32C2×#6668C2211yescharacteristic
33C4×#1204—28no—
34C4×#1204C418yescomplemented
35#91115C2212yes—
36C5×C3⋊#95C2212yes—
37C3×#3963C2212yes—
38C4⋊C4×Dic6—51nomaximal, Hall
39C2×C4×Dic10—31no—
40C2×C4×Dic10—32no—
41C4×C5×C4⋊C4—31no—
42#65265—31no—
43#65265—31no—
44C4⋊C4×Dic5—31no—
45C4⋊C4×Dic5D311yescharacteristic
46#65224—31no—
47#65217—32no—
48#64936—32no—
49#64936—32no—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×Dic15C2312yes—
53C22×Dic15C2311yescharacteristic
54C22×Dic15C2311yescharacteristic
55C22×C5×Dic3C2312yes—
56C22×C5×Dic3C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C22×C3×Dic5C2312yes—
59C22×C3×Dic5C2311yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C2×#1204C2×C414yes—
62C2×#1204D414yes—
63C2×#1204Q814yes—
64C3×C5×C4⋊C4Q814yescomplemented
65C4⋊Dic15—22no—
66C4×Dic15—24no—
67C4×Dic15C2×C414yes—
68C5×C4⋊Dic3—22no—
69C4×C5×Dic3—24no—
70C4×C5×Dic3C2×C414yes—
71C3×C4⋊Dic5—22no—
72C3×C4×Dic5—24no—
73C3×C4×Dic5C2×C414yes—
74Dic3⋊Dic5—24no—
75Dic3⋊Dic5—42no—
76Dic3⋊Dic5C2×C414yes—
77#6669—24no—
78#6669—42no—
79#6669C2×C414yes—
80#6668—24no—
81#6668—42no—
82#6668C2×C414yes—
83C2×C4×Dic6—51no—
84C2×C4×Dic6—52no—
85C3×C4×C4⋊C4—51no—
86#4975—51no—
87#4975—51no—
88Dic3×C4⋊C4—51no—
89Dic3×C4⋊C4D511yescharacteristic
90#4935—51no—
91#4928—52no—
92#4647—52no—
93#4647—52no—
94C22×Dic10—31no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C4⋊C4C2×D311yescharacteristic
97C2×C42×C5—31no—
98C2×C42×C5—32no—
99C2×C4⋊Dic5—31no—
100C2×C4⋊Dic5—31no—
101C2×C4⋊Dic5—32no—
102C2×C4⋊Dic5—32no—
103C2×C4⋊Dic5C2×D311yescharacteristic
104C2×#683—31no—
105C2×#683—31no—
106C2×#683—32no—
107C2×#683—32no—
108C2×#683—32no—
109C2×#683—32no—
110C2×C4×Dic5—31no—
111C2×C4×Dic5—32no—
112C2×C4×Dic5C2×D312yes—
113C2×C4×Dic5C2×D311yescharacteristic
114C4×Dic10—38no—
115C4×Dic10—68no—
116C5×#95—32no—
117#3963—32no—
118#3963C2×D312yes—
119Q8×C4⋊C4—151noSylow
120C23×C3×C5C2411yescharacteristic
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5C22×C412yes—
124C2×C3×C4×C5C2×Q814yes—
125C2×C3×C4×C5C2×Q812yes—
126C2×C3×C4×C5#4112yes—
127C2×Dic15—21no—
128C2×Dic15—21no—
129C2×Dic15—22no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15C22×C412yes—
133C2×Dic15C22×C412yes—
134C2×Dic15C2×D412yes—
135C2×Dic15C2×Q812yes—
136C2×C5×Dic3—21no—
137C2×C5×Dic3—21no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3C22×C412yes—
142C2×C5×Dic3C22×C412yes—
143C2×C5×Dic3C2×D412yes—
144C2×C5×Dic3C2×Q812yes—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5C22×C412yes—
151C2×C3×Dic5C22×C412yes—
152C2×C3×Dic5C2×D412yes—
153C2×C3×Dic5C2×Q812yes—
154#1204C4⋊C4116yesdirect product factor
155C22×Dic6—51no—
156C2×C3×C4⋊C4—51no—
157C2×C3×C4⋊C4C2×D511yescharacteristic
158C2×C3×C42—51no—
159C2×C3×C42—52no—
160C2×C4⋊Dic3—51no—
161C2×C4⋊Dic3—51no—
162C2×C4⋊Dic3—52no—
163C2×C4⋊Dic3—52no—
