Finite Group Databasebeta

← #10697760

Subgroups — #10697760

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#498C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic
3C2×#5083C211yesmaximal, characteristic
4C2×#4546C211yesmaximal, characteristic, complemented
5C2×#4547C211yesmaximal, characteristic, complemented
6#69787C211yesmaximal, characteristic
7#67654C211yesmaximal, characteristic, complemented
8#10271—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C22×#215—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#498—22no—
14C3×#498—22no—
15C2×C3×#106—22no—
16C2×C3×C4×C8C2211yescharacteristic
17C2×#835—21no—
18C2×(C2×D3)⋊C8—22no—
19#5084—22no—
20#5084—22no—
21#5083—22no—
22#5083—22no—
23C2×#809—21no—
24C2×#808—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4547—22no—
29#4546—22no—
30C2×#498D311yescharacteristic, complemented
31C2×#476—31no—
32C2×#461—31no—
33#1954—31no—
34C2×#333—31no—
35C2×#332—31no—
36#1277—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×C3×C8D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D412yes—
49C3×C4⋊Q8D412yes—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#215—24no—
56C2×#215—42no—
57C2×#215—42no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3D411yescharacteristic
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62C22×C3⋊C8—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C3×#106—28no—
68C3×C4×C8—21no—
69C3×C4×C8—21no—
70#835—24no—
71(C2×D3)⋊C8—44no—
72#809—24no—
73#808—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79C4⋊(C3⋊C8)D412yes—
80C22×SD16—61no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×C4⋊Q8C2×D311yescharacteristic
83C2×C4×D4—31no—
84#498—22no—
85#498—22no—
86#476—62no—
87#476—62no—
88#461—62no—
89#461—62no—
90C2×#106—22no—
91C2×#106—61no—
92C2×C4⋊C8—31no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8—31no—
95C2×#103—61no—
96C2×#102—61no—
97C2×C22⋊C8—62no—
98C2×C4×C8C2×D311yescharacteristic
99#333—62no—
100#332—62no—
101C23×D3—21no—
102C2×C3×Q8—22no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×Dic3—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12SD1612yes—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C2×C4×D3—42no—
126C2×C4×D3—42no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8SD1614yescomplemented
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142#215—48no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—42no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×D4—31no—
158C23×C4—61no—
159C2×SD16—64no—
160C2×SD16—122no—
161C2×SD16—122no—
162C22×C8—61no—
163C22×C8—61no—
164C22×C8—61no—
165C22×C8#6712yes—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8#6712yes—
173C4⋊Q8#6712yes—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C22⋊C4—61no—
184C2×C42C22×D311yescharacteristic
185#106—28no—
186#106—64no—
187C4⋊C8—32no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—64no—
194#102—64no—
195C22⋊C8—124no—
196C4×C8—21no—
197C4×C8—21no—
198C23×C3#12111yescharacteristic
199C22×D3—22no—
200C22×D3—41no—
201C22×D3—42no—
202C22×D3—41no—
203C22×D3—42no—
204C3×Q8—42no—
205C3×Q8—42no—
206C3×Q8—82no—
207C3×Q8—82no—
208C3×Q8—81no—
209C3×Q8—81no—
210C2×C3×C4—22no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4—82no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×C3×C4C2×SD1611yescharacteristic
223C2×C3×C4C2×SD1612yes—
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4#13711yescharacteristic
227C2×C3×C4#13712yes—
228C2×Dic3—41no—
229C2×Dic3—42no—
230D12—22no—
231D12—22no—
232D12—41no—
233D12—41no—
234C4×D3—82no—
235C4×D3—81no—
236C4×D3—81no—
237C3×C8—24no—
238C3⋊C8—42no—
239C3⋊C8—42no—
240C3⋊C8—81no—
241C24—61no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×D4—32no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—61no—
260C22×C4—121no—
261C22×C4—122no—
262C22×C4—122no—
263C22×C4—121no—
264C22×C4D3×D411yescharacteristic
265C22×C4#23711yescharacteristic
266C22×C4C2×#6711yescharacteristic
267SD16—128no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—62no—
271C2×C8—62no—
272C2×C8—121no—
273C2×C8—121no—
274C2×C8—121no—
275C2×C8—121no—
276C2×C8—121no—
277C2×C8—122no—
278C2×C8#21514yescomplemented
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42D3×D411yescharacteristic
292C42D3×D411yescharacteristic
293C42C2×#6711yescharacteristic
294C42C2×#6711yescharacteristic
295C22×C3#39211yescharacteristic
296C22×C3#46112yes—
297C22×C3#46511yescharacteristic
298C22×C3#46811yescharacteristic
299C22×C3#47112yes—
300C2×D3—44no—
301C2×D3—42no—
302C2×D3—82no—
303C2×D3—81no—
304C2×D3—81no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4—82no—
309C3×C4#44811yescharacteristic
310C3×C4#44812yes—
311C3×C4#44911yescharacteristic
312C3×C4#45011yescharacteristic
313C3×C4#45211yescharacteristic
314C3×C4#49211yescharacteristic
315C3×C4#49711yescharacteristic
316Dic3—81no—
317C23—62no—
318C23—121no—
319C23—122no—
320C23—121no—
321C23—122no—
322C23#93811yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—82no—
326Q8—82no—
327Q8—81no—
328Q8—81no—
329D4—62no—
330D4—62no—
331D4—121no—
332D4—121no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—121no—
340C2×C4—122no—
341C2×C4—242no—
342C2×C4—241no—
343C2×C4—241no—
344C2×C4D3×SD1611yescharacteristic
345C2×C4#91411yescharacteristic
346C2×C4#91511yescharacteristic
347C2×C4#91611yescharacteristic
348C2×C4#93711yescharacteristic
349C2×C4#93711yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4#93911yescharacteristic
352C2×C4#94011yescharacteristic
353C2×C4C2×#21512yes—
354C2×C4#94212yes—
355C2×C4#95311yescharacteristic
356C8—24no—
357C8—122no—
358C8—122no—
359C8—241no—
360C2×C3#167112yes—
361C2×C3#169112yes—
362C2×C3#200011yescharacteristic
363C2×C3#201311yescharacteristic
364C2×C3#201911yescharacteristic
365D3—82no—
366C22—122no—
367C22—124no—
368C22—241no—
369C22—241no—
370C22—242no—
371C22#508312yes—
372C22#517111yescharacteristic
373C22#517411yescharacteristic
374C22#517712yes—
375C22#522311yescharacteristic
376C4—21no—
377C4—21no—
378C4—82no—
379C4—82no—
380C4—241no—
381C4#508511yescharacteristic
382C4#508611yescharacteristic
383C4#516911yescharacteristic
384C4#517011yescharacteristic
385C4#517211yescharacteristic
386C4#517311yescharacteristic
387C4#522712yes—
388C3#1027111yescharacteristic, complemented, Sylow
389C2—242no—
390C2#7134412yes—
391C2#7134512yes—
392C2#7179111yescharacteristic
393C2#7205411yescharacteristic
394C2#7205911yescharacteristic
subgroupG/H
centerC23#938
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socleC23×C3#121
prime radicalC22×D12D4
hypercenterC2×#498D3
hypocenterC3#10271
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
1. C2×C3×#498C2semidirect
4. C2×#4546C2semidirect
5. C2×#4547C2semidirect
7. #67654C2semidirect
30. C2×#498D3semidirect
129. C2×C3×C8SD16semidirect
278. C2×C8#215semidirect
388. C3#10271semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4982
C2×#50842
C2×#50832
C2×#45462
C2×#45472
#697872
#676542
#102713