Finite Group Databasebeta

← #10697786

Subgroups — #10697786

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic
3C2×#5087C211yesmaximal, characteristic
4C2×#4546C211yesmaximal, characteristic, complemented
5C2×#4547C211yesmaximal, characteristic, complemented
6#69787C211yesmaximal, characteristic
7#67654C211yesmaximal, characteristic, complemented
8#10275—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#487—22no—
13C3×#487—22no—
14C2×C3×#103—22no—
15C2×C3×C4×C8C2211yescharacteristic
16C2×(C2×D3)⋊C8—22no—
17#5087—22no—
18#5087—22no—
19#5084—22no—
20#5084—22no—
21C2×#809—21no—
22C2×#809—21no—
23C2×#808—21no—
24C2×#807—21no—
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4547—22no—
29#4546—22no—
30C2×#487D311yescharacteristic, complemented
31C2×#476—31no—
32C2×#474—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×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C3×#103—28no—
65C3×C4×C8—21no—
66C3×C4×C8—21no—
67(C2×D3)⋊C8—44no—
68#809—24no—
69#809—24no—
70#808—24no—
71#807—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4⋊(C3⋊C8)D412yes—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4×D4—31no—
81#487—22no—
82#487—22no—
83#476—62no—
84#476—62no—
85#474—62no—
86#474—62no—
87C2×#107—61no—
88C2×#106—61no—
89C2×C4⋊C8—31no—
90C2×C4⋊C8—31no—
91C2×C4⋊C8—31no—
92C2×#103—22no—
93C2×#102—61no—
94C2×#102—61no—
95C2×C22⋊C8—62no—
96C2×C4×C8C2×D311yescharacteristic
97#333—62no—
98#332—62no—
99C23×D3—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C2×C3×Q8—82no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12Q1612yes—
119C2×C4×D3—41no—
120C2×C4×D3—41no—
121C2×C4×D3—42no—
122C2×C4×D3—42no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—42no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×D4—31no—
153C23×C4—61no—
154C22×C8—61no—
155C22×C8—61no—
156C22×C8—61no—
157C22×C8#6712yes—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8#6712yes—
165C4⋊Q8#6712yes—
166C4×D4—61no—
167C4×D4—61no—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—31no—
175C2×C22⋊C4—61no—
176C2×C42C22×D311yescharacteristic
177#107—64no—
178#106—64no—
179C4⋊C8—32no—
180C4⋊C8—32no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185#103—28no—
186#102—64no—
187#102—64no—
188C22⋊C8—124no—
189C4×C8—21no—
190C4×C8—21no—
191C23×C3#12111yescharacteristic
192C22×D3—22no—
193C22×D3—41no—
194C22×D3—42no—
195C22×D3—41no—
196C22×D3—42no—
197C3×Q8—44no—
198C3×Q8—82no—
199C3×Q8—82no—
200C3×Q8—81no—
201C3×Q8—81no—
202C2×C3×C4—22no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—82no—
206C2×C3×C4—82no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×C3×C4C2×Q1611yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13512yes—
216C2×C3×C4AGL(1,ℤ/8)12yes—
217C2×C3×C4#13711yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×Dic3—41no—
220C2×Dic3—42no—
221D12—22no—
222D12—22no—
223D12—41no—
224D12—41no—
225C4×D3—82no—
226C4×D3—81no—
227C4×D3—81no—
228C3×C8—42no—
229C3⋊C8—42no—
230C3⋊C8—42no—
231C3⋊C8—81no—
232C24—61no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×D4—32no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—61no—
249C22×C4—121no—
250C22×C4—122no—
251C22×C4—122no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4#23711yescharacteristic
255C22×C4C2×#6711yescharacteristic
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—121no—
262C2×C8—121no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—122no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42D3×D411yescharacteristic
280C42D3×D411yescharacteristic
281C42C2×#6711yescharacteristic
282C42C2×#6711yescharacteristic
283C22×C3#39211yescharacteristic
284C22×C3#46312yes—
285C22×C3#46711yescharacteristic
286C22×C3#47011yescharacteristic
287C22×C3#47112yes—
288C2×D3—44no—
289C2×D3—42no—
290C2×D3—82no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—82no—
296C3×C4—82no—
297C3×C4#44912yes—
298C3×C4#45111yescharacteristic
299C3×C4#45111yescharacteristic
300C3×C4#45211yescharacteristic
301C3×C4#45211yescharacteristic
302C3×C4#49511yescharacteristic
303C3×C4#49611yescharacteristic
304Dic3—81no—
305C23—62no—
306C23—122no—
307C23—121no—
308C23—121no—
309C23—122no—
310C23#93811yescharacteristic
311Q8—44no—
312Q8—82no—
313Q8—82no—
314Q8—81no—
315Q8—81no—
316D4—62no—
317D4—62no—
318D4—121no—
319D4—121no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—121no—
326C2×C4—122no—
327C2×C4—242no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4D3×Q1611yescharacteristic
331C2×C4#91811yescharacteristic
332C2×C4#91811yescharacteristic
333C2×C4#91911yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#94011yescharacteristic
339C2×C4#94912yes—
340C2×C4#95012yes—
341C2×C4#95311yescharacteristic
342C8—42no—
343C8—122no—
344C8—122no—
345C8—241no—
346C2×C3#168312yes—
347C2×C3#170312yes—
348C2×C3#200111yescharacteristic
349C2×C3#201211yescharacteristic
350C2×C3#201911yescharacteristic
351D3—82no—
352C22—124no—
353C22—122no—
354C22—241no—
355C22—242no—
356C22—241no—
357C22#505712yes—
358C22#517712yes—
359C22#518811yescharacteristic
360C22#518911yescharacteristic
361C22#522311yescharacteristic
362C4—21no—
363C4—21no—
364C4—82no—
365C4—82no—
366C4—241no—
367C4#505811yescharacteristic
368C4#505911yescharacteristic
369C4#518611yescharacteristic
370C4#518611yescharacteristic
371C4#518711yescharacteristic
372C4#518711yescharacteristic
373C4#523812yes—
374C3#1027511yescharacteristic, complemented, Sylow
375C2—242no—
376C2#7123312yes—
377C2#7123512yes—
378C2#7172811yescharacteristic
379C2#7205911yescharacteristic
380C2#7207111yescharacteristic
subgroupG/H
centerC23#938
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC23×C3#121
prime radicalC22×D12D4
hypercenterC2×#487D3
hypocenterC3#10275
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
1. C2×C3×#487C2semidirect
4. C2×#4546C2semidirect
5. C2×#4547C2semidirect
7. #67654C2semidirect
30. C2×#487D3semidirect
374. C3#10275semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×#50842
C2×#50872
C2×#45462
C2×#45472
#697872
#676542
#102753