Finite Group Databasebeta

← #10697145

Subgroups — #10697145

Order 768.

646 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C3×#3036C211yesmaximal, characteristic, complemented
2#83302C211yesmaximal, characteristic, complemented
3C2×#5015C211yesmaximal, characteristic, complemented
4#71800C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic, complemented
6#70502C211yesmaximal, characteristic, complemented
7#67714C211yesmaximal, characteristic, complemented
8#10176—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C22×#213—21no—
13C2×#874C2211yescharacteristic
14C3×#463—22no—
15C3×#461—21no—
16C3×#459—21no—
17C3×#423C2211yescharacteristic
18C3×#417—21no—
19#5088—21no—
20#5073—21no—
21#5036—22no—
22#5015—22no—
23#5015—22no—
24#5015—22no—
25C2×#809—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29D4⋊(C3⋊C8)—22no—
30#4525—22no—
31#4485—22no—
32#3205—31no—
33#3036D311yescharacteristic, complemented
34C2×#459—31no—
35#1545—31no—
36#1480—31no—
37#1460—31no—
38#1301—31no—
39C2×C3×#41—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—21no—
45C3×AGL(1,ℤ/8)—42no—
46C3×AGL(1,ℤ/8)—42no—
47C2×C3×SD16—22no—
48C2×C3×D8—22no—
49C2×C3×M16—21no—
50C3×C4⋊Q8D411yescharacteristic
51C3×#127D411yescharacteristic
52C3×#124—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C42C2311yescharacteristic
64C2×#213—24no—
65C2×#213—24no—
66C2×#213—42no—
67C2×#213—42no—
68C22×C3⋊C8—21no—
69#874—21no—
70#874—21no—
71#874—21no—
72#874D412yes—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×#103—22no—
76C3×#102—22no—
77#830—21no—
78#830D412yes—
79#809—24no—
80#809—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C2×AGL(1,ℤ/8)—21no—
88C22×D8—61no—
89#532C2×D311yescharacteristic
90C2×#127—31no—
91C2×C4×D4C2×D311yescharacteristic
92#493—61no—
93#486—61no—
94#463—22no—
95#461—21no—
96#459—21no—
97#459—62no—
98#459—62no—
99#459—62no—
100#447—62no—
101#431—31no—
102#431—31no—
103#423C2×D311yescharacteristic
104C2×C4⋊C8—31no—
105#417—21no—
106C2×#102—61no—
107#331—62no—
108D4⋊C8—62no—
109#323—62no—
110C24×C3—21no—
111C23×D3—41no—
112C3×#41—41no—
113C3×#41—41no—
114C3×#41—41no—
115C3×#41—41no—
116C2×C3×Q8—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C2×C3×D4D812yescomplemented
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—82no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×D12—22no—
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—41no—
139C2×D12—41no—
140C2×D12—41no—
141C2×D12—41no—
142C2×D12—41no—
143C2×D12—41no—
144C2×D12—41no—
145C2×D12—82no—
146C2×D12—82no—
147C3×SD16—44no—
148C3×D8—44no—
149C3×M16—41no—
150C3×M16—41no—
151C2×C3×C8—22no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164#213—48no—
165#213—84no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×#41—21no—
172C22×D4—61no—
173C22×D4—61no—
174C22×D4—61no—
175C22×D4#6711yescharacteristic
176C23×C4—21no—
177AGL(1,ℤ/8)—42no—
178AGL(1,ℤ/8)—42no—
179C2×SD16—22no—
180C2×D8—22no—
181C2×D8—64no—
182C2×D8—64no—
183C2×D8—122no—
184C2×D8—122no—
185C2×M16—21no—
186C22×C8—61no—
187C4⋊Q8#6711yescharacteristic
188#127—32no—
189#127—61no—
190#127—61no—
191#127—61no—
192#127#6711yescharacteristic
193#124—21no—
194#121—21no—
195#121—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C42C22×D311yescharacteristic
205C4⋊C8—21no—
206C4⋊C8—32no—
207C4⋊C8—61no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210C4⋊C8#6712yes—
211#103—22no—
212#102—22no—
213#102—64no—
214#102—64no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C4×C8—61no—
218C4×C8—61no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3#12111yescharacteristic
226C22×D3—42no—
227C22×D3—82no—
228C22×D3—82no—
229C22×D3—81no—
230C22×D3—81no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×D4—22no—
234C3×D4—22no—
