Finite Group Databasebeta

← #10698096

Subgroups — #10698096

Order 768.

1044 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 89 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×#2433C211yesmaximal, characteristic
2#84050C211yesmaximal, characteristic, complemented
3#82631C211yesmaximal, characteristic, complemented
4#72129C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#69445C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10329—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×#127—21no—
11C2×C3×#121C2211yescharacteristic
12C2×C3×#120—22no—
13C2×#936—21no—
14C2×#924C2211yescharacteristic
15C3×#392—22no—
16C3×#390—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5247—22no—
19#5222—22no—
20#5221—21no—
21#5218—21no—
22C2×#833—22no—
23C4×#217C2211yescharacteristic
24#5207C2211yescharacteristic
25#5207C2211yescharacteristic
26#5042—22no—
27#5040—21no—
28#5033—21no—
29#4970—21no—
30#4537—22no—
31#4452—22no—
32#3193—31no—
33#2435—31no—
34#2433D311yescharacteristic
35#1783—31no—
36#1508—31no—
37#1507—31no—
38#1297—31no—
39C25×C3D412yes—
40C22×C3×D4—21no—
41C22×C3×D4—24no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—21no—
48C22×C3×D4—44no—
49C22×C3×D4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×Dic6D411yescharacteristic, complemented
52C3×#127—44no—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#120—48no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C42—21no—
63C2×#217—24no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#217—24no—
67#936—44no—
68C2×#214—22no—
69C2×C4⋊Dic3D411yescharacteristic, complemented
70#924—21no—
71#924—21no—
72#924—21no—
73#924—21no—
74C2×Dic3⋊C4—21no—
75C2×C4×Dic3—21no—
76C2×#208—21no—
77C2×#208D412yescomplemented
78C3×#95D412yes—
79#833—44no—
80#832—22no—
81C3⋊#95D412yescomplemented
82#830—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#810—22no—
87#807—22no—
88C23×D4C2×D311yescharacteristic
89C2×#127—21no—
90C2×#124—61no—
91C2×C22⋊Q8—31no—
92C2×#121C2×D311yescharacteristic
93C2×#120—22no—
94#484—61no—
95#480—61no—
96#468—62no—
97#450—61no—
98#448—61no—
99C2×#100—62no—
100#407—31no—
101#407—31no—
102#394—62no—
103#392—22no—
104#390—21no—
105#390—61no—
106#379—62no—
107C4×C22⋊C4—31no—
108C4×C22⋊C4C2×D311yescharacteristic
109#328—62no—
110#327—62no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—22no—
114C24×C3—21no—
115C24×C3—44no—
116C24×C3—44no—
117C24×C3—44no—
118C24×C3C2×D411yescharacteristic
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—48no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C2×C3×D4—48no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—84no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—42no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×Dic3—22no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6—42no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—48no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C42—42no—
163#217—48no—
164#217—42no—
165#217—41no—
166#217—41no—
167#217—84no—
168#214—82no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173C4×Dic3—42no—
174#208—42no—
175#208—42no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—41no—
178C25#6712yes—
179C22×Q8—31no—
180C22×D4—21no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—44no—
189C22×D4#6711yescharacteristic
190C23×C4C22×D311yescharacteristic
191C2×SD16—62no—
192C2×M16—32no—
193C2×M16—61no—
194#127—44no—
195#124—124no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204#120—48no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—64no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—64no—
215C2×C42—21no—
216C2×C42—61no—
217#107—62no—
218#103—62no—
219#102—62no—
220#100—124no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225#95—32no—
226#95#6712yes—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—24no—
230C23×C3—24no—
231C23×C3—24no—
232C23×C3—41no—
233C23×C3—44no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—44no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—44no—
242C23×C3—44no—
243C23×C3—41no—
244C23×C3—42no—
245C23×C3—82no—
246C23×C3—82no—
247C23×C3—816no—
248C23×C3#12011yescharacteristic
249C23×C3#12011yescharacteristic
250C23×C3#12111yescharacteristic
251C3×D4—44no—
252C3×D4—44no—
253C3×D4—44no—
254C3×D4—44no—
255C3×D4—42no—
256C3×D4—81no—
257C3×D4—816no—
258C3×D4—82no—
259C3×D4—82no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12711yescharacteristic
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282Dic6—42no—
283Dic6—81no—
284C3⋊C8—42no—
285C3⋊C8—81no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24C2×#6711yescharacteristic
294C2×Q8—61no—
295C2×Q8—61no—
296C2×Q8—122no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—84no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—62no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4D3×D411yescharacteristic
326C22×C4D3×D411yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4#23711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C2×#6711yescharacteristic
331SD16—242no—
332M16—122no—
333M16—122no—
334C2×C8—62no—
335C2×C8—121no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—61no—
339C4⋊C4—61no—
340C4⋊C4—121no—
341C4⋊C4—121no—
342C22⋊C4—24no—
343C22⋊C4—48no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—128no—
347C22⋊C4—122no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C22⋊C4—244no—
351C42—42no—
352C42—122no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—44no—
361C22×C3—48no—
362C22×C3—48no—
363C22×C3—44no—
364C22×C3—84no—
365C22×C3—84no—
366C22×C3—84no—
367C22×C3—84no—
368C22×C3—84no—
369C22×C3—82no—
370C22×C3—82no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3#39211yescharacteristic
375C22×C3#45312yes—
376C22×C3#45512yes—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—44no—
381C3×C4—81no—
382C3×C4B(4,𝔽2)12yes—
383Dic3—82no—
384Dic3—81no—
385Dic3—81no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—41no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—44no—
401C23—44no—
402C23—41no—
403C23—42no—
404C23—82no—
405C23—82no—
406C23—816no—
407C23#93711yescharacteristic
408C23#93911yescharacteristic
409C23#95311yescharacteristic
410Q8—122no—
411Q8—241no—
412D4—44no—
413D4—44no—
414D4—44no—
415D4—44no—
416D4—42no—
417D4—81no—
418D4—816no—
419D4—82no—
420D4—82no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4#93711yescharacteristic
440C2×C4#93811yescharacteristic
441C2×C4#93911yescharacteristic
442C2×C4#94011yescharacteristic
443C8—122no—
444C8—241no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—41no—
448C2×C3—84no—
449C2×C3—161no—
450C2×C3#200211yescharacteristic
451C2×C3#200411yescharacteristic
452C2×C3#202111yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—44no—
461C22—48no—
462C22—48no—
463C22—44no—
464C22—84no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—82no—
470C22—82no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22#507712yes—
475C22#522311yescharacteristic
476C22#524212yes—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—44no—
481C4—81no—
482C4—242no—
483C4—241no—
484C4—241no—
485C4#503212yes—
486C3#1032911yescharacteristic, complemented, Sylow
487C2—22no—
488C2—22no—
489C2—41no—
490C2—84no—
491C2—161no—
492C2#7167111yescharacteristic
493C2#7176911yescharacteristic
494C2#7185711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2433C2
socleC22×C3#392
hypercenter#2433D3
hypocenterC3#10329
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2433

Complements

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

NH ≅ G/Ntype
2. #84050C2semidirect
3. #82631C2semidirect
4. #72129C2semidirect
5. #71673C2semidirect
6. #69445C2semidirect
7. #67618C2semidirect
51. C22×Dic6D4semidirect
69. C2×C4⋊Dic3D4semidirect
77. C2×#208D4semidirect
81. C3⋊#95D4semidirect
486. C3#10329semidirect

Maximal subgroups

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

subgroupindex
C3×#24332
#840502
#826312
#721292
#716732
#694452
#676182
#103293