Finite Group Databasebeta

← #10697716

Subgroups — #10697716

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5083C211yesmaximal, characteristic
2C2×#5078C211yesmaximal, characteristic, complemented
3C2×#4841C211yesmaximal, characteristic
4C2×C3×#332C211yesmaximal, characteristic, complemented
5#70470C211yesmaximal, characteristic
6C2×#4486C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#10264—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C22×#204—21no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×#835—21no—
17C2×(C2×D3)⋊C8—22no—
18C2×#816—21no—
19#5083—22no—
20#5083—22no—
21#5078—22no—
22#5078—22no—
23C2×#807—22no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4841—22no—
27#4841—22no—
28C3×#332—22no—
29#4486—22no—
30C2×#500—31no—
31C2×#461—31no—
32C2×#461—31no—
33#1954—31no—
34C2×#332—31no—
35C2×#332D311yescharacteristic, complemented
36#1277—31no—
37C22×C3×Q8—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—24no—
51C2×#215—42no—
52C2×#215—42no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C22×C3⋊C8—21no—
58C22×C3⋊C8D412yes—
59C2×#204—24no—
60C2×#204—42no—
61C2×#204—42no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74#835—24no—
75(C2×D3)⋊C8—44no—
76#816—24no—
77#807—28no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C22×SD16—61no—
83C22×SD16—61no—
84C2×C4⋊Q8—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86C2×C4×D4—31no—
87#500—62no—
88#500—62no—
89#461—62no—
90#461—62no—
91#461—62no—
92#461—62no—
93C2×#107—62no—
94C2×C4⋊C8—31no—
95C2×C4⋊C8C2×D311yescharacteristic
96C2×C4⋊C8C2×D311yescharacteristic
97C2×#103—61no—
98C2×#103—61no—
99C2×C22⋊C8—62no—
100C2×C4×C8—31no—
101#332—22no—
102#332—62no—
103C23×D3—21no—
104C2×C3×Q8—22no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—41no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12D812yes—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—42no—
125C2×C4×D3—42no—
126C2×Dic6—22no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6—41no—
130C2×Dic6—41no—
131C2×Dic6—41no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—42no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145#215—48no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153C4⋊Dic3—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8SD1614yescomplemented
160#204—48no—
161C22×Q8—21no—
162C22×Q8—61no—
163C22×D4—31no—
164C23×C4—61no—
165C2×SD16—64no—
166C2×SD16—64no—
167C2×SD16—122no—
168C2×SD16—122no—
169C2×SD16—122no—
170C2×SD16—122no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—32no—
174C22×C8—61no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—32no—
179C4⋊Q8—61no—
180C4⋊Q8—61no—
181C4⋊Q8—61no—
182C4⋊Q8D1212yes—
183C4×D4—61no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42C22×D311yescharacteristic
194#107—68no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200C4⋊C8#6712yes—
201#103—64no—
202#103—64no—
203C22⋊C8—124no—
204C4×C8—61no—
205C4×C8—61no—
206C23×C3#12111yescharacteristic
207C22×D3—22no—
208C22×D3—41no—
209C22×D3—42no—
210C22×D3—41no—
211C22×D3—42no—
212C3×Q8—42no—
213C3×Q8—42no—
214C3×Q8—81no—
215C3×Q8—81no—
216C2×C3×C4—22no—
217C2×C3×C4—42no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12711yescharacteristic
226C2×C3×C4C2×D811yescharacteristic
227C2×C3×C4C2×SD1612yes—
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4#13712yes—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237D12—22no—
238D12—22no—
239D12—41no—
240D12—41no—
241C4×D3—82no—
242C4×D3—81no—
243C4×D3—81no—
244Dic6—42no—
245Dic6—42no—
