Finite Group Databasebeta

← #10695792

Subgroups — #10695792

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×#3006C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic, complemented
3#82630C211yesmaximal, characteristic, complemented
4#72136C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic, complemented
6#69972C211yesmaximal, characteristic, complemented
7#67687C211yesmaximal, characteristic, complemented
8#9975—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#951—21no—
13C2×#924C2211yescharacteristic
14C3×#449—22no—
15C3×#448—21no—
16C3×#447—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19#5239—22no—
20#5232—21no—
21#5228—21no—
22#5218—21no—
23#5218—21no—
24#5207C2211yescharacteristic
25C2×#830—22no—
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5041—22no—
29#5040—21no—
30#5034—21no—
31#4971—21no—
32#4525—22no—
33#4469—22no—
34#3187—31no—
35#3007—31no—
36#3006D311yescharacteristic
37#1780—31no—
38#1472—31no—
39#1471—31no—
40#1293—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4D411yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×Dic6D411yescharacteristic
54C3×AGL(1,ℤ/8)—44no—
55C2×C3×SD16—22no—
56C2×C3×D8—22no—
57C2×C3×M16—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×#121—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×#217—21no—
65#951—44no—
66C2×#216—22no—
67C2×#214—22no—
68C2×#214—22no—
69C2×C4⋊Dic3D411yescharacteristic
70#924—21no—
71#924—21no—
72#924—21no—
73#924—21no—
74C2×Dic3⋊C4—21no—
75C2×#208—21no—
76C22×C3⋊C8—21no—
77C22×C3⋊C8D412yes—
78C3×#103—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8D412yes—
82#835—22no—
83#832—22no—
84#832—22no—
85#830—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830—44no—
91#830D412yes—
92#810—22no—
93#808—22no—
94C22×#41C2×D311yescharacteristic
95C2×#137—61no—
96C2×AGL(1,ℤ/8)—21no—
97C2×C22⋊Q8—31no—
98C2×#121C2×D311yescharacteristic
99#483—61no—
100#481—61no—
101#465—62no—
102#452—62no—
103#451—61no—
104#450—61no—
105#450—61no—
106#449—22no—
107#448—21no—
108#448—61no—
109#447—21no—
110#417—21no—
111#417—61no—
112#407—31no—
113#407C2×D311yescharacteristic
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—62no—
117#324—62no—
118#323—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3#4111yescharacteristic
122C3×#41—24no—
123C3×#41—24no—
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—42no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C2×C3×D4—41no—
145C2×C3×D4—82no—
146C22×C3×C4—24no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—41no—
152C22×C3×C4—41no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—42no—
164C3×SD16—44no—
165C3×D8—44no—
166C3×M16—42no—
167C2×C3×C8—22no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172#217—41no—
173#217—41no—
174#216—44no—
175#214—44no—
176#214—82no—
177C4⋊Dic3—21no—
178C4⋊Dic3—21no—
179Dic3⋊C4—41no—
180Dic3⋊C4—41no—
181#208—42no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—42no—
185C2×C3⋊C8—42no—
186C2×C3⋊C8—41no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—44no—
192C22×Q8—31no—
193C22×Q8#6711yescharacteristic
194C22×D4—21no—
195C22×D4—21no—
196C22×D4#6711yescharacteristic
197C22×D4#6711yescharacteristic
198C23×C4—21no—
199C23×C4C22×D311yescharacteristic
200#137—124no—
201AGL(1,ℤ/8)—44no—
202C2×Q16—62no—
203C2×SD16—22no—
204C2×SD16—62no—
205C2×SD16—62no—
206C2×D8—22no—
207C2×M16—21no—
208C2×M16—61no—
209C22×C8—32no—
210C22×C8—61no—
211C22⋊Q8—61no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215#121—21no—
216#121—21no—
217#121—21no—
218#121—21no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4#6711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—61no—
224#106—62no—
225#103—22no—
226#103—62no—
227#103—62no—
228#102—22no—
229#102—62no—
230#102—62no—
231C22⋊C8—21no—
232C22⋊C8—32no—
233C22⋊C8—61no—
