Finite Group Databasebeta

← 3−1+2×GL(2,ℤ/4) #72276470

Subgroups — 3−1+2×GL(2,ℤ/4) #72276470

Order 2592.

606 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C22×A4×3−1+2C211yesmaximal, characteristic, complemented
2C2×S4×3−1+2C211yesmaximal, characteristic, complemented
33−1+2×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C32×GL(2,ℤ/4)C311yesmaximal, characteristic
5C9×GL(2,ℤ/4)C313yesmaximal, complemented
63−1+2×#120—31nomaximal
7C2×A4×3−1+2—21no—
8C2×A4×3−1+2C2211yescharacteristic
9S4×3−1+2—21no—
10#67×3−1+2—41nomaximal
11C22×C32×A4C2×C311yescharacteristic
12C2×C32×S4C2×C311yescharacteristic
13C32×SL(2,ℤ/4)C2×C311yescharacteristic
14C24×3−1+2D311yescharacteristic, complemented
15C22×#1094—21no—
16C22×C9⋊A4—23no—
17C22×C9×A4C2×C313yescomplemented
18C2×C9×S4C2×C313yescomplemented
19C2×D4×3−1+2—31no—
20C2×D4×3−1+2—31no—
21C2×D4×3−1+2—31no—
22C9×SL(2,ℤ/4)C2×C313yescomplemented
23C22⋊C4×3−1+2—31no—
24C22⋊C4×3−1+2—31no—
25C22⋊C4×3−1+2—31no—
26C22×C3×3−1+2—41no—
27C2×D3×3−1+2—41no—
28A4×3−1+2D411yescharacteristic, complemented
29Dic3×3−1+2—41no—
30C3×GL(2,ℤ/4)—31no—
31C3×GL(2,ℤ/4)C3211yescharacteristic
32C32×#120—31no—
33C9×#120—33no—
34C2×C32×A4—21no—
35C2×C32×A4C22×C311yescharacteristic
36C32×S4—21no—
37C32×#67—41no—
38C23×3−1+2—21no—
39C23×3−1+2—31no—
40C23×3−1+2—31no—
41C23×3−1+2—31no—
42C23×3−1+2—61no—
43C23×3−1+2C2×D311yescharacteristic
44C2×#1094—22no—
45C2×#1094—21no—
46C2×C9⋊A4—26no—
47C2×C9⋊A4—23no—
48C2×C9×A4—23no—
49C2×C9×A4C22×C313yes—
50C9×S4—23no—
51D4×3−1+2—61no—
52D4×3−1+2—62no—
53D4×3−1+2—61no—
54D4×3−1+2—61no—
55D4×3−1+2—61no—
56C2×C4×3−1+2—31no—
57C2×C4×3−1+2—31no—
58C2×C4×3−1+2—31no—
59C9×#67—43no—
60C2×C3×3−1+2—41no—
61C2×C3×3−1+2—81no—
62D3×3−1+2—81no—
63C24×C32C3×D311yescharacteristic
64C22×C3×A4—21no—
65C22×C3×A4—31no—
66C22×C3×A4—61no—
67C22×C3×A4C2×C3211yescharacteristic
68C2×C3×S4—31no—
69C2×C3×S4C2×C3211yescharacteristic
70C2×C32×D4—31no—
71C2×C32×D4—31no—
72C2×C32×D4—31no—
73C3×SL(2,ℤ/4)—31no—
74C3×SL(2,ℤ/4)C2×C3211yescharacteristic
75C24×C9C3×D313yescomplemented
76C22×C22⋊C9—23no—
77C32×C22⋊C4—31no—
78C32×C22⋊C4—31no—
79C32×C22⋊C4—31no—
80C2×D4×C9—33no—
81C2×D4×C9—33no—
82C2×D4×C9—33no—
83C9×C22⋊C4—33no—
84C9×C22⋊C4—33no—
85C9×C22⋊C4—33no—
86C22×C33—41no—
87C2×C32×D3—41no—
88C32×A4C3×D411yescharacteristic
89C32×Dic3—41no—
90C22×3−1+2—31no—
91C22×3−1+2—31no—
92C22×3−1+2—32no—
93C22×3−1+2—31no—
94C22×3−1+2—61no—
95C22×3−1+2—61no—
96C22×3−1+2—61no—
97C22×3−1+2—61no—
98C22×3−1+2—61no—
99C22×3−1+2—61no—
100C22×3−1+2—81no—
101C22×3−1+2—83no—
102C22×3−1+2#6711yescharacteristic, complemented
103C22×3−1+2S411yescharacteristic, complemented
104C22×C3×C9—43no—
105C2×D3×C9—43no—
106#1094—21no—
107C9⋊A4—23no—
108C9×A4C3×D413yescomplemented
109C4×3−1+2—61no—
110C4×3−1+2—61no—
111C4×3−1+2—61no—
112C9×Dic3—43no—
113GL(2,ℤ/4)3−1+211yescharacteristic, direct product factor
114C3×#120—31no—
115C3×#120—91no—
116C3×3−1+2—41noSylow
117C23×C32—21no—
118C23×C32—31no—
119C23×C32—31no—
120C23×C32—31no—
121C23×C32—61no—
122C23×C32C2×C3×D311yescharacteristic
123C2×C3×A4—21no—
124C2×C3×A4—22no—
125C2×C3×A4—21no—
126C2×C3×A4—31no—
127C2×C3×A4—61no—
128C2×C3×A4—62no—
129C2×C3×A4—61no—
130C2×C3×A4C22×C3211yescharacteristic
131C3×S4—21no—
132C3×S4—61no—
133C32×D4—61no—
134C32×D4—62no—
135C32×D4—61no—
136C32×D4—61no—
137C32×D4—61no—
138C2×C32×C4—31no—
139C2×C32×C4—31no—
140C2×C32×C4—31no—
141C3×#67—41no—
142C3×#67—121no—
143C23×C9—23no—
144C23×C9—33no—
145C23×C9—33no—
146C23×C9—33no—
147C23×C9—63no—
148C23×C9C2×C3×D313yes—
149C2×C22⋊C9—26no—
150C2×C22⋊C9—23no—
151D4×C9—63no—
152D4×C9—66no—
153D4×C9—63no—
154D4×C9—63no—
155D4×C9—63no—
156C2×C4×C9—33no—
157C2×C4×C9—33no—
158C2×C4×C9—33no—
159C2×C33—41no—
160C2×C33—81no—
161C32×D3—81no—
162C2×3−1+2—21no—
163C2×3−1+2—31no—
164C2×3−1+2—31no—
165C2×3−1+2—61no—
166C2×3−1+2—86no—
167C2×3−1+2—82no—
168C2×3−1+2—81no—
169C2×3−1+2—83no—
170C2×3−1+2—121no—
171C2×3−1+2C2×S411yescharacteristic
172C2×C3×C9—43no—
173C2×C3×C9—83no—
174D3×C9—83no—
175C24×C3—31no—
176C24×C3C32×D311yescharacteristic
177C22×A4—21no—
178C22×A4—61no—
179C22×A4C2×3−1+211yescharacteristic, complemented
180C2×S4C2×3−1+211yescharacteristic, complemented
181C2×C3×D4—31no—
182C2×C3×D4—31no—
183C2×C3×D4—31no—
184C2×C3×D4—91no—
185C2×C3×D4—91no—
