Finite Group Databasebeta

← #10697633

Subgroups — #10697633

Order 768.

604 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#83179C211yesmaximal, characteristic
2#83169C211yesmaximal, characteristic
3#79068C211yesmaximal, characteristic, complemented
4C3×#1545C211yesmaximal, characteristic
5#70792C211yesmaximal, characteristic, complemented
6#70767C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10250—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#936—21no—
11#5470C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5222—22no—
16#5221—22no—
17#5221—21no—
18#5218—22no—
19#5121—21no—
20#5120—21no—
21#5076—21no—
22#5075—22no—
23#5070—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4851—21no—
27#4850—21no—
28#4829—22no—
29C3×#331—22no—
30#4467—22no—
31#4456—22no—
32#3208—31no—
33#3078—31no—
34#2548—31no—
35#1545D311yescharacteristic
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#217—24no—
50C2×#217—24no—
51#936—42no—
52#936—42no—
53C2×#214—24no—
54#926—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57#924—21no—
58#924—21no—
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76#832—24no—
77C3⋊#95D412yes—
78#830—21no—
79#818—22no—
80#817—22no—
81#816—22no—
82#816—22no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×#127C2×D311yescharacteristic
87C2×#124—61no—
88C2×C4×Q8—31no—
89#492—61no—
90#486—61no—
91#479—62no—
92#477—61no—
93#475—61no—
94#460—62no—
95#450—62no—
96#431—31no—
97#431C2×D311yescharacteristic
98#428—61no—
99#423C2×D311yescharacteristic
100#419—61no—
101#394—62no—
102#390—61no—
103#390—62no—
104C4×C4⋊C4—31no—
105#331—22no—
106#328—62no—
107Q8⋊C8—62no—
108C24×C3—22no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—44no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
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—82no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—22no—
126C22×Dic3—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6SD1612yescomplemented
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—48no—
144#217—41no—
145#217—84no—
146#214—84no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—42no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C2×C3⋊C8—22no—
164C22×Q8—31no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4D1212yes—
168C2×SD16—64no—
169C2×M16—21no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172#127—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#125—61no—
177#124—122no—
178#124—122no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187#117—61no—
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—21no—
199C4⋊C8—21no—
200C4⋊C8—32no—
201C4⋊C8#6712yes—
202#103—62no—
203#103—62no—
204#102—64no—
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×D4—44no—
217C3×D4—82no—
218C3×D4—82no—
219C3×D4—82no—
220C3×D4—81no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×SD1611yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246Dic6—22no—
247Dic6—22no—
248Dic6—41no—
249Dic6—41no—
250Dic6—42no—
251Dic6—81no—
252C3×C8—42no—
253C3×C8—81no—
254C3⋊C8—42no—
255C24—22no—
256C2×Q8—32no—
257C2×Q8—61no—
258C2×Q8—61no—
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×C4C2×D1211yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282SD16—244no—
283M16—41no—
284M16—41no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—62no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—122no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—128no—
300C22⋊C4—121no—
301C22⋊C4—244no—
302C42—121no—
303C42—121no—
304C42—121no—
305C42—121no—
306C42—121no—
307C42#23511yescharacteristic
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
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#44811yescharacteristic
325C3×C4#44911yescharacteristic
326C3×C4#45512yes—
327C3×C4#46511yescharacteristic
328C3×C4#46811yescharacteristic
329Dic3—42no—
330Dic3—82no—
331Dic3—81no—
332Dic3—81no—
333Dic3—81no—
334C23—44no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23#88211yescharacteristic
340Q8—62no—
341Q8—62no—
342Q8—121no—
343Q8—121no—
344Q8—122no—
345Q8—241no—
346D4—44no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4#88211yescharacteristic
367C2×C4#88411yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4#91111yescharacteristic
370C2×C4#91211yescharacteristic
371C2×C4D3×SD1611yescharacteristic
372C2×C4#91511yescharacteristic
373C2×C4#93011yescharacteristic
374C2×C4#93711yescharacteristic
375C2×C4#94011yescharacteristic
376C8—42no—
377C8—81no—
378C8—122no—
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#477611yescharacteristic
393C22#478311yescharacteristic
394C22#496011yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—122no—
399C4—242no—
400C4—241no—
401C4—241no—
402C4—241no—
403C4#474612yes—
404C4#477811yescharacteristic
405C4#478011yescharacteristic
406C4#515811yescharacteristic
407C4#517011yescharacteristic
408C3#1025011yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—162no—
412C2#7008411yescharacteristic
413C2#7020511yescharacteristic
414C2#7033111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1545C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#1545D3
hypocenterC3#10250
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
3. #79068C2semidirect
5. #70792C2semidirect
6. #70767C2semidirect
7. #67715C2semidirect
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
#831792
#831692
#790682
C3×#15452
#707922
#707672
#677152
#102503