164C2×C4⋊Dic3C2×D511yescharacteristic
165C2×Dic3⋊C4—51no—
166C2×Dic3⋊C4—51no—
167C2×Dic3⋊C4—52no—
168C2×Dic3⋊C4—52no—
169C2×Dic3⋊C4—52no—
170C2×Dic3⋊C4—52no—
171C2×C4×Dic3—51no—
172C2×C4×Dic3—52no—
173C2×C4×Dic3C2×D512yes—
174C2×C4×Dic3C2×D511yescharacteristic
175C4×Dic6—58no—
176C4×Dic6—108no—
177C3×#95—52no—
178C3⋊#95—52no—
179C3⋊#95C2×D512yes—
180C22×C4×C5—31no—
181C22×C4×C5—31no—
182C22×C4×C5—32no—
183C22×C4×C5C22×D312yes—
184C22×C4×C5C22×D311yescharacteristic
185C22×Dic5—31no—
186C22×Dic5—31no—
187C22×Dic5—32no—
188C22×Dic5C22×D312yes—
189C22×Dic5C22×D311yescharacteristic
190C22×Dic5C22×D311yescharacteristic
191C2×Dic10—34no—
192C2×Dic10—34no—
193C2×Dic10—34no—
194C5×C4⋊C4—62no—
195C5×C4⋊C4Dic614yescomplemented
196C42×C5—34no—
197C42×C5—64no—
198C4⋊Dic5—22no—
199C4⋊Dic5—34no—
200C4⋊Dic5—64no—
201C4⋊Dic5—62no—
202C4⋊Dic5—122no—
203#683—34no—
204#683—34no—
205#683—64no—
206#683—64no—
207#683—122no—
208#683—122no—
209C4×Dic5—24no—
210C4×Dic5—34no—
211C4×Dic5—64no—
212C4×Dic5C4×D314yes—
213C2×C4×Q8—151no—
214C2×C4×Q8—152no—
215#389—152no—
216#389—152no—
217#389—152no—
218#384—151no—
219#384—151no—
220#384—151no—
221C4×C4⋊C4—31no—
222C4×C4⋊C4—51no—
223C4×C4⋊C4—151no—
224C22×C3×C5C23×C411yescharacteristic
225C22×C3×C5C22×D411yescharacteristic
226C22×C3×C5C22×Q811yescharacteristic
227C22×C3×C5C22×Q811yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)11yescharacteristic
230C22×C3×C52−1+411yescharacteristic
231C3×C4×C5—24no—
232C3×C4×C5C4×Q814yescomplemented
233Dic15—22no—
234Dic15—42no—
235Dic15C2×C4⋊C414yes—
236C3×Dic5—22no—
237C3×Dic5—42no—
238C3×Dic5C2×C4⋊C414yes—
239C5×Dic3—22no—
240C5×Dic3—42no—
241C5×Dic3C2×C4⋊C414yes—
242C22×C3×C4—51no—
243C22×C3×C4—51no—
244C22×C3×C4—52no—
245C22×C3×C4C22×D512yes—
246C22×C3×C4C22×D511yescharacteristic
247C22×Dic3—51no—
248C22×Dic3—51no—
249C22×Dic3—52no—
250C22×Dic3C22×D512yes—
251C22×Dic3C22×D511yescharacteristic
252C22×Dic3C22×D511yescharacteristic
253C2×Dic6—54no—
254C2×Dic6—54no—
255C2×Dic6—54no—
256C3×C4⋊C4—102no—
257C3×C4⋊C4Dic1014yescomplemented
258C3×C42—54no—
259C3×C42—104no—
260C4⋊Dic3—22no—
261C4⋊Dic3—54no—
262C4⋊Dic3—104no—
263C4⋊Dic3—102no—
264C4⋊Dic3—202no—
265Dic3⋊C4—54no—
266Dic3⋊C4—54no—
267Dic3⋊C4—104no—
268Dic3⋊C4—104no—
269Dic3⋊C4—202no—
270Dic3⋊C4—202no—
271C4×Dic3—24no—
272C4×Dic3—54no—
273C4×Dic3—104no—
274C4×Dic3C4×D514yes—
275C23×C5C23×D311yescharacteristic
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—32no—
279C2×C4×C5—32no—
280C2×C4×C5—32no—
281C2×C4×C5—32no—
282C2×C4×C5—61no—
283C2×C4×C5—61no—
284C2×C4×C5—62no—
285C2×C4×C5—62no—
286C2×C4×C5—62no—
287C2×C4×C5C2×Dic614yes—
288C2×C4×C5C2×Dic612yes—
289C2×C4×C5C2×C4×D312yes—
290C2×C4×C5#23512yes—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—32no—
297C2×Dic5—32no—
298C2×Dic5—32no—
299C2×Dic5—32no—
300C2×Dic5—61no—
