Finite Group Databasebeta

← C2×#76129 #11430700

Subgroups — C2×#76129 #11430700

Order 768.

2246 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#889C212yesmaximal
2C2×C3×#389C211yesmaximal, characteristic
3C2×#4983C212yesmaximal
4C2×#4982C212yesmaximal
5C2×#4976C212yesmaximal
6C2×#4928C211yesmaximal, characteristic
7C2×#4928C211yesmaximal, characteristic
8#76129C2116yesmaximal, direct product factor
9C2×#4657C212yesmaximal
10C2×#4654C212yesmaximal
11C2×#2726—31nomaximal, Sylow
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×C4×Dic3C2212yes—
27C2×#889—216no—
28C2×#889C22116yes—
29C3×#389C2218yes—
30C2×C3×#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36#4983C22116yes—
37#4982C22116yes—
38#4976C22116yes—
39#4928C2218yes—
40#4928C2218yes—
41#4657C22116yes—
42#4654C22116yes—
43C22×#125—32no—
44#2726—316no—
45C2×#398—32no—
46C2×#398—32no—
47C2×#398—32no—
48C2×#389—31no—
49C2×#389—31no—
50C2×#389—32no—
51C2×#389D311yescharacteristic
52C2×#387—32no—
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C2×C3×C4⋊C4—28no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—42no—
65C2×C3×C4⋊C4C2318yes—
66C2×C3×C4⋊C4C2314yes—
67C2×C3×C4⋊C4C2314yes—
68C2×C3×C42—42no—
69C2×C3×C42C2314yes—
70C2×C4⋊Dic3—42no—
71C2×C4⋊Dic3—42no—
72C2×C4⋊Dic3D418yes—
73C2×C4⋊Dic3Q818yes—
74C2×C4⋊Dic3C2314yes—
75C2×C4⋊Dic3C2314yes—
76C2×C4⋊Dic3C2314yes—
77C2×C4⋊Dic3C2314yes—
78C2×Dic3⋊C4—28no—
79C2×Dic3⋊C4—28no—
80C2×Dic3⋊C4—28no—
81C2×Dic3⋊C4—28no—
82C2×Dic3⋊C4—28no—
83C2×Dic3⋊C4—44no—
84C2×Dic3⋊C4C2318yes—
85C2×Dic3⋊C4C2318yes—
86C2×Dic3⋊C4C2318yes—
87C2×Dic3⋊C4C2318yes—
88C2×Dic3⋊C4C2318yes—
89C2×Dic3⋊C4C2318yes—
90C2×C4×Dic3—28no—
91C2×C4×Dic3C2318yes—
92#889—264no—
93C3×#95C2318yes—
94C3⋊#95C2318yes—
95C3⋊#95C2318yes—
96C3⋊#95C2318yes—
97C3⋊#95C2318yes—
98C3⋊#95C2318yes—
99C2×#125—316no—
100C2×#125—616no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—32no—
104C22×C4⋊C4—32no—
105C22×C4⋊C4—32no—
106C22×C4⋊C4—32no—
107C22×C4⋊C4—32no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C4⋊C4C2×D312yes—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C42—32no—
115C22×C42C2×D311yescharacteristic
116#398—316no—
117#398—316no—
118#398—316no—
119#389—316no—
120#389—38no—
121#389—38no—
122#389C2×D318yes—
123#387—316no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95C2×D312yes—
130C24×C3C2411yescharacteristic
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4C2×D414yes—
144C22×C3×C4C2×Q814yes—
145C22×C3×C4#4114yes—
146C22×C3×C4C2414yes—
147C22×C3×C4C2414yes—
148C22×C3×C4C2412yes—
149C22×C3×C4C2412yes—
150C22×C3×C4C2412yes—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3C2×D418yes—
162C22×Dic3C2×Q818yes—
163C22×Dic3C2414yes—
164C22×Dic3C2414yes—
165C22×Dic3C2414yes—
166C22×Dic3C2414yes—
167C3×C4⋊C4—216no—
168C3×C4⋊C4—28no—
169C3×C4⋊C4—44no—
170C3×C42—44no—
171C4⋊Dic3—44no—
172C4⋊Dic3—44no—
173C4⋊Dic3C2×D4116yes—
174C4⋊Dic3C2×Q8116yes—
175Dic3⋊C4—216no—
176Dic3⋊C4—216no—
177Dic3⋊C4—216no—
178Dic3⋊C4—216no—
179Dic3⋊C4—216no—
180Dic3⋊C4—48no—
181C4×Dic3—216no—
182C23×C4—32no—
183C23×C4—32no—
184C23×C4—32no—
185C23×C4—32no—
186C23×C4C22×D312yes—
187C23×C4C22×D312yes—
188C23×C4C22×D311yescharacteristic
189C23×C4C22×D311yescharacteristic
190C23×C4C22×D311yescharacteristic
191#125—664no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—38no—
195C2×C4⋊C4—38no—
196C2×C4⋊C4—34no—
197C2×C4⋊C4—34no—
198C2×C4⋊C4—38no—
199C2×C4⋊C4—38no—
200C2×C4⋊C4—38no—
201C2×C4⋊C4—38no—
202C2×C4⋊C4—38no—
203C2×C4⋊C4—38no—
204C2×C4⋊C4—34no—
205C2×C4⋊C4—34no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—68no—
208C2×C4⋊C4—68no—
209C2×C4⋊C4—68no—
210C2×C4⋊C4—68no—
211C2×C4⋊C4—68no—
212C2×C4⋊C4—124no—
213C2×C4⋊C4—122no—
214C2×C4⋊C4—122no—
215C2×C4⋊C4C22×D318yes—
216C2×C4⋊C4C22×D314yes—
217C2×C4⋊C4C22×D314yes—
218C2×C42—38no—
219C2×C42—42no—
220C2×C42—68no—
221C2×C42C22×D314yes—
222#95—38no—
223#95—38no—
224#95—38no—
225#95—38no—
226#95—38no—
