Finite Group Databasebeta

← #10696436

Subgroups — #10696436

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#498C211yesmaximal, characteristic, complemented
2#83240C211yesmaximal, characteristic
3C2×#5087C211yesmaximal, characteristic
4#71778C211yesmaximal, characteristic, complemented
5C2×#4547C211yesmaximal, characteristic, complemented
6C2×#4459C211yesmaximal, characteristic
7#67655C211yesmaximal, characteristic, complemented
8#10074—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#947—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#498—22no—
14C3×#498—22no—
15C2×C3×#106—22no—
16C2×C3×C4×C8C2211yescharacteristic
17#5229—21no—
18C2×#835—22no—
19#5087—22no—
20#5087—22no—
21#4992—22no—
22#4987—22no—
23C2×#809—21no—
24C2×#807—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4547—22no—
29#4459—22no—
30C2×#498D311yescharacteristic, complemented
31C2×#474—31no—
32#2547—31no—
33#1981—31no—
34C2×#332—31no—
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
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⋊C4D412yes—
54C2×C3×C42C2311yescharacteristic
55#947—42no—
56#947—42no—
57C2×(C2×D3)⋊C4—24no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3D412yes—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4×Dic3—21no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C3×#106—28no—
69C3×C4×C8—21no—
70C3×C4×C8D412yes—
71#835—44no—
72C3⋊#95—24no—
73C3⋊#95—21no—
74#809—24no—
75#807—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×#124—61no—
83C2×C4×D4—31no—
84#498—22no—
85#498—22no—
86#474—62no—
87#474—62no—
88C2×#107—61no—
89C2×#106—22no—
90C2×C4⋊C8—31no—
91C2×C4⋊C8—31no—
92C2×#103—62no—
93C2×#102—61no—
94C2×C4×C8C2×D311yescharacteristic
95#397—62no—
96#391—61no—
97#380—62no—
98C4×C4⋊C4—31no—
99#332—62no—
100D4⋊C8—62no—
101C23×D3—21no—
102C2×C3×Q8—22no—
103C2×C3×Q8—41no—
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—42no—
111C2×C3×Q8—42no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—22no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12SD1612yes—
125C2×C4×D3—41no—
126C2×C4×D3—41no—
127C2×C4×D3—42no—
128C2×C4×D3—42no—
129C2×C3×C8—24no—
130C2×C3×C8—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143(C2×D3)⋊C4—48no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—22no—
149C4⋊Dic3—22no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—42no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×D4—31no—
160C23×C4—61no—
161C22×C8—21no—
162C22×C8—61no—
163C22×C8—61no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8#6712yes—
172#124—122no—
173#124—122no—
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—31no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4#6712yes—
184C2×C22⋊C4—64no—
185C2×C22⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—64no—
189#106—28no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194#103—124no—
195#102—64no—
196C4×C8—21no—
197C4×C8#6712yes—
198#95—64no—
199#95—61no—
200C23×C3#12011yescharacteristic
201C22×D3—22no—
202C22×D3—41no—
203C22×D3—42no—
204C22×D3—41no—
205C22×D3—42no—
206C3×Q8—44no—
207C3×Q8—81no—
208C3×Q8—81no—
209C3×Q8—81no—
210C3×Q8—81no—
211C3×Q8—81no—
212C3×Q8—81no—
213C2×C3×C4—22no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—82no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×C3×C4C2×SD1611yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4#13512yes—
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)12yes—
230C2×C3×C4#13711yescharacteristic
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236D12—22no—
237D12—22no—
238D12—41no—
239D12—41no—
240C4×D3—82no—
241C4×D3—81no—
242C4×D3—81no—
243C3×C8—24no—
244C3⋊C8—42no—
245C3⋊C8—81no—
246C24—61no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×D4—32no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—61no—
266C22×C4—62no—
267C22×C4—122no—
268C22×C4—122no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4C2×#6711yescharacteristic
274C2×C8—24no—
275C2×C8—22no—
276C2×C8—62no—
277C2×C8—121no—
278C2×C8—121no—
279C2×C8—122no—
280C2×C8—121no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C22⋊C4—128no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—121no—
297C42—121no—
298C42D3×D411yescharacteristic
299C42#23711yescharacteristic
300C42C2×#6711yescharacteristic
301C42C2×#6711yescharacteristic
302C22×C3#39211yescharacteristic
303C22×C3#44912yes—
304C22×C3#45011yescharacteristic
305C22×C3#45211yescharacteristic
306C22×C3#45612yes—
307C2×D3—44no—
308C2×D3—42no—
309C2×D3—82no—
310C2×D3—81no—
311C2×D3—81no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4—82no—
316C3×C4#44811yescharacteristic
317C3×C4#44911yescharacteristic
318C3×C4#46312yes—
319C3×C4#46511yescharacteristic
320C3×C4#46811yescharacteristic
321C3×C4#49511yescharacteristic
322C3×C4#49611yescharacteristic
323Dic3—82no—
324Dic3—81no—
325C23—62no—
326C23—122no—
327C23—121no—
328C23—121no—
329C23—122no—
330C23#93711yescharacteristic
331Q8—44no—
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338D4—62no—
339D4—62no—
340D4—121no—
341D4—121no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—122no—
353C2×C4—242no—
354C2×C4—241no—
355C2×C4—241no—
356C2×C4D3×SD1611yescharacteristic
357C2×C4#91412yes—
358C2×C4#91411yescharacteristic
359C2×C4#91511yescharacteristic
360C2×C4#91612yes—
361C2×C4#91611yescharacteristic
362C2×C4#93711yescharacteristic
363C2×C4#93811yescharacteristic
364C2×C4#93911yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4#94011yescharacteristic
367C2×C4#95311yescharacteristic
368C8—24no—
369C8—122no—
370C8—241no—
371C2×C3#165212yes—
372C2×C3#165412yes—
373C2×C3#200011yescharacteristic
374C2×C3#200511yescharacteristic
375C2×C3#201211yescharacteristic
376D3—82no—
377C22—122no—
378C22—124no—
379C22—241no—
380C22—241no—
381C22—242no—
382C22#509512yes—
383C22#516911yescharacteristic
384C22#517212yes—
385C22#517311yescharacteristic
386C22#522311yescharacteristic
387C4—21no—
388C4—21no—
389C4—82no—
390C4—82no—
391C4—241no—
392C4—242no—
393C4#516512yes—
394C4#516811yescharacteristic
395C4#517011yescharacteristic
396C4#517111yescharacteristic
397C4#517211yescharacteristic
398C4#517411yescharacteristic
399C4#517611yescharacteristic
400C3#1007411yescharacteristic, complemented, Sylow
401C2—242no—
402C2#7134312yes—
403C2#7134612yes—
404C2#7179511yescharacteristic
405C2#7205411yescharacteristic
406C2#7205511yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#498D3
hypocenterC3#10074
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. #71778C2semidirect
5. C2×#4547C2semidirect
7. #67655C2semidirect
30. C2×#498D3semidirect
400. C3#10074semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4982
#832402
C2×#50872
#717782
C2×#45472
C2×#44592
#676552
#100743