301C2×Dic5—61no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5—62no—
305C2×Dic5C2×C4×D312yes—
306C2×Dic5C2×C4×D312yes—
307C2×Dic5D3×D412yes—
308C2×Dic5D3×Q812yes—
309Dic10—316no—
310C22×Q8—151no—
311C4×Q8—158no—
312C4×Q8—308no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—51no—
315C2×C4⋊C4—151no—
316C2×C4⋊C4—151no—
317C2×C4⋊C4—151no—
318C2×C4⋊C4—152no—
319C2×C4⋊C4—152no—
320C2×C4⋊C4—152no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4—152no—
323C2×C4⋊C4—152no—
324C2×C4⋊C4—152no—
325C2×C4⋊C4D3×D511yescharacteristic
326C2×C42—31no—
327C2×C42—32no—
328C2×C42—51no—
329C2×C42—52no—
330C2×C42—151no—
331C2×C42—152no—
332#95—32no—
333#95—52no—
334#95—152no—
335C2×C3×C5C22×C4⋊C411yescharacteristic
336C2×C3×C5C2×C4×Q811yescharacteristic
337C2×C3×C5#52011yescharacteristic
338C2×C3×C5D4×Q811yescharacteristic
339C2×C3×C5#55011yescharacteristic
340C2×C3×C5#55711yescharacteristic
341C2×C3×C5Q8211yescharacteristic
342C23×C3C23×D511yescharacteristic
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—52no—
346C2×C3×C4—52no—
347C2×C3×C4—52no—
348C2×C3×C4—52no—
349C2×C3×C4—101no—
350C2×C3×C4—101no—
351C2×C3×C4—102no—
352C2×C3×C4—102no—
353C2×C3×C4—102no—
354C2×C3×C4C2×Dic1014yes—
355C2×C3×C4C2×Dic1012yes—
356C2×C3×C4C2×C4×D512yes—
357C2×C3×C4#70912yes—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—52no—
364C2×Dic3—52no—
365C2×Dic3—52no—
366C2×Dic3—52no—
367C2×Dic3—101no—
368C2×Dic3—101no—
369C2×Dic3—102no—
370C2×Dic3—102no—
371C2×Dic3—102no—
372C2×Dic3C2×C4×D512yes—
373C2×Dic3C2×C4×D512yes—
374C2×Dic3D4×D512yes—
375C2×Dic3Q8×D512yes—
376Dic6—516no—
377C22×C5C22×Dic611yescharacteristic
378C22×C5C22×C4×D311yescharacteristic
379C22×C5C2×#23511yescharacteristic
380C22×C5C2×D3×D411yescharacteristic
381C22×C5C2×D3×Q811yescharacteristic
382C22×C5#100911yescharacteristic
383C22×C5#101011yescharacteristic
384C4×C5—24no—
385C4×C5—34no—
386C4×C5—62no—
387C4×C5—122no—
388C4×C5C4×Dic614yescomplemented
389Dic5—22no—
390Dic5—34no—
391Dic5—42no—
392Dic5—62no—
393Dic5—122no—
394Dic5D3×C4⋊C414yes—
395C2×Q8—154no—
396C2×Q8—154no—
397C2×Q8—154no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—32no—
401C22×C4—51no—
402C22×C4—51no—
403C22×C4—52no—
404C22×C4—151no—
405C22×C4—151no—
406C22×C4—152no—
407C22×C4C2×D3×D512yes—
408C22×C4C2×D3×D511yescharacteristic
409C4⋊C4—62no—
410C4⋊C4—102no—
411C4⋊C4—154no—
412C4⋊C4—154no—
413C4⋊C4—154no—
414C4⋊C4—304no—
415C4⋊C4—304no—
416C4⋊C4—304no—
417C4⋊C4—302no—
418C4⋊C4—602no—
419C4⋊C4—602no—
420C4⋊C4—602no—
421C4⋊C4#120414yesdirect product factor
422C42—34no—
423C42—54no—
424C42—64no—
425C42—104no—
426C42—154no—
427C42—304no—
428C3×C5Q8×C4⋊C411yescharacteristic, complemented, Hall
429C22×C3C22×Dic1011yescharacteristic
430C22×C3C22×C4×D511yescharacteristic
431C22×C3C2×#70911yescharacteristic
432C22×C3C2×D4×D511yescharacteristic
