Finite Group Databasebeta

← #10690670

Subgroups — #10690670

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#82517C211yesmaximal, characteristic, complemented
3#71639C211yesmaximal, characteristic, complemented
4#71601C211yesmaximal, characteristic, complemented
5C2×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#67668C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#9097—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)C411yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#442—21no—
13C3×#440—21no—
14C3×C4×SD16—21no—
15C3×C4×D8—21no—
16C3×#428C411yescharacteristic
17C3×#418C411yescharacteristic
18C3×#417C411yescharacteristic
19C3×C4×M16C2211yescharacteristic
20#5024—21no—
21#5012—21no—
22#5011—22no—
23#4925—21no—
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C4×#208—21no—
28C2×C4×C3⋊C8C2211yescharacteristic
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—21no—
32D4⋊(C3⋊C8)—21no—
33#4525—22no—
34#2965—31no—
35C4×AGL(1,ℤ/8)D311yescharacteristic
36#1477—31no—
37C2×D4⋊C8—31no—
38#1460—31no—
39#1283—31no—
40#1278—31no—
41C2×C3×#41C2×C411yescharacteristic
42C22×C3×D4C2×C411yescharacteristic
43C23×C3×C4—21no—
44C3×AGL(1,ℤ/8)—22no—
45C3×AGL(1,ℤ/8)—22no—
46C2×C3×SD16—21no—
47C2×C3×D8—21no—
48C2×C3×M16C2×C411yescharacteristic
49C2×C3×M16C2×C411yescharacteristic
50C3×C4×Q8D411yescharacteristic
51C3×C4×Q8D411yescharacteristic
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4D411yescharacteristic
59C3×C4×D4D411yescharacteristic
60C3×#117—21no—
61C3×#117C2×C411yescharacteristic
62C2×C3×C4⋊C4C2×C411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#208—22no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×#107—21no—
70C3×#106—21no—
71C3×#103—21no—
72C3×#102—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×#97—21no—
76C3×C4×C8—21no—
77#830—22no—
78#830—21no—
79#830D412yes—
80C4⋊(C3⋊C8)—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85#803—22no—
86C4×C3⋊C8—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D411yescharacteristic
90C4×C3⋊C8D411yescharacteristic
91C2×AGL(1,ℤ/8)Dic311yescharacteristic
92C4×#41C2×D311yescharacteristic
93C2×C4×D4C2×D311yescharacteristic
94#446—61no—
95#442—21no—
96#440—21no—
97C4×SD16—21no—
98C4×D8—21no—
99#436—61no—
100#435—62no—
101#431—31no—
102#431—31no—
103#431—61no—
104#428Dic311yescharacteristic
105C2×C4⋊C8—31no—
106#418Dic311yescharacteristic
107#417Dic311yescharacteristic
108C4×M16—61no—
109C4×M16C2×D311yescharacteristic
110C2×C4×C8—31no—
111Q8⋊C8—61no—
112D4⋊C8—61no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115#323—62no—
116C24×C3—21no—
117C3×#41—21no—
118C3×#41—21no—
119C3×#41—21no—
120C3×#41—21no—
121C2×C3×Q8C22⋊C411yescharacteristic
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4C22⋊C411yescharacteristic
129C2×C3×D4M1612yescomplemented
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—81no—
137C22×C3×C4—81no—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C3×SD16—24no—
142C3×D8—24no—
143C3×M16—21no—
144C3×M16—21no—
145C3×M16—21no—
146C3×M16M1612yescomplemented
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4C22⋊C411yescharacteristic
154C3×C4⋊C4C22⋊C411yescharacteristic
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C42—21no—
161C3×C42—21no—
162C3×C42—21no—
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167#208—44no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—42no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—81no—
175C2×#41C2×Dic311yescharacteristic
176C22×D4C2×Dic311yescharacteristic
177C23×C4—21no—
178AGL(1,ℤ/8)—22no—
179AGL(1,ℤ/8)—22no—
180C2×SD16—21no—
181C2×D8—21no—
182C2×M16—62no—
183C2×M16C2×Dic311yescharacteristic
184C2×M16C2×Dic311yescharacteristic
185C22×C8—61no—
186C22×C8—61no—
187C4×Q8#6711yescharacteristic
188C4×Q8#6711yescharacteristic
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4#6711yescharacteristic
196C4×D4#6711yescharacteristic
197#117—21no—
198#117C2×Dic311yescharacteristic
199C2×C4⋊C4C2×Dic311yescharacteristic
200C2×C22⋊C4—21no—
201C2×C42—21no—
202C2×C42C22×D311yescharacteristic
203#107—21no—
204#106—21no—
205C4⋊C8—62no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210#103—21no—
211#102—21no—
212#102—21no—
213#102—21no—
214C22⋊C8—32no—
215C22⋊C8—62no—
216C22⋊C8—61no—
217#97—21no—
