Finite Group Databasebeta

← #10695790

Subgroups — #10695790

Order 768.

742 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3187C211yesmaximal, characteristic, complemented
2#82735C211yesmaximal, characteristic, complemented
3#82624C211yesmaximal, characteristic
4#71691C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic, complemented
6#69956C211yesmaximal, characteristic, complemented
7#67679C211yesmaximal, characteristic, complemented
8#9975—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#942—21no—
13C2×#932—21no—
14C3×#483—21no—
15C3×#481—21no—
16C3×#465—22no—
17C3×#450—21no—
18C3×#448—21no—
19C2×C3×C22⋊C8C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×(C2×D3)⋊C8—22no—
24#5050—21no—
25#5047—22no—
26#5046—21no—
27#5040—21no—
28#5037—22no—
29#5036—21no—
30#4966—21no—
31#4966—21no—
32#4528—22no—
33#4525—22no—
34#3187D311yescharacteristic, complemented
35#3007—31no—
36#3006—31no—
37#1780—31no—
38#1472—31no—
39#1471—31no—
40#1293—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—44no—
51C22×D12D411yescharacteristic
52C22×C4×D3—21no—
53C22×Dic6D411yescharacteristic
54C2×C3×SD16—22no—
55C22×C3×C8D412yes—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×#121—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×#121—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×#216—22no—
70#942—44no—
71C2×#215—22no—
72C2×#214—22no—
73#932—44no—
74C2×#213—22no—
75C2×#208—21no—
76C2×#208—21no—
77C22×C3⋊C8—21no—
78C3×#106—22no—
79C3×#103—22no—
80C3×#102—22no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830—21no—
87#830D412yes—
88#830D412yes—
89(C2×D3)⋊C8—44no—
90#810—22no—
91#810—22no—
92#809—22no—
93#809—22no—
94C22×#41—31no—
95C2×#137—61no—
96C2×AGL(1,ℤ/8)—61no—
97C2×C22⋊Q8C2×D311yescharacteristic
98C2×#121C2×D311yescharacteristic
99#483—21no—
100#481—21no—
101#465—22no—
102#452—62no—
103#451—61no—
104#450—21no—
105#450—61no—
106#449—62no—
107#448—21no—
108#448—61no—
109#447—61no—
110#417—61no—
111#417—61no—
112#407—31no—
113#407—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—62no—
116C2×C22⋊C8C2×D311yescharacteristic
117#324—62no—
118#323—62no—
119C24×C3—21no—
120C24×C3#4111yescharacteristic
121C23×D3—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—42no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—82no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C2×#67—42no—
142C2×#67—42no—
143C2×#67—41no—
144C2×#67—41no—
145C22×Dic3—21no—
146#235—24no—
147#235—24no—
148#235—42no—
149#235—42no—
150#235—42no—
151#235—42no—
152#235—42no—
153#235—42no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12—42no—
157C2×C4×D3—24no—
158C2×C4×D3—42no—
159C2×C4×D3—41no—
160C2×C4×D3—41no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—42no—
164C3×SD16—82no—
165C2×C3×C8—22no—
166C2×C3×C8—42no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177#216—44no—
178#215—44no—
179#214—44no—
180#213—44no—
181#208—42no—
182#208—42no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—42no—
186C2×C3⋊C8—41no—
187C2×#41—61no—
188C2×#41—61no—
189C2×#41—61no—
190C2×#41—61no—
191C2×#41—124no—
192C22×Q8—31no—
193C22×Q8#6711yescharacteristic
194C22×D4—21no—
195C22×D4—31no—
196C22×D4—61no—
197C22×D4#6711yescharacteristic
198C23×C4—61no—
199C23×C4C22×D311yescharacteristic
200#137—124no—
201AGL(1,ℤ/8)—124no—
202C2×Q16—62no—
203C2×SD16—22no—
204C2×SD16—62no—
205C2×SD16—62no—
206C2×D8—62no—
207C2×M16—61no—
208C2×M16—61no—
209C22×C8—61no—
210C22×C8#6712yes—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215#121—21no—
216#121—21no—
217#121—21no—
218#121—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4#6711yescharacteristic
221C2×C4⋊C4#6711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224#106—22no—
225#103—22no—
226#103—62no—
227#103—62no—
228#102—22no—
229#102—62no—
230#102—62no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—32no—
234C22⋊C8—32no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8—61no—
238C22⋊C8—61no—
239C22⋊C8—124no—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—81no—