186C2×C3×D4—91no—
187SL(2,ℤ/4)C2×3−1+211yescharacteristic, complemented
188C3×C22⋊C4—31no—
189C3×C22⋊C4—31no—
190C3×C22⋊C4—31no—
191C3×C22⋊C4—91no—
192C3×C22⋊C4—91no—
193C3×C22⋊C4—91no—
194C22×C32—31no—
195C22×C32—31no—
196C22×C32—32no—
197C22×C32—31no—
198C22×C32—41no—
199C22×C32—61no—
200C22×C32—61no—
201C22×C32—61no—
202C22×C32—61no—
203C22×C32—61no—
204C22×C32—61no—
205C22×C32—81no—
206C22×C32—121no—
207C22×C32—241no—
208C22×C32C3×#6711yescharacteristic
209C22×C32C3×S411yescharacteristic
210C2×C3×D3—41no—
211C2×C3×D3—121no—
212C3×A4—21no—
213C3×A4—31no—
214C3×A4—61no—
215C3×A4C32×D411yescharacteristic
216C32×C4—61no—
217C32×C4—61no—
218C32×C4—61no—
219C3×Dic3—41no—
220C3×Dic3—121no—
221C22×C9—33no—
222C22×C9—33no—
223C22×C9—36no—
224C22×C9—33no—
225C22×C9—63no—
226C22×C9—63no—
227C22×C9—63no—
228C22×C9—63no—
229C22×C9—63no—
230C22×C9—63no—
231C22×C9—83no—
232C22×C9C3×#6713yescomplemented
233C22×C9C3×S413yescomplemented
234C22⋊C9—23no—
235C4×C9—63no—
236C4×C9—63no—
237C4×C9—63no—
238#120—31noSylow
239C33—41no—
2403−1+2—81no—
2413−1+2—83no—
2423−1+2GL(2,ℤ/4)11yescharacteristic, direct product factor
243C3×C9—43no—
244C23×C3—21no—
245C23×C3—31no—
246C23×C3—31no—
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—91no—
252C23×C3—91no—
253C23×C3—91no—
254C23×C3—181no—
255C23×C3C2×C32×D311yescharacteristic
256C2×A4—22no—
257C2×A4—21no—
258C2×A4—21no—
259C2×A4—62no—
260C2×A4—61no—
261C2×A4C22×3−1+211yescharacteristic
262S4—21no—
263C3×D4—61no—
264C3×D4—62no—
265C3×D4—61no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—182no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—91no—
277C2×C3×C4—91no—
278C2×C3×C4—91no—
279#67—41no—
280C2×C32—21no—
281C2×C32—31no—
282C2×C32—31no—
283C2×C32—41no—
284C2×C32—61no—
285C2×C32—81no—
286C2×C32—82no—
287C2×C32—81no—
288C2×C32—121no—
289C2×C32—121no—
290C2×C32—241no—
291C2×C32—241no—
292C2×C32—242no—
293C2×C32C2×C3×S411yescharacteristic
294C3×D3—81no—
295C3×D3—241no—
296C2×C9—23no—
297C2×C9—33no—
298C2×C9—33no—
299C2×C9—63no—
300C2×C9—86no—
301C2×C9—83no—
302C2×C9—123no—
303C2×C9C2×C3×S413yes—
304C24D3×3−1+211yescharacteristic, complemented
305C2×D4—31no—
306C2×D4—31no—
307C2×D4—31no—
308C22⋊C4—31no—
309C22⋊C4—31no—
310C22⋊C4—31no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—32no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—31no—
317C22×C3—41no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3—61no—
321C22×C3—61no—
322C22×C3—61no—
323C22×C3—61no—
324C22×C3—81no—
325C22×C3—91no—
326C22×C3—91no—
327C22×C3—91no—
328C22×C3—92no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3—181no—
335C22×C3—241no—
336C22×C3C32×#6711yescharacteristic
337C22×C3C32×S411yescharacteristic
338C2×D3—41no—
339A4—21no—
340A4—61no—
341A4D4×3−1+211yescharacteristic, complemented
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—181no—
346C3×C4—181no—
347C3×C4—181no—
348Dic3—41no—
349C32—41no—
350C32—81no—
351C32—121no—
352C32—241no—
353C32C3×GL(2,ℤ/4)11yescharacteristic
354C9—83no—
355C9C3×GL(2,ℤ/4)13yescomplemented
356C23—21no—
357C23—31no—
358C23—31no—
359C23—31no—
360C23—61no—
361C23C2×D3×3−1+211yescharacteristic
362D4—61no—
363D4—62no—
364D4—61no—
365D4—61no—
366D4—61no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C3—21no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—41no—
375C2×C3—61no—
376C2×C3—61no—
377C2×C3—81no—
378C2×C3—82no—
379C2×C3—81no—
380C2×C3—91no—
381C2×C3—91no—
382C2×C3—121no—
383C2×C3—181no—
384C2×C3—242no—
385C2×C3—241no—
386C2×C3—361no—
387C2×C3C2×C32×S411yescharacteristic
388D3—81no—
389C22—31no—
390C22—31no—
391C22—32no—
392C22—31no—
393C22—61no—
394C22—61no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22—61no—
399C22#67×3−1+211yescharacteristic, complemented
400C22S4×3−1+211yescharacteristic, complemented
401C4—61no—
402C4—61no—
403C4—61no—
404C3—31no—
405C3—41no—
406C3—81no—
407C3—241no—
408C3C32×GL(2,ℤ/4)11yescharacteristic
409C2—21no—
410C2—31no—
411C2—31no—
412C2—61no—
413C2—121no—
414C2C2×S4×3−1+211yescharacteristic
subgroupG/H
center, FrattiniC2×C3C2×C32×S4
commutator, cosocleC2×C3×A4C22×C32
FittingC24×3−1+2D3
socleC23×C3C2×C32×D3
prime radicalC32×GL(2,ℤ/4)C3
hypercenterC22×3−1+2S4
hypocenterA4D4×3−1+2
LCSG ▷ C2×C3×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C22×3−1+2