218#97—62no—
219C4×C8—21no—
220C4×C8—31no—
221C4×C8—31no—
222C4×C8—61no—
223C4×C8—61no—
224C4×C8—61no—
225C23×C3—21no—
226C23×C3—22no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—81no—
230C23×C3—81no—
231C23×C3C4×D411yescharacteristic
232C3×Q8—21no—
233C3×Q8—21no—
234C3×D4—21no—
235C3×D4—22no—
236C3×D4—21no—
237C3×D4—22no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—82no—
255C2×C3×C4—81no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4C2×C22⋊C411yescharacteristic
259C2×C3×C4C4×D411yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12311yescharacteristic
264C2×C3×C4C2×M1611yescharacteristic
265C2×C3×C4C2×M1611yescharacteristic
266C2×C3×C4#13111yescharacteristic
267C2×C3×C4#13111yescharacteristic
268C3×C8—22no—
269C3×C8—41no—
270C3⋊C8—22no—
271C3⋊C8—41no—
272C3⋊C8—42no—
273C3⋊C8—81no—
274C3⋊C8—81no—
275C24—21no—
276#41—21no—
277#41—21no—
278#41—21no—
279#41—21no—
280C2×Q8#21711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4#20812yescomplemented
288C2×D4#21711yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4D3×D411yescharacteristic
298C22×C4#23711yescharacteristic
299C22×C4C22×Dic311yescharacteristic
300SD16—24no—
301D8—24no—
302M16—21no—
303M16—21no—
304M16—21no—
305M16—124no—
306M16#20812yescomplemented
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—62no—
312C2×C8—122no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—241no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4#21711yescharacteristic
321C4⋊C4#21711yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42D3×D411yescharacteristic
331C42#23711yescharacteristic
332C42C2×#6711yescharacteristic
333C42C2×#6711yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—42no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3#38611yescharacteristic
345C22×C3#43511yescharacteristic
346C22×C3#43611yescharacteristic
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4—41no—
353C3×C4—81no—
354C3×C4#40711yescharacteristic
355C3×C4#40811yescharacteristic
356C3×C4#43511yescharacteristic
357C3×C4#43611yescharacteristic
358C3×C4#47111yescharacteristic
359C3×C4#47211yescharacteristic
360C23—21no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—81no—
365C23—81no—
366C23D4×Dic311yescharacteristic
367Q8—21no—
368Q8—21no—
369D4—21no—
370D4—22no—
371D4—21no—
372D4—22no—
373D4—41no—
374D4—41no—
375D4—41no—
376D4—41no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C2×#20811yescharacteristic
394C2×C4C2×#20811yescharacteristic
395C2×C4D4×Dic311yescharacteristic
396C2×C4#93511yescharacteristic
397C2×C4#93711yescharacteristic
398C2×C4#93911yescharacteristic
399C2×C4#93911yescharacteristic
400C2×C4#94811yescharacteristic
401C2×C4#94811yescharacteristic
402C2×C4C2×#21711yescharacteristic
403C8—22no—
404C8—41no—
405C8—62no—
406C8—121no—
407C8—122no—
408C8—241no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—81no—
413C2×C3—82no—
414C2×C3#187011yescharacteristic
415C2×C3#190611yescharacteristic
416C2×C3#190711yescharacteristic
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—82no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22#501111yescharacteristic
428C22#501211yescharacteristic
429C22#522011yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—81no—
437C4#501111yescharacteristic
438C4#501211yescharacteristic
439C4#519911yescharacteristic
440C4#520011yescharacteristic
441C4#520711yescharacteristic
442C4#523411yescharacteristic
443C3#909711yescharacteristic, complemented, Sylow
444C2—21no—
445C2—21no—
446C2—81no—
447C2—82no—
448C2#7161111yescharacteristic
449C2#7209111yescharacteristic
450C2#7211411yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×AGL(1,ℤ/8)C2
socleC22×C3#386
prime radicalC2×C3×AGL(1,ℤ/8)C4
cosocleC2×C3×C42C23
hypercenterC4×AGL(1,ℤ/8)D3
hypocenterC3#9097
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #82517C2semidirect
3. #71639C2semidirect
4. #71601C2semidirect
5. C2×D4⋊(C3⋊C8)C2semidirect
6. #67668C2semidirect
7. #67667C2semidirect
129. C2×C3×D4M16semidirect
146. C3×M16M16semidirect
287. C2×D4#208semidirect
306. M16#208semidirect
443. C3#9097semidirect

Maximal subgroups

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

subgroupindex
C3×C4×AGL(1,ℤ/8)2
#825172
#716392
#716012
C2×D4⋊(C3⋊C8)2
#676682
#676672
#90973