246C23×C3—81no—
247C23×C3#12111yescharacteristic
248C23×C3#12111yescharacteristic
249C23×C3#12111yescharacteristic
250C22×D3—22no—
251C22×D3—41no—
252C22×D3—42no—
253C22×D3—41no—
254C22×D3—42no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×D4—42no—
258C3×D4—42no—
259C3×D4—82no—
260C3×D4—82no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—81no—
272C2×C3×C4—81no—
273C2×C3×C4#12011yescharacteristic
274C2×C3×C4#12011yescharacteristic
275C2×C3×C4#12011yescharacteristic
276C2×C3×C4#12711yescharacteristic
277#67—44no—
278#67—44no—
279C2×Dic3—22no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282D12—22no—
283D12—22no—
284D12—41no—
285D12—41no—
286C4×D3—44no—
287C4×D3—44no—
288Dic6—22no—
289Dic6—22no—
290Dic6—41no—
291Dic6—41no—
292C3×C8—42no—
293C3⋊C8—42no—
294C3⋊C8—42no—
295C3⋊C8—81no—
296C24—21no—
297C24—61no—
298C24#23711yescharacteristic
299#41—64no—
300#41—64no—
301#41—122no—
302#41—122no—
303#41—122no—
304#41—122no—
305#41—122no—
306#41—122no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—42no—
310C2×Q8—61no—
311C2×Q8—61no—
312C2×Q8—122no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—82no—
321C2×D4—122no—
322C2×D4—122no—
323C2×D4—122no—
324C2×D4—121no—
325C2×D4—121no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—64no—
331C22×C4—61no—
332C22×C4—122no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4D3×D411yescharacteristic
336C22×C4D3×D411yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4C2×#6711yescharacteristic
339C22×C4C2×#6711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341Q16—124no—
342SD16—82no—
343SD16—124no—
344SD16—124no—
345D8—124no—
346M16—122no—
347M16—122no—
348C2×C8—22no—
349C2×C8—42no—
350C2×C8—62no—
351C2×C8—62no—
352C2×C8—122no—
353C2×C8—121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—82no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3—162no—
377C22×C3#39211yescharacteristic
378C22×C3#47112yes—
379C2×D3—44no—
380C2×D3—42no—
381C2×D3—82no—
382C2×D3—81no—
383C2×D3—81no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4—81no—
390C3×C4B(4,𝔽2)12yes—
391Dic3—42no—
392C23—22no—
393C23—41no—
394C23—41no—
395C23—42no—
396C23—41no—
397C23—62no—
398C23—81no—
399C23—81no—
400C23—121no—
401C23—122no—
402C23—122no—
403C23—121no—
404C23#93811yescharacteristic
405C23#93911yescharacteristic
406C23#93911yescharacteristic
407Q8—42no—
408Q8—42no—
409Q8—62no—
410Q8—62no—
411Q8—121no—
412Q8—121no—
413D4—42no—
414D4—42no—
415D4—62no—
416D4—62no—
417D4—82no—
418D4—82no—
419D4—121no—
420D4—121no—
421D4—124no—
422D4—124no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—62no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—124no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—124no—
440C2×C4#93711yescharacteristic
441C2×C4#93711yescharacteristic
442C2×C4#94011yescharacteristic
443C2×C4#95311yescharacteristic
444C8—42no—
445C8—122no—
446C8—122no—
447C8—241no—
448C2×C3—22no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—161no—
452C2×C3#201711yescharacteristic
453C2×C3#201911yescharacteristic
454C2×C3#202111yescharacteristic
455D3—82no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—124no—
469C22—122no—
470C22—162no—
471C22—242no—
472C22—241no—
473C22—241no—
474C22#517712yes—
475C22#522311yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—122no—
483C4#524412yes—
484C3#997511yescharacteristic, complemented, Sylow
485C2—22no—
486C2—41no—
487C2—41no—
488C2—161no—
489C2—242no—
490C2#7185111yescharacteristic
491C2#7205911yescharacteristic
492C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3187C2
socleC22×C3#392
prime radical#82624C2
hypercenter#3187D3
hypocenterC3#9975
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
1. C3×#3187C2semidirect
2. #82735C2semidirect
4. #71691C2semidirect
5. #71676C2semidirect
6. #69956C2semidirect
7. #67679C2semidirect
34. #3187D3semidirect
484. C3#9975semidirect

Maximal subgroups

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

subgroupindex
C3×#31872
#827352
#826242
#716912
#716762
#699562
#676792
#99753