Finite Group Databasebeta

← #10696542

Subgroups — #10696542

Order 768.

598 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 83 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#83306C211yesmaximal, characteristic
2#83288C211yesmaximal, characteristic, complemented
3#73428C211yesmaximal, characteristic, complemented
4C3×#1505C211yesmaximal, characteristic, complemented
5#70824C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10090—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—22no—
11C2×#875C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#5092—21no—
16#5089—21no—
17#5076—21no—
18#5074—21no—
19#5060—22no—
20#5040—21no—
21#5036—21no—
22#5016—22no—
23#4939—21no—
24#4929—21no—
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4674—22no—
28#4673—22no—
29C3×#327—22no—
30#4481—22no—
31#4464—22no—
32#3213—31no—
33#3152—31no—
34#2640—31no—
35#1558—31no—
36#1527—31no—
37#1505D311yescharacteristic, complemented
38#1300—31no—
39C2×C3×#41—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D412yes—
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#951—48no—
53C2×#216—24no—
54C2×#214—24no—
55C2×#214—22no—
56C2×#213—22no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—24no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×#208—21no—
63#875—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#95—21no—
75C3×#95D412yes—
76C3⋊#95—24no—
77#830—21no—
78#810—22no—
79#809—22no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803D412yes—
83C4×C3⋊C8—21no—
84C2×#137—62no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87C2×#124—31no—
88#497—62no—
89#496—61no—
90#494—61no—
91#492—61no—
92#489—61no—
93#463—62no—
94#448—61no—
95#447—61no—
96#432—31no—
97#432C2×D311yescharacteristic
98C4×M16—31no—
99#397—62no—
100#393—62no—
101#391—61no—
102#390—61no—
103#376C2×D311yescharacteristic
104#333—62no—
105#329—62no—
106#327—22no—
107C23×D3—21no—
108C3×#41—44no—
109C2×C3×Q8—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C22×C3×C4—22no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—22no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—42no—
124C2×Dic6—21no—
125C2×Dic6—22no—
126C2×Dic6—22no—
127C2×Dic6—21no—
128C2×Dic6—44no—
129C2×Dic6—42no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—42no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#216—48no—
141#214—48no—
142#214—82no—
143#213—82no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—84no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153#208—24no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×#41—21no—
157C22×Q8—31no—
158C22×Q8—32no—
159C22×D4—31no—
160#137—128no—
161C2×Q16—64no—
162C2×SD16—64no—
163C2×SD16—62no—
164C2×D8—62no—
165C2×M16—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8D1211yescharacteristic
171#127D1211yescharacteristic
172#124—21no—
173#124—61no—
174#124—61no—
175#124—61no—
176#124—61no—
177#121—21no—
178#121—21no—
179C4×D4—21no—
180C4×D4—21no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—64no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—32no—
190#103—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194#97—21no—
195#97—32no—
196#97—61no—
197C4×C8—61no—
198#95—21no—
199#95—64no—
200#95#6712yes—
201C23×C3—41no—
202C23×C3#12011yescharacteristic
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—41no—
206C22×D3—42no—
207C3×Q8—42no—
208C3×D4—42no—
209C3×D4—42no—
210C3×D4—81no—
211C3×D4—82no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—82no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12711yescharacteristic
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—44no—
230D12—42no—
231D12—81no—
232Dic6—44no—
233Dic6—42no—
234Dic6—82no—
235Dic6—81no—
236C3×C8—81no—
237C3⋊C8—24no—
238C3⋊C8—42no—
239C24—61no—
240#41—44no—
241C2×Q8—21no—
242C2×Q8—61no—
243C2×Q8—62no—
244C2×Q8—61no—
245C2×Q8—62no—
246C2×Q8—122no—
247C2×Q8—124no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—122no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—61no—
258C22×C4—62no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—128no—
263SD16—128no—
264SD16—242no—
265D8—242no—
266M16—64no—
267C2×C8—41no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—244no—
283C42—42no—
284C42C2×D1211yescharacteristic
285C42#23511yescharacteristic
286C42D3×D411yescharacteristic
287C42D3×D411yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—81no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3#39211yescharacteristic
294C22×C3#45312yes—
295C2×D3—84no—
296C2×D3—81no—
297C2×D3—81no—
298C2×D3—81no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—82no—
305C3×C4—81no—
306C3×C4#45512yes—
307Dic3—82no—
308Dic3—81no—
309C23—41no—
310C23—122no—
311C23—122no—
312C23—122no—
313C23—121no—
314C23#93711yescharacteristic
315Q8—42no—
316Q8—124no—
317Q8—122no—
318Q8—242no—
319Q8—241no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—82no—
324D4—122no—
325D4—241no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—82no—
334C2×C4—122no—
335C2×C4—124no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4#88211yescharacteristic
339C2×C4#88211yescharacteristic
340C2×C4#88411yescharacteristic
341C2×C4#88411yescharacteristic
342C2×C4#93011yescharacteristic
343C2×C4#94011yescharacteristic
344C8—64no—
345C8—81no—
346C8—122no—
347C2×C3—22no—
348C2×C3—161no—
349C2×C3#200211yescharacteristic
350C2×C3#200411yescharacteristic
351C2×C3#200911yescharacteristic
352D3—161no—
353C22—21no—
354C22—21no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—244no—
359C22—241no—
360C22—241no—
361C22—241no—
362C22#496011yescharacteristic
363C22#519812yes—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—82no—
370C4—81no—
371C4—242no—
372C4—241no—
373C4#482512yes—
374C3#1009011yescharacteristic, complemented, Sylow
375C2—22no—
376C2—161no—
377C2—481no—
378C2#7005311yescharacteristic
379C2#7032411yescharacteristic
380C2#7038611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#392
prime radical#83306C2
hypercenter#1505D3
hypocenterC3#10090
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
2. #83288C2semidirect
3. #73428C2semidirect
4. C3×#1505C2semidirect
5. #70824C2semidirect
6. #70804C2semidirect
7. #67615C2semidirect
37. #1505D3semidirect
374. C3#10090semidirect

Maximal subgroups

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

subgroupindex
#833062
#832882
#734282
C3×#15052
#708242
#708042
#676152
#100903