246Dic6—81no—
247Dic6—81no—
248C3×C8—42no—
249C3×C8—81no—
250C3⋊C8—24no—
251C3⋊C8—42no—
252C24—61no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—62no—
260C2×Q8—121no—
261C2×Q8—121no—
262C2×Q8—121no—
263C2×Q8—121no—
264C2×Q8—121no—
265C2×D4—32no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C22×C4—41no—
272C22×C4—61no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—122no—
276C22×C4—122no—
277C22×C4—121no—
278C22×C4#23511yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281SD16—128no—
282SD16—128no—
283C2×C8—22no—
284C2×C8—34no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—62no—
292C2×C8—121no—
293C2×C8—121no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—61no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—61no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C42C2×D1211yescharacteristic
307C42D3×D411yescharacteristic
308C42D3×D411yescharacteristic
309C42C2×#6711yescharacteristic
310C22×C3#39211yescharacteristic
311C22×C3#46112yes—
312C22×C3#46611yescharacteristic
313C22×C3#46911yescharacteristic
314C22×C3#47312yes—
315C2×D3—44no—
316C2×D3—42no—
317C2×D3—82no—
318C2×D3—81no—
319C2×D3—81no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—82no—
323C3×C4#44711yescharacteristic
324C3×C4#44711yescharacteristic
325C3×C4#44812yes—
326C3×C4#44911yescharacteristic
327C3×C4#44911yescharacteristic
328C3×C4#49211yescharacteristic
329C3×C4#49711yescharacteristic
330Dic3—82no—
331Dic3—81no—
332C23—62no—
333C23—121no—
334C23—122no—
335C23—121no—
336C23—122no—
337C23#88411yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342Q8—122no—
343Q8—122no—
344Q8—241no—
345Q8—241no—
346D4—62no—
347D4—62no—
348D4—121no—
349D4—121no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4—242no—
362C2×C4#88211yescharacteristic
363C2×C4#88211yescharacteristic
364C2×C4#88411yescharacteristic
365C2×C4D3×D811yescharacteristic
366C2×C4#91111yescharacteristic
367C2×C4D3×SD1612yes—
368C2×C4#91411yescharacteristic
369C2×C4#91611yescharacteristic
370C2×C4#91812yes—
371C2×C4#93011yescharacteristic
372C2×C4#93711yescharacteristic
373C2×C4#94011yescharacteristic
374C8—42no—
375C8—64no—
376C8—81no—
377C8—122no—
378C2×C3#167312yes—
379C2×C3#169312yes—
380C2×C3#199911yescharacteristic
381C2×C3#201311yescharacteristic
382C2×C3#202011yescharacteristic
383D3—82no—
384C22—122no—
385C22—124no—
386C22—242no—
387C22—241no—
388C22—241no—
389C22#475512yes—
390C22#477511yescharacteristic
391C22#478211yescharacteristic
392C22#481812yes—
393C22#496011yescharacteristic
394C4—21no—
395C4—21no—
396C4—82no—
397C4—242no—
398C4—241no—
399C4#477311yescharacteristic
400C4#478111yescharacteristic
401C4#480612yes—
402C4#515611yescharacteristic
403C4#517211yescharacteristic
404C4#517511yescharacteristic
405C4#519011yescharacteristic
406C3#1026411yescharacteristic, complemented, Sylow
407C2—242no—
408C2#6844312yes—
409C2#6845112yes—
410C2#7012511yescharacteristic
411C2#7020411yescharacteristic
412C2#7028211yescharacteristic
subgroupG/H
centerC23#884
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#121
prime radicalC22×D12D4
hypercenterC2×#332D3
hypocenterC3#10264
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. C2×#5078C2semidirect
4. C2×C3×#332C2semidirect
6. C2×#4486C2semidirect
7. #67703C2semidirect
35. C2×#332D3semidirect
159. C2×C3⋊C8SD16semidirect
406. C3#10264semidirect

Maximal subgroups

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

subgroupindex
C2×#50832
C2×#50782
C2×#48412
C2×C3×#3322
#704702
C2×#44862
#677032
#102643