Finite Group Databasebeta

← #10697631

Subgroups — #10697631

Order 768.

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

normal subgroups only

This group has 604 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×#2548C211yesmaximal, characteristic
2#83169C211yesmaximal, characteristic
3#82431C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic, complemented
6#69471C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10250—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×C4×Q8C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#394—22no—
14C3×#390—22no—
15C3×#390—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17#5225—21no—
18#5218—22no—
19#5076—21no—
20#5075—22no—
21#5070—21no—
22#5022—22no—
23#5021—21no—
24#5020—21no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4544—22no—
30#4535—22no—
31#4452—22no—
32#3208—31no—
33#3078—31no—
34#2548D311yescharacteristic
35#1545—31no—
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D412yes—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#124—42no—
48C3×#124—42no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C42—21no—
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#214—24no—
62#926—21no—
63#924—21no—
64#924—21no—
65C2×#208—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#95D412yes—
72#835—22no—
73#835—22no—
74#832—24no—
75#830—21no—
76#830—21no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×#127C2×D311yescharacteristic
87C2×#124—21no—
88C2×C4×Q8C2×D311yescharacteristic
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#431—31no—
98#428—61no—
99#423—31no—
100#419—61no—
101#394—22no—
102#390—22no—
103#390—21no—
104C4×C4⋊C4C2×D311yescharacteristic
105#331—62no—
106#328—62no—
107Q8⋊C8—62no—
108C24×C3—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8SD1612yescomplemented
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—44no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—82no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C2×Dic6—21no—
134C2×Dic6—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—48no—
141C3×C22⋊C4—84no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—41no—
151#214—84no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C4×Dic3—41no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C22×Q8#6711yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4#6712yes—
168C2×SD16—64no—
169C2×M16—61no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172#127—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#125—61no—
177#124—42no—
178#124—42no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—61no—
186C4×Q8—61no—
187#117—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#106—62no—
198C4⋊C8—32no—
199C4⋊C8—32no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202#103—62no—
203#103—62no—
204#102—64no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C4×C8—61no—
208C4×C8—61no—
209#95#6712yes—
210C23×C3—44no—
211C23×C3—44no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3#12011yescharacteristic
216C3×Q8—22no—
217C3×Q8—22no—
218C3×Q8—41no—
219C3×Q8—41no—
220C3×D4—44no—
221C3×D4—82no—
222C3×D4—82no—
223C3×D4—82no—
224C3×D4—81no—
225C3×D4—81no—
226C3×D4—81no—
227C3×D4—81no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×C3×C4C2×SD1611yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4#13711yescharacteristic
248C2×Dic3—41no—
249C2×Dic3—41no—
250Dic6—42no—
251Dic6—81no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C3⋊C8—81no—
255C24—22no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8#21412yescomplemented
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—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4C2×#6711yescharacteristic
280C22×C4C2×#6711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282SD16—244no—
283M16—121no—
284M16—121no—
285C2×C8—62no—
286C2×C8—62no—
287C2×C8—121no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—48no—
300C22⋊C4—84no—
301C22⋊C4—121no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—121no—
307C42D3×D411yescharacteristic
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42#23711yescharacteristic
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—42no—
325C3×C4—82no—
326C3×C4—81no—
327C3×C4#44811yescharacteristic
328C3×C4#44911yescharacteristic
329C3×C4#45512yes—
330C3×C4#46511yescharacteristic
331C3×C4#46811yescharacteristic
332Dic3—81no—
333Dic3—81no—
334C23—44no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23#95311yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
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—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4#93211yescharacteristic
367C2×C4C2×#21411yescharacteristic
368C2×C4#93711yescharacteristic
369C2×C4#93711yescharacteristic
370C2×C4#93811yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#93911yescharacteristic
373C2×C4#94011yescharacteristic
374C2×C4#95011yescharacteristic
375C2×C4#95111yescharacteristic
376C8—122no—
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#521811yescharacteristic
393C22#522311yescharacteristic
394C22#523911yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—82no—
400C4—81no—
401C4—241no—
402C4—241no—
403C4#503711yescharacteristic
404C4#504011yescharacteristic
405C4#504111yescharacteristic
406C4#504211yescharacteristic
407C4#506112yes—
408C3#1025011yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—162no—
412C2#7168511yescharacteristic
413C2#7173111yescharacteristic
414C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2548C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenter#2548D3
hypocenterC3#10250
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
3. #82431C2semidirect
5. #71621C2semidirect
6. #69471C2semidirect
7. #67620C2semidirect
114. C2×C3×Q8SD16semidirect
263. C2×Q8#214semidirect
408. C3#10250semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
#831692
#824312
#717592
#716212
#694712
#676202
#102503