Finite Group Databasebeta

← #10697629

Subgroups — #10697629

Order 768.

604 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3208C211yesmaximal, characteristic
2#83367C211yesmaximal, characteristic, complemented
3#83179C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic
6#70766C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10250—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×#936—21no—
12C2×C4×Dic6C2211yescharacteristic
13C3×#492—21no—
14C3×#486—21no—
15C3×#460—22no—
16C3×#450—22no—
17C3×#431C2211yescharacteristic
18#5222—22no—
19#5221—22no—
20#5221—21no—
21#5094—21no—
22#5091—21no—
23#5063—22no—
24#5004—21no—
25#4995—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4544—22no—
30#4538—22no—
31#4484—22no—
32#3208D311yescharacteristic
33#3078—31no—
34#2548—31no—
35#1545—31no—
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C22×Dic6D411yescharacteristic
43C2×C3×SD16—24no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#125—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#217—24no—
58C2×#217—24no—
59#936—42no—
60#936—42no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×Dic3⋊C4—21no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8D412yes—
72C3×#102—24no—
73C3×C22⋊C8—21no—
74C3×C4×C8—21no—
75C3⋊#95D412yes—
76#830—21no—
77#810—22no—
78#810—22no—
79#808—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×#127C2×D311yescharacteristic
87C2×#124—61no—
88C2×C4×Q8—31no—
89#492—21no—
90#486—21no—
91#479—62no—
92#477—61no—
93#475—61no—
94#460—22no—
95#450—22no—
96#431—31no—
97#431C2×D311yescharacteristic
98#428—61no—
99#423—31no—
100#419—61no—
101#394—62no—
102#390—61no—
103#390—62no—
104C4×C4⋊C4—31no—
105#331—62no—
106#328—62no—
107Q8⋊C8—62no—
108C24×C3—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—44no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—82no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6SD1612yescomplemented
135C3×SD16—84no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—48no—
150#217—84no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—42no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159C4×Dic3—41no—
160#208—41no—
161#208—41no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C22×Q8—31no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4#6712yes—
168C2×SD16—24no—
169C2×M16—61no—
170C4⋊Q8#6711yescharacteristic
171C4⋊Q8#6711yescharacteristic
172#127—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#125—21no—
177#124—122no—
178#124—122no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187#117—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—64no—
193C2×C42—61no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#106—62no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8#6712yes—
202#103—62no—
203#103—62no—
204#102—24no—
205C22⋊C8—21no—
206C22⋊C8—61no—
207C4×C8—21no—
208C4×C8—61no—
209#95—32no—
210C23×C3—44no—
211C23×C3—44no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3#12011yescharacteristic
216C3×Q8—42no—
217C3×Q8—81no—
218C3×D4—44no—
219C3×D4—82no—
220C3×D4—82no—
221C3×D4—82no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×C3×C4C2×SD1611yescharacteristic
237C2×C3×C4#13511yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4#13711yescharacteristic
240C2×Dic3—22no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248Dic6—22no—
249Dic6—22no—
250Dic6—41no—
251Dic6—41no—
252C3×C8—42no—
253C3⋊C8—42no—
254C3⋊C8—81no—
255C24—22no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—32no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—82no—
276C22×C4—61no—
277C22×C4—62no—
278C22×C4—61no—
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4C2×#6711yescharacteristic
282SD16—84no—
283M16—121no—
284M16—121no—
285C2×C8—22no—
286C2×C8—62no—
287C2×C8—121no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—122no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—41no—
300C22⋊C4—128no—
301C22⋊C4—244no—
302C42—41no—
303C42—121no—
304C42—121no—
305C42—121no—
306C42—121no—
307C42D3×D411yescharacteristic
308C42#23711yescharacteristic
309C42C2×#6711yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—84no—
314C22×C3—84no—
315C22×C3—82no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3#39211yescharacteristic
319C22×C3#45011yescharacteristic
320C22×C3#45211yescharacteristic
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—81no—
325C3×C4—81no—
326C3×C4#44811yescharacteristic
327C3×C4#44911yescharacteristic
328C3×C4#45512yes—
329C3×C4#46511yescharacteristic
330C3×C4#46811yescharacteristic
331Dic3—42no—
332Dic3—82no—
333Dic3—81no—
334C23—44no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23#93711yescharacteristic
340Q8—42no—
341Q8—62no—
342Q8—62no—
343Q8—81no—
344Q8—121no—
345Q8—121no—
346D4—44no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—62no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—121no—
366C2×C4D3×SD1611yescharacteristic
367C2×C4#91411yescharacteristic
368C2×C4#91511yescharacteristic
369C2×C4#91611yescharacteristic
370C2×C4#93711yescharacteristic
371C2×C4#93811yescharacteristic
372C2×C4#93911yescharacteristic
373C2×C4#93911yescharacteristic
374C2×C4#94011yescharacteristic
375C2×C4#95311yescharacteristic
376C8—42no—
377C8—122no—
378C8—241no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—162no—
382C2×C3#200011yescharacteristic
383C2×C3#200411yescharacteristic
384C2×C3#200811yescharacteristic
385C22—21no—
386C22—21no—
387C22—84no—
388C22—84no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22#516911yescharacteristic
393C22#517311yescharacteristic
394C22#522311yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—81no—
399C4—81no—
400C4—122no—
401C4—242no—
402C4—241no—
403C4#517011yescharacteristic
404C4#517111yescharacteristic
405C4#517211yescharacteristic
406C4#517411yescharacteristic
407C4#524212yes—
408C3#1025011yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—162no—
412C2#7185711yescharacteristic
413C2#7205411yescharacteristic
414C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3208C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#3208D3
hypocenterC3#10250
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
2. #83367C2semidirect
4. #71835C2semidirect
6. #70766C2semidirect
7. #67746C2semidirect
134. C2×Dic6SD16semidirect
408. C3#10250semidirect

Maximal subgroups

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

subgroupindex
C3×#32082
#833672
#831792
#718352
#717592
#707662
#677462
#102503