Finite Group Databasebeta

← #10692397

Subgroups — #10692397

Order 768.

774 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2341C211yesmaximal, characteristic, complemented
2#84082C211yesmaximal, characteristic
3#72138C211yesmaximal, characteristic, complemented
4#69528C211yesmaximal, characteristic, complemented
5#69444C211yesmaximal, characteristic, complemented
6#67626C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#9368—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×C4×Q8—21no—
11C2×#949C412yescomplemented
12C2×#930C2211yescharacteristic
13C3×#390—21no—
14C3×#390—22no—
15C3×#380—21no—
16C3×#380C2211yescharacteristic
17C3×C4×C22⋊C4—21no—
18C3×C4×C22⋊C4C2211yescharacteristic
19#5238—22no—
20C2×#837—22no—
21#5232C412yescomplemented
22#5227—21no—
23#5217—21no—
24C4×#217C2211yescharacteristic
25#5207C2211yescharacteristic
26#5207C2211yescharacteristic
27C2×#806—22no—
28#4451—21no—
29#4451—21no—
30#3006—31no—
31#2341D311yescharacteristic, complemented
32#1964—31no—
33#1789—31no—
34#1507—31no—
35#1305—31no—
36#1297—31no—
37C2×C3×#41—21no—
38C2×C3×#41—21no—
39C2×C3×#41—44no—
40C2×C3×#41C2×C412yes—
41C2×C3×#41D412yes—
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4—21no—
44C22×C3×D4D411yescharacteristic
45C23×C3×C4—21no—
46C23×C3×C4C2311yescharacteristic
47C22×#67—21no—
48C22×D12C2×C411yescharacteristic
49C3×C4×Q8—44no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4D412yes—
56C2×C3×C42—22no—
57C2×C3×C42—21no—
58C2×#217—21no—
59#949—44no—
60C2×#215—22no—
61C2×#213—22no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3C2×C411yescharacteristic
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C2×#208C2×C412yes—
71C3×#95—21no—
72C3×#95D411yescharacteristic
73C3×#95D411yescharacteristic
74#837—44no—
75#835—22no—
76#832—22no—
77C3⋊#95—21no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#806—44no—
83C22×#41C2×D311yescharacteristic
84C2×AGL(1,ℤ/8)—32no—
85C2×#121—31no—
86C2×C4×Q8—21no—
87#449—62no—
88#448—61no—
89#447—61no—
90C2×#104—62no—
91#417—32no—
92C2×#99—62no—
93#407—31no—
94#407—31no—
95#390—21no—
96#390—22no—
97#380—21no—
98#380C2×D311yescharacteristic
99C4×C22⋊C4—21no—
100C4×C22⋊C4—31no—
101C4×C22⋊C4C2×D311yescharacteristic
102#327—61no—
103#327—61no—
104C24×C3—21no—
105C24×C3C2×D411yescharacteristic
106C23×D3—21no—
107C3×#41—24no—
108C3×#41—24no—
109C3×#41—42no—
110C3×#41—42no—
111C3×#41—44no—
112C3×#41—44no—
113C2×C3×Q8—24no—
114C2×C3×Q8—21no—
115C2×C3×Q8C22⋊C412yes—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C2×C3×D4—44no—
121C2×C3×D4C22⋊C412yes—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—44no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×#67—42no—
137C2×#67—42no—
138C2×#67—41no—
139C2×#67—41no—
140C22×Dic3—21no—
141C22×Dic3—41no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—42no—
145C3×C4⋊C4—44no—
146C3×C4⋊C4—42no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—44no—
149C3×C22⋊C4—42no—
150C3×C22⋊C4—82no—
151C3×C42—44no—
152C3×C42—42no—
153#217—22no—
154#217—41no—
155#215—44no—
156#213—44no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4×Dic3—42no—
162#208—42no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—41no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—44no—
169C2×#41C4×D312yes—
170C2×#41D1212yes—
171C22×Q8#6711yescharacteristic
172C22×D4—21no—
173C22×D4—31no—
174C22×D4—61no—
175C22×D4#6711yescharacteristic
176C23×C4—21no—
177C23×C4C22×D311yescharacteristic
178AGL(1,ℤ/8)—124no—
179C2×SD16—62no—
180C2×D8—62no—
181C2×M16—32no—
182C2×M16—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#121—61no—
187C4×Q8—44no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4#6712yes—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—61no—
200#104—124no—
201#103—62no—
202#102—62no—
203#99—124no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#95—21no—
209#95—61no—
210#95#6711yescharacteristic
211#95#6711yescharacteristic
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—22no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3C4×D411yescharacteristic