227#95C22×D318yes—
228C23×C3C22×D412yes—
229C23×C3C22×Q812yes—
230C23×C3C2×#4112yes—
231C23×C3H(5,𝔽2)12yes—
232C23×C32−1+412yes—
233C23×C32−1+412yes—
234C23×C32−1+412yes—
235C23×C3C2511yescharacteristic
236C2×C3×C4—28no—
237C2×C3×C4—28no—
238C2×C3×C4—28no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—48no—
252C2×C3×C4—48no—
253C2×C3×C4C22×D414yes—
254C2×C3×C4C22×Q814yes—
255C2×C3×C4C2×#4114yes—
256C2×Dic3—216no—
257C2×Dic3—216no—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—48no—
269C2×Dic3—48no—
270C2×Dic3C22×D418yes—
271C2×Dic3C22×Q818yes—
272C24C23×D311yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—38no—
286C22×C4—38no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4—34no—
290C22×C4—34no—
291C22×C4—64no—
292C22×C4—64no—
293C22×C4—64no—
294C22×C4—64no—
295C22×C4—64no—
296C22×C4—64no—
297C22×C4—64no—
298C22×C4—64no—
299C22×C4—64no—
300C22×C4—64no—
301C22×C4#23514yes—
302C22×C4D3×D414yes—
303C22×C4D3×Q814yes—
304C22×C4C23×D314yes—
305C22×C4C23×D314yes—
306C22×C4C23×D312yes—
307C22×C4C23×D312yes—
308C22×C4C23×D312yes—
309C4⋊C4—28no—
310C4⋊C4—216no—
311C4⋊C4—316no—
312C4⋊C4—316no—
313C4⋊C4—44no—
314C4⋊C4—616no—
315C4⋊C4—616no—
316C4⋊C4—616no—
317C4⋊C4—616no—
318C4⋊C4—616no—
319C4⋊C4—128no—
320C4⋊C4—124no—
321C4⋊C4—124no—
322C42—44no—
323C42—616no—
324C22×C3#53714yes—
325C22×C3#53914yes—
326C22×C3#54414yes—
327C22×C3#54714yes—
328C22×C3D4×Q814yes—
329C22×C3#55714yes—
330C22×C3#56414yes—
331C22×C3C23×D411yescharacteristic
332C22×C3C23×Q811yescharacteristic
333C22×C3C22×#4111yescharacteristic
334C22×C3C2×H(5,𝔽2)11yescharacteristic
335C22×C3C2×2−1+411yescharacteristic
336C22×C3C2×2−1+411yescharacteristic
337C22×C3C2×2−1+411yescharacteristic
338C3×C4—24no—
339C3×C4—24no—
340C3×C4—24no—
341C3×C4—44no—
342C3×C4—44no—
343Dic3—28no—
344Dic3—28no—
345Dic3—44no—
346Dic3—44no—
347C23C2×#23512yes—
348C23C2×D3×D412yes—
349C23#100412yes—
350C23C2×D3×Q812yes—
351C23#100712yes—
352C23#101012yes—
353C23#101012yes—
354C23C24×D311yescharacteristic
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—38no—
371C2×C4—38no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—64no—
375C2×C4—64no—
376C2×C4—64no—
377C2×C4—64no—
378C2×C4—616no—
379C2×C4—616no—
380C2×C4—64no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4—64no—
384C2×C4—64no—
385C2×C4—64no—
386C2×C4—128no—
387C2×C4—128no—
388C2×C4C2×#23514yes—
389C2×C4C2×D3×D414yes—
390C2×C4C2×D3×Q814yes—
391C2×C3#272618yescomplemented
392C2×C3C2×#53711yescharacteristic
393C2×C3C2×#53911yescharacteristic
394C2×C3C2×#54411yescharacteristic
395C2×C3C2×#54711yescharacteristic
396C2×C3C2×D4×Q811yescharacteristic
397C2×C3C2×#55711yescharacteristic
398C2×C3C2×#56411yescharacteristic
399C22#551514yes—
400C22#555114yes—
401C22#556714yes—
402C22#559714yes—
403C22#562314yes—
404C22C3⋊#56414yes—
405C22#572714yes—
406C22C22×#23511yescharacteristic
407C22C22×D3×D411yescharacteristic
408C22C2×#100411yescharacteristic
409C22C22×D3×Q811yescharacteristic
410C22C2×#100711yescharacteristic
411C22C2×#101011yescharacteristic
412C22C2×#101011yescharacteristic
413C4—24no—
414C4—24no—
415C4—24no—
416C4—44no—
417C4—44no—
418C4—68no—
419C4—68no—
420C4—124no—
421C4—124no—
422C3C2×#272611yescharacteristic, complemented, Sylow
423C2#7612918yesdirect product factor
424C2C2×#551511yescharacteristic
425C2C2×#555111yescharacteristic
426C2C2×#556711yescharacteristic
427C2C2×#559711yescharacteristic
428C2C2×#562311yescharacteristic
429C2C2×C3⋊#56411yescharacteristic
430C2C2×#572711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#389C2
socle, prime radicalC24×C3C24
hypercenterC2×#389D3
hypocenterC3C2×#2726
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
8. #76129C2direct (N × H)
391. C2×C3#2726semidirect
422. C3C2×#2726semidirect
423. C2#76129direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8892
C2×C3×#3892
C2×#49832
C2×#49822
C2×#49762
C2×#49282
#761292
C2×#46572
C2×#46542
C2×#27263