Finite Group Databasebeta

← #10696422

Subgroups — #10696422

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic, complemented
2#83260C211yesmaximal, characteristic
3C2×#5087C211yesmaximal, characteristic
4#71778C211yesmaximal, characteristic, complemented
5C2×#4546C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic
7#67655C211yesmaximal, characteristic, complemented
8#10072—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
16#5231—22no—
17C2×#835—22no—
18#5087—22no—
19#5087—22no—
20#4988—21no—
21#4987—22no—
22#4977—21no—
23C2×#809—21no—
24C2×#807—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4546—22no—
29#4459—22no—
30C2×#487D311yescharacteristic, complemented
31C2×#474—31no—
32#2563—31no—
33#1981—31no—
34C2×#333—31no—
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8D412yes—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×C3×C8—21no—
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⋊Q8—21no—
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×(C2×D3)⋊C4—24no—
57C2×C4⋊Dic3D412yes—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4×Dic3—21no—
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C3×#103—28no—
67C3×C4×C8—21no—
68C3×C4×C8D412yes—
69#835—44no—
70C3⋊#95—24no—
71C3⋊#95—21no—
72#809—24no—
73#807—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4×D4—31no—
81#487—22no—
82#487—22no—
83#474—62no—
84#474—62no—
85C2×#107—61no—
86C2×C4⋊C8—31no—
87C2×C4⋊C8—31no—
88C2×#103—22no—
89C2×#103—62no—
90C2×#102—61no—
91C2×C4×C8C2×D311yescharacteristic
92#393—62no—
93#390—61no—
94#382—61no—
95#380—62no—
96C4×C4⋊C4—31no—
97#333—62no—
98D4⋊C8—62no—
99C23×D3—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—44no—
104C2×C3×Q8—41no—
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
107C22×C3×C4—22no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—22no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12Q1612yes—
120C2×C4×D3—41no—
121C2×C4×D3—41no—
122C2×C4×D3—42no—
123C2×C4×D3—42no—
124C2×C3×C8—22no—
125C2×C3×C8—42no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
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×C42#4111yescharacteristic
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—84no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—42no—
152C22×Q8—21no—
153C22×Q8#6712yes—
154C22×D4—31no—
155C23×C4—61no—
156C22×C8—21no—
157C22×C8—61no—
158C22×C8—61no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8#6712yes—
167C4×D4—61no—
168C4×D4—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—32no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—64no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181#107—64no—
182C4⋊C8—32no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186#103—28no—
187#103—124no—
188#102—64no—
189C4×C8—21no—
190C4×C8#6712yes—
191#95—64no—
192#95—61no—
193C23×C3#12011yescharacteristic
194C22×D3—22no—
195C22×D3—41no—
196C22×D3—42no—
197C22×D3—41no—
198C22×D3—42no—
199C3×Q8—44no—
200C3×Q8—44no—
201C3×Q8—82no—
202C3×Q8—81no—
203C3×Q8—81no—
204C2×C3×C4—22no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—44no—
208C2×C3×C4—82no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12711yescharacteristic
215C2×C3×C4C2×Q1611yescharacteristic
216C2×C3×C4#13512yes—
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4AGL(1,ℤ/8)12yes—
219C2×C3×C4#13711yescharacteristic
220C2×C3×C4#13711yescharacteristic
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226D12—22no—
227D12—22no—
228D12—41no—
229D12—41no—
230C4×D3—82no—
231C4×D3—81no—
232C4×D3—81no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—81no—
236C24—61no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×D4—32no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—61no—
253C22×C4—62no—
254C22×C4—122no—
255C22×C4—122no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—122no—
267C2×C8—121no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—62no—
277C4⋊C4—62no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—244no—
283C42—121no—
284C42—121no—
285C42D3×D411yescharacteristic
286C42#23711yescharacteristic
287C42C2×#6711yescharacteristic
288C42C2×#6711yescharacteristic
289C22×C3#39211yescharacteristic
290C22×C3#44912yes—
291C22×C3#45111yescharacteristic
292C22×C3#45211yescharacteristic
293C22×C3#45512yes—
294C2×D3—44no—
295C2×D3—42no—
296C2×D3—82no—
297C2×D3—81no—
298C2×D3—81no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4—82no—
303C3×C4#45111yescharacteristic
304C3×C4#45211yescharacteristic
305C3×C4#46312yes—
306C3×C4#46711yescharacteristic
307C3×C4#47011yescharacteristic
308C3×C4#49511yescharacteristic
309C3×C4#49611yescharacteristic
310Dic3—82no—
311Dic3—81no—
312C23—62no—
313C23—122no—
314C23—121no—
315C23—121no—
316C23—122no—
317C23#93711yescharacteristic
318Q8—44no—
319Q8—44no—
320Q8—82no—
321Q8—81no—
322Q8—81no—
323D4—62no—
324D4—62no—
325D4—121no—
326D4—121no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—82no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—242no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4#91412yes—
341C2×C4#91612yes—
342C2×C4D3×Q1611yescharacteristic
343C2×C4#91811yescharacteristic
344C2×C4#91811yescharacteristic
345C2×C4#91911yescharacteristic
346C2×C4#93711yescharacteristic
347C2×C4#93811yescharacteristic
348C2×C4#93911yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#94011yescharacteristic
351C2×C4#95311yescharacteristic
352C8—42no—
353C8—122no—
354C8—241no—
355C2×C3#164412yes—
356C2×C3#164612yes—
357C2×C3#200111yescharacteristic
358C2×C3#200411yescharacteristic
359C2×C3#201211yescharacteristic
360D3—82no—
361C22—124no—
362C22—122no—
363C22—241no—
364C22—242no—
365C22—241no—
366C22#506112yes—
367C22#517212yes—
368C22#518611yescharacteristic
369C22#518711yescharacteristic
370C22#522311yescharacteristic
371C4—21no—
372C4—21no—
373C4—82no—
374C4—82no—
375C4—242no—
376C4—241no—
377C4#516512yes—
378C4#516811yescharacteristic
379C4#517611yescharacteristic
380C4#518611yescharacteristic
381C4#518711yescharacteristic
382C4#518811yescharacteristic
383C4#518911yescharacteristic
384C3#1007211yescharacteristic, complemented, Sylow
385C2—242no—
386C2#7123112yes—
387C2#7123812yes—
388C2#7173111yescharacteristic
389C2#7205511yescharacteristic
390C2#7207111yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#487D3
hypocenterC3#10072
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. #71778C2semidirect
5. C2×#4546C2semidirect
7. #67655C2semidirect
30. C2×#487D3semidirect
384. C3#10072semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
#832602
C2×#50872
#717782
C2×#45462
C2×#44592
#676552
#100723