235C3×D4—42no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—42no—
241C3×D4—81no—
242C3×D4—81no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—82no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—82no—
251C2×C3×C4—81no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12711yescharacteristic
258C2×C3×C4C2×D811yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
262D12—44no—
263D12—42no—
264D12—42no—
265D12—82no—
266D12—82no—
267D12—81no—
268D12—81no—
269D12—82no—
270C3×C8—42no—
271C3⋊C8—42no—
272C3⋊C8—42no—
273C3⋊C8—81no—
274C24—21no—
275C24—121no—
276#41—41no—
277#41—41no—
278#41—41no—
279#41—41no—
280C2×Q8—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—242no—
301C2×D4—242no—
302C2×D4#21312yescomplemented
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4D3×D411yescharacteristic
309C22×C4#23711yescharacteristic
310C22×C4C2×#6711yescharacteristic
311SD16—44no—
312D8—44no—
313D8—128no—
314D8—244no—
315M16—41no—
316M16—41no—
317C2×C8—22no—
318C2×C8—62no—
319C2×C8—62no—
320C2×C8—121no—
321C2×C8—121no—
322C2×C8—121no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C42C2×#6711yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—42no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3#39211yescharacteristic
346C22×C3#46611yescharacteristic
347C22×C3#46911yescharacteristic
348C2×D3—82no—
349C2×D3—82no—
350C2×D3—82no—
351C2×D3—162no—
352C2×D3—162no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—81no—
357C3×C4—81no—
358C3×C4#44711yescharacteristic
359C3×C4#44711yescharacteristic
360C3×C4#44911yescharacteristic
361C3×C4#44911yescharacteristic
362C3×C4#47212yes—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—81no—
368C23—81no—
369C23—122no—
370C23—242no—
371C23—242no—
372C23—241no—
373C23—241no—
374C23#93911yescharacteristic
375Q8—41no—
376Q8—41no—
377D4—22no—
378D4—22no—
379D4—42no—
380D4—42no—
381D4—41no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—81no—
386D4—81no—
387D4—124no—
388D4—122no—
389D4—122no—
390D4—242no—
391D4—242no—
392D4—242no—
393D4—241no—
394D4—241no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4C2×#21311yescharacteristic
405C2×C4#93211yescharacteristic
406C2×C4#93711yescharacteristic
407C2×C4#93711yescharacteristic
408C2×C4#93811yescharacteristic
409C2×C4#93911yescharacteristic
410C2×C4#94011yescharacteristic
411C2×C4#94911yescharacteristic
412C2×C4#95011yescharacteristic
413C2×C4#95311yescharacteristic
414C8—42no—
415C8—122no—
416C8—122no—
417C8—241no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—82no—
421C2×C3—161no—
422C2×C3#199911yescharacteristic
423C2×C3#200811yescharacteristic
424C2×C3#201811yescharacteristic
425D3—162no—
426C22—21no—
427C22—21no—
428C22—42no—
429C22—82no—
430C22—82no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—242no—
437C22—242no—
438C22—242no—
439C22—482no—
440C22—482no—
441C22#503811yescharacteristic
442C22#503911yescharacteristic
443C22#522311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4#503611yescharacteristic
450C4#503711yescharacteristic
451C4#519912yes—
452C4#521711yescharacteristic
453C4#523811yescharacteristic
454C3#1017611yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—82no—
458C2—161no—
459C2—482no—
460C2#7168411yescharacteristic
461C2#7209511yescharacteristic
462C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#10176
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. C3×#3036C2semidirect
2. #83302C2semidirect
3. C2×#5015C2semidirect
4. #71800C2semidirect
5. #71601C2semidirect
6. #70502C2semidirect
7. #67714C2semidirect
33. #3036D3semidirect
126. C2×C3×D4D8semidirect
302. C2×D4#213semidirect
454. C3#10176semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#833022
C2×#50152
#718002
#716012
#705022
#677142
#101763