Finite Group Databasebeta

← #10695626

Subgroups — #10695626

Order 768.

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

normal subgroups only

This group has 786 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×#3187C211yesmaximal, characteristic
2#82760C211yesmaximal, characteristic, complemented
3C2×#5040C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic, complemented
5#71686C211yesmaximal, characteristic
6#69963C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9949—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#944—21no—
13C22×#214—21no—
14C3×#483—21no—
15C3×#481—21no—
16C3×#465—22no—
17C3×#450—21no—
18C3×#448—21no—
19C2×C3×C22⋊C8C2211yescharacteristic
20C2×#834—22no—
21#5212C2211yescharacteristic
22C4×#217C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#816—22no—
25#5040—22no—
26#5040—22no—
27#4979—22no—
28#4973—21no—
29C2×#810—21no—
30#4946—22no—
31#4538—22no—
32#3187D311yescharacteristic
33C2×#448—31no—
34#2434—31no—
35#1964—31no—
36#1787—31no—
37#1520—31no—
38#1282—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—44no—
50C22×Dic6D412yes—
51C2×C3×SD16—22no—
52C22×C3×C8D412yes—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×#217—21no—
67C2×#217D412yes—
68#944—44no—
69C2×#214—24no—
70C2×#214—44no—
71C2×C4⋊Dic3—21no—
72C2×Dic3⋊C4—24no—
73C2×C4×Dic3—21no—
74C22×C3⋊C8—21no—
75C3×#106—22no—
76C3×#103—22no—
77C3×#102—22no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80#834—44no—
81C3⋊#95—24no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yes—
86#816—44no—
87#810—24no—
88C23×Q8—31no—
89C22×SD16—61no—
90C2×C4⋊Q8—61no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#121C2×D311yescharacteristic
93#483—21no—
94#481—21no—
95#465—22no—
96#450—21no—
97#448—21no—
98#448—62no—
99#448—62no—
100C2×#103—61no—
101C2×#103—62no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8C2×D311yescharacteristic
105#395—62no—
106#391—61no—
107#385—62no—
108#380—31no—
109C4×C22⋊C4—31no—
110#328—62no—
111C24×C3—21no—
112C24×C3C2×D411yescharacteristic
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—42no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—44no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C22×Dic3—41no—
139C2×Dic6—24no—
140C2×Dic6—24no—
141C2×Dic6—22no—
142C2×Dic6—22no—
143C2×Dic6—24no—
144C2×Dic6—44no—
145C2×Dic6—44no—
146C2×Dic6—44no—
147C2×Dic6—42no—
148C2×Dic6—42no—
149C2×Dic6—42no—
150C2×Dic6—41no—
151C2×Dic6—41no—
152C3×SD16—82no—
153C2×C3×C8—22no—
154C2×C3×C8—42no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165#217—22no—
166#217—42no—
167#217SD1614yescomplemented
168#214—48no—
169C4⋊Dic3—42no—
170Dic3⋊C4—48no—
171C4×Dic3—42no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—42no—
174C22×Q8—32no—
175C22×Q8—61no—
176C22×Q8—61no—
177C22×Q8—61no—
178C22×Q8—61no—
179C22×Q8—61no—
180C22×Q8—124no—
181C22×Q8#6711yescharacteristic
182C22×D4—21no—
183C22×D4#6711yescharacteristic
184C23×C4—61no—
185C23×C4C22×D311yescharacteristic
186C2×SD16—22no—
187C2×SD16—64no—
188C2×SD16—124no—
189C22×C8—61no—
190C22×C8#6712yes—
191C4⋊Q8—124no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196#121—21no—
197#121—21no—
198#121—21no—
199#121—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—64no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4#6711yescharacteristic
204C2×C4⋊C4#6711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—32no—
208C2×C22⋊C4—61no—
209C2×C42—61no—
210#106—22no—
211#103—22no—
212#103—64no—
213#103—124no—
214#102—22no—
215#99—124no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—32no—
219C22⋊C8—61no—
220#95—64no—
221#95—61no—
222#95—61no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3#12011yescharacteristic
230C23×C3#12111yescharacteristic
231C23×C3#12711yescharacteristic
232C23×C3C2×SD1612yes—
233C23×C3#13712yes—
234C3×Q8—42no—
235C3×Q8—81no—
236C3×D4—42no—
237C3×D4—42no—
238C3×D4—82no—
239C3×D4—81no—
240C3×D4—81no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—42no—
260C2×Dic3—44no—
261C2×Dic3—44no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—81no—
266C2×Dic3—81no—
267C2×Dic3—81no—
268Dic6—24no—
269Dic6—24no—
270Dic6—42no—
271Dic6—42no—
272Dic6—44no—
273Dic6—44no—
274C3×C8—42no—
275C3⋊C8—42no—
276C24—21no—
277C24D3×D411yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—42no—
281C2×Q8—64no—
282C2×Q8—64no—
283C2×Q8—64no—
284C2×Q8—62no—
285C2×Q8—62no—
286C2×Q8—122no—
287C2×Q8—122no—
288C2×Q8—122no—
289C2×Q8—124no—
290C2×Q8—124no—
291C2×Q8—124no—
292C2×Q8—121no—
293C2×Q8—121no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—41no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—124no—
308C22×C4—121no—
309C22×C4—121no—
310C22×C4—121no—
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4#23711yescharacteristic
314C22×C4C2×#6711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C22×C4C2×#6711yescharacteristic
317SD16—82no—
318SD16—128no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—62no—
322C2×C8—122no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—122no—
330C4⋊C4—128no—
331C22⋊C4—34no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—62no—
337C22⋊C4—122no—
338C42—122no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—162no—
351C22×C3#39211yescharacteristic
352C22×C3#44812yes—
353C22×C3#46112yes—
354C22×C3#49211yescharacteristic
355C22×C3#49711yescharacteristic
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4—81no—
362C3×C4B(4,𝔽2)12yes—
363Dic3—44no—
364Dic3—44no—
365Dic3—161no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23D3×SD1612yes—
373C23#91512yes—
374C23#93711yescharacteristic
375C23#93911yescharacteristic
376C23#94011yescharacteristic
377Q8—42no—
378Q8—64no—
379Q8—64no—
380Q8—81no—
381Q8—122no—
382Q8—122no—
383Q8—124no—
384Q8—124no—
385D4—42no—
386D4—42no—
387D4—81no—
388D4—82no—
389D4—81no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—64no—
403C2×C4—64no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—124no—
408C2×C4—124no—
409C2×C4—122no—
410C2×C4—241no—
411C2×C4—241no—
412C2×C4—241no—
413C2×C4#93711yescharacteristic
414C2×C4#93811yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4#95311yescharacteristic
417C8—42no—
418C8—122no—
419C2×C3—22no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—161no—
424C2×C3#200911yescharacteristic
425C2×C3#201311yescharacteristic
426C2×C3#202111yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—162no—
439C22#516212yes—
440C22#516711yescharacteristic
441C22#517012yes—
442C22#517511yescharacteristic
443C22#522311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—124no—
451C4—124no—
452C4—481no—
453C4#524412yes—
454C3#994911yescharacteristic, complemented, Sylow
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2#7185111yescharacteristic
461C2#7205711yescharacteristic
462C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3187C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#3187D3
hypocenterC3#9949
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
2. #82760C2semidirect
4. #71690C2semidirect
6. #69963C2semidirect
7. #67677C2semidirect
167. #217SD16semidirect
454. C3#9949semidirect

Maximal subgroups

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

subgroupindex
C3×#31872
#827602
C2×#50402
#716902
#716862
#699632
#676772
#99493