234C22⋊C8—61no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8—61no—
238C22⋊C8—124no—
239C22⋊C8#6712yes—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—81no—
251C23×C3—81no—
252C23×C3#12111yescharacteristic
253C23×C3#12111yescharacteristic
254C23×C3#12111yescharacteristic
255C3×Q8—22no—
256C3×Q8—22no—
257C3×Q8—41no—
258C3×Q8—41no—
259C3×D4—22no—
260C3×D4—22no—
261C3×D4—41no—
262C3×D4—41no—
263C3×D4—42no—
264C3×D4—44no—
265C3×D4—44no—
266C3×D4—42no—
267C3×D4—82no—
268C3×D4—82no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—44no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—44no—
280C2×C3×C4—41no—
281C2×C3×C4—81no—
282C2×C3×C4#12011yescharacteristic
283C2×C3×C4#12011yescharacteristic
284C2×C3×C4#12011yescharacteristic
285C2×C3×C4#12711yescharacteristic
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—81no—
290Dic6—42no—
291Dic6—42no—
292C3×C8—42no—
293C3⋊C8—42no—
294C3⋊C8—42no—
295C3⋊C8—81no—
296C24—21no—
297C24—21no—
298C24#23711yescharacteristic
299#41—24no—
300#41—24no—
301#41—42no—
302#41—42no—
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
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—21no—
316C2×D4—21no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—82no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4D3×D411yescharacteristic
336C22×C4D3×D411yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4C2×#6711yescharacteristic
339C22×C4C2×#6711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341Q16—124no—
342SD16—44no—
343SD16—124no—
344SD16—242no—
345D8—44no—
346M16—42no—
347M16—122no—
348C2×C8—22no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—122no—
352C2×C8—122no—
353C2×C8—121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—61no—
357C4⋊C4—61no—
358C4⋊C4—121no—
359C4⋊C4—121no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—121no—
363C22⋊C4—121no—
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—44no—
373C22×C3—42no—
374C22×C3—82no—
375C22×C3—82no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—81no—
379C22×C3—81no—
380C22×C3—81no—
381C22×C3—162no—
382C22×C3#39211yescharacteristic
383C22×C3#47112yes—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—42no—
388C3×C4—81no—
389C3×C4B(4,𝔽2)12yes—
390Dic3—81no—
391Dic3—81no—
392C23—22no—
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—81no—
403C23—81no—
404C23#93811yescharacteristic
405C23#93911yescharacteristic
406C23#93911yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—41no—
410Q8—41no—
411Q8—122no—
412Q8—122no—
413D4—22no—
414D4—22no—
415D4—41no—
416D4—41no—
417D4—42no—
418D4—44no—
419D4—44no—
420D4—42no—
421D4—82no—
422D4—82no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—81no—
436C2×C4—122no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—241no—
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—82no—
452C2×C3—161no—
453C2×C3#201711yescharacteristic
454C2×C3#201911yescharacteristic
455C2×C3#202111yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—44no—
465C22—42no—
466C22—82no—
467C22—82no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—162no—
474C22#520012yes—
475C22#522311yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—81no—
481C4—241no—
482C4—241no—
483C4#503212yes—
484C3#997511yescharacteristic, complemented, Sylow
485C2—22no—
486C2—41no—
487C2—41no—
488C2—82no—
489C2—161no—
490C2#7167111yescharacteristic
491C2#7209711yescharacteristic
492C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3006C2
socleC22×C3#392
hypercenter#3006D3
hypocenterC3#9975
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
2. #84084C2semidirect
3. #82630C2semidirect
4. #72136C2semidirect
5. #71676C2semidirect
6. #69972C2semidirect
7. #67687C2semidirect
484. C3#9975semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840842
#826302
#721362
#716762
#699722
#676872
#99753