433C22×C3C2×Q8×D511yescharacteristic
434C22×C3#414911yescharacteristic
435C22×C3#415011yescharacteristic
436C3×C4—24no—
437C3×C4—54no—
438C3×C4—102no—
439C3×C4—202no—
440C3×C4C4×Dic1014yescomplemented
441Dic3—22no—
442Dic3—42no—
443Dic3—54no—
444Dic3—102no—
445Dic3—202no—
446Dic3D5×C4⋊C414yes—
447C2×C5C2×C4×Dic611yescharacteristic
448C2×C5C2×D3×C4⋊C411yescharacteristic
449C2×C5#552311yescharacteristic
450C2×C5D4×Dic611yescharacteristic
451C2×C5#555311yescharacteristic
452C2×C5Q8×Dic611yescharacteristic
453C2×C5#556711yescharacteristic
454C23C22×D3×D511yescharacteristic
455Q8—1516no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—32no—
462C2×C4—52no—
463C2×C4—52no—
464C2×C4—52no—
465C2×C4—52no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—101no—
472C2×C4—101no—
473C2×C4—102no—
474C2×C4—102no—
475C2×C4—102no—
476C2×C4—152no—
477C2×C4—152no—
478C2×C4—152no—
479C2×C4—152no—
480C2×C4—301no—
481C2×C4—301no—
482C2×C4—302no—
483C2×C4—302no—
484C2×C4—302no—
485C2×C4#677112yes—
486C2×C4C4×D3×D512yes—
487C2×C4C2×#120414yes—
488C2×C4C2×#120412yes—
489C2×C3C2×C4×Dic1011yescharacteristic
490C2×C3C2×D5×C4⋊C411yescharacteristic
491C2×C3#6585711yescharacteristic
492C2×C3D4×Dic1011yescharacteristic
493C2×C3#6588711yescharacteristic
494C2×C3Q8×Dic1011yescharacteristic
495C2×C3#6590111yescharacteristic
496C5C4⋊C4×Dic611yescharacteristic, complemented, Sylow
497C22C2×#677111yescharacteristic
498C22C2×C4×D3×D511yescharacteristic
499C22#9202011yescharacteristic
500C22D3×D4×D511yescharacteristic
501C22D3×Q8×D511yescharacteristic
502C22#9203511yescharacteristic
503C22C22×#120411yescharacteristic
504C4—24no—
505C4—34no—
506C4—54no—
507C4—62no—
508C4—102no—
509C4—122no—
510C4—154no—
511C4—202no—
512C4—302no—
513C4—602no—
514C4C4×#120414yescomplemented
515C3C4⋊C4×Dic1011yescharacteristic, complemented, Sylow
516C2#1175367011yescharacteristic
517C2#1175368711yescharacteristic
518C2#1175382511yescharacteristic
519C2D3×D5×C4⋊C411yescharacteristic
520C2C2×C4×#120411yescharacteristic
521C2D4×#120411yescharacteristic
522C2Q8×#120411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, prime radical, cosocleC23×C3×C5C24
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5Q8×C4⋊C4
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
34. C4×#1204C4semidirect
64. C3×C5×C4⋊C4Q8semidirect
154. #1204C4⋊C4direct (N × H)
195. C5×C4⋊C4Dic6semidirect
232. C3×C4×C5C4×Q8semidirect
257. C3×C4⋊C4Dic10semidirect
388. C4×C5C4×Dic6semidirect
421. C4⋊C4#1204direct (N × H)
428. C3×C5Q8×C4⋊C4semidirect
440. C3×C4C4×Dic10semidirect
496. C5C4⋊C4×Dic6semidirect
514. C4C4×#1204semidirect
515. C3C4⋊C4×Dic10semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12042
C4⋊C4×Dic152
C5×Dic3×C4⋊C42
C3×C4⋊C4×Dic52
#117488662
#117488612
#117488342
#117488042
#117488032
#117488022
C4⋊C4×Dic103
C4⋊C4×Dic65