Complements

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

NH ≅ G/Ntype
1. C22×A4×3−1+2C2semidirect
2. C2×S4×3−1+2C2semidirect
3. 3−1+2×SL(2,ℤ/4)C2semidirect
5. C9×GL(2,ℤ/4)C3semidirect
14. C24×3−1+2D3semidirect
17. C22×C9×A4C2×C3semidirect
18. C2×C9×S4C2×C3semidirect
22. C9×SL(2,ℤ/4)C2×C3semidirect
28. A4×3−1+2D4semidirect
75. C24×C9C3×D3semidirect
102. C22×3−1+2#67semidirect
103. C22×3−1+2S4semidirect
108. C9×A4C3×D4semidirect
113. GL(2,ℤ/4)3−1+2direct (N × H)
179. C22×A4C2×3−1+2semidirect
180. C2×S4C2×3−1+2semidirect
187. SL(2,ℤ/4)C2×3−1+2semidirect
232. C22×C9C3×#67semidirect
233. C22×C9C3×S4semidirect
242. 3−1+2GL(2,ℤ/4)direct (N × H)
304. C24D3×3−1+2semidirect
341. A4D4×3−1+2semidirect
355. C9C3×GL(2,ℤ/4)semidirect
399. C22#67×3−1+2semidirect
400. C22S4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C22×A4×3−1+22
C2×S4×3−1+22
3−1+2×SL(2,ℤ/4)2
C32×GL(2,ℤ/4)3
C9×GL(2,ℤ/4)3
3−1+2×#1203
#67×3−1+24