222C23×C3#12011yescharacteristic
223C23×C3#12111yescharacteristic
224C22×D3—41no—
225C22×D3—42no—
226C22×D3—81no—
227C22×D3—81no—
228C3×Q8—24no—
229C3×Q8—22no—
230C3×Q8#9914yescomplemented
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—44no—
234C3×D4—44no—
235C3×D4—42no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—44no—
246C2×C3×C4—44no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4C2×C22⋊C411yescharacteristic
256C2×C3×C4C4×D411yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12311yescharacteristic
259#67—82no—
260#67—81no—
261#67—81no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—41no—
265C2×Dic3—81no—
266C2×Dic3—81no—
267D12—42no—
268D12—81no—
269C3⋊C8—42no—
270C3⋊C8—81no—
271C24—21no—
272C24—61no—
273C24C2×#6711yescharacteristic
274#41—24no—
275#41—24no—
276#41—42no—
277#41—42no—
278#41—44no—
279#41—44no—
280C2×Q8—24no—
281C2×Q8—21no—
282C2×Q8(C2×D3)⋊C412yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—44no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—122no—
291C2×D4—122no—
292C2×D4—122no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4(C2×D3)⋊C412yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—61no—
305C22×C4—121no—
306C22×C4C2×C4×D311yescharacteristic
307C22×C4C2×D1211yescharacteristic
308C22×C4#23511yescharacteristic
309C22×C4#23511yescharacteristic
310C22×C4C2×#6711yescharacteristic
311C22×C4C2×#6711yescharacteristic
312SD16—124no—
313D8—124no—
314M16—122no—
315M16—122no—
316C2×C8—62no—
317C2×C8—121no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—62no—
326C22⋊C4—82no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C22⋊C4—121no—
330C42—44no—
331C42—42no—
332C42—122no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—82no—
347C22×C3—82no—
348C22×C3#38611yescharacteristic
349C22×C3#45311yescharacteristic
350C22×C3#45411yescharacteristic
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—162no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—24no—
358C3×C4—21no—
359C3×C4—42no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4C2×#9912yes—
363Dic3—82no—
364Dic3—81no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—122no—
375C23—121no—
376C23—241no—
377C23—241no—
378C23C4×#6711yescharacteristic
379C23#93011yescharacteristic
380C23#95311yescharacteristic
381Q8—24no—
382Q8—22no—
383Q8#80614yescomplemented
384D4—22no—
385D4—22no—
386D4—44no—
387D4—44no—
388D4—42no—
389D4—122no—
390D4—241no—
391D4—242no—
392D4—241no—
393D4—241no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—121no—
416C2×C4—241no—
417C2×C4—241no—
418C2×C4C2×(C2×D3)⋊C411yescharacteristic
419C2×C4C4×#6711yescharacteristic
420C2×C4#92911yescharacteristic
421C2×C4#93011yescharacteristic
422C8—122no—
423C8—241no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—82no—
429C2×C3#188311yescharacteristic
430C2×C3#189811yescharacteristic
431C2×C3#190611yescharacteristic
432D3—161no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—82no—
447C22—82no—
448C22—241no—
449C22—241no—
450C22—241no—
451C22—482no—
452C22#521311yescharacteristic
453C22#524111yescharacteristic
454C22#524511yescharacteristic
455C4—21no—
456C4—24no—
457C4—21no—
458C4—21no—
459C4—42no—
460C4—82no—
461C4—81no—
462C4—242no—
463C4—241no—
464C4C2×#80612yes—
465C3#936811yescharacteristic, complemented, Sylow
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—82no—
471C2—481no—
472C2#7154311yescharacteristic
473C2#7164511yescharacteristic
474C2#7166711yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2341C2
socleC22×C3#386
prime radical#84082C2
cosocleC23×C3×C4C23
hypercenter#2341D3
hypocenterC3#9368
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
1. C3×#2341C2semidirect
3. #72138C2semidirect
4. #69528C2semidirect
5. #69444C2semidirect
6. #67626C2semidirect
7. #67618C2semidirect
11. C2×#949C4semidirect
21. #5232C4semidirect
31. #2341D3semidirect
230. C3×Q8#99semidirect
383. Q8#806semidirect
465. C3#9368semidirect

Maximal subgroups

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

subgroupindex
C3×#23412
#840822
#721382
#695282
#694442
#676262
#676182
#93683