Finite Group Databasebeta

← #10696396

Subgroups — #10696396

Order 768.

670 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×#5080C211yesmaximal, characteristic, complemented
3#73800C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic, complemented
5C2×#4482C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67619C211yesmaximal, characteristic, complemented
8#10068—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#215—21no—
11C2×#897—22no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16C2×#835—21no—
17#5083—22no—
18#5083—22no—
19#5080—22no—
20#5080—22no—
21C2×#808—22no—
22#4939—21no—
23#4935—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4677—22no—
27C3×#340—22no—
28#4483—22no—
29#4482—22no—
30C2×#498—31no—
31C2×#461—31no—
32#2542—31no—
33#1988D311yescharacteristic, complemented
34C2×#332—31no—
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#215—24no—
52C2×#215—42no—
53C2×#215—42no—
54C2×(C2×D3)⋊C4—24no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D412yes—
59C2×C4⋊Dic3D411yescharacteristic
60C22×C3⋊C8—21no—
61C22×C3⋊C8—21no—
62#897—44no—
63#897—44no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68#869—21no—
69#869—21no—
70#869—21no—
71#869—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×#95—21no—
75#835—24no—
76#808—28no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79C4×C3⋊C8—21no—
80C4×C3⋊C8D412yes—
81C22×SD16—61no—
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×C22⋊Q8—62no—
85C2×C4×D4—31no—
86#498—62no—
87#498—62no—
88#461—62no—
89#461—62no—
90C2×#106—62no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93C2×#103—61no—
94C2×C4×C8—31no—
95#399—62no—
96#390—61no—
97#384—61no—
98C4×C4⋊C4C2×D311yescharacteristic
99#340—22no—
100#332—62no—
101D4⋊C8—62no—
102C23×D3—21no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C22×C3×C4—22no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—22no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12SD1612yes—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—42no—
125C2×C4×D3—42no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×C3×C8—41no—
131C2×C3×C8—42no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4SD1614yescomplemented
138C3×C42—41no—
139C3×C42—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144#215—48no—
145(C2×D3)⋊C4—48no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—22no—
151C4⋊Dic3—21no—
152C4⋊Dic3—48no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C2×C3⋊C8—24no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C22×Q8—21no—
161C22×Q8—61no—
162C22×D4—31no—
163C23×C4—61no—
164C2×SD16—64no—
165C2×SD16—122no—
166C2×SD16—122no—
167C22×C8—21no—
168C22×C8—61no—
169C22×C8—61no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8D1212yes—
178C22⋊Q8—124no—
179C22⋊Q8—124no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—64no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4#6712yes—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—61no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195#106—68no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—32no—
199C4⋊C8—61no—
200#103—64no—
201C4×C8—32no—
202C4×C8—61no—
203#95—21no—
204C23×C3#12011yescharacteristic
205C22×D3—22no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—42no—
209C22×D3—42no—
210C3×Q8—42no—
211C3×Q8—42no—
212C3×Q8—81no—
213C3×Q8—81no—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4C2×SD1612yes—
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)12yes—
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—42no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—44no—
237D12—22no—
238D12—22no—
239D12—41no—
240D12—41no—
241C4×D3—82no—
242C4×D3—81no—
243C4×D3—81no—
244Dic6—82no—
245Dic6—81no—
246Dic6—81no—
247C3×C8—81no—
248C3⋊C8—24no—
249C3⋊C8—42no—
250C24—61no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—121no—
258C2×Q8—121no—
259C2×Q8—122no—
260C2×Q8—122no—
261C2×D4—32no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—62no—
270C22×C4—61no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4—122no—
274C22×C4—122no—
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4C2×#6711yescharacteristic
278SD16—128no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—64no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—121no—
285C2×C8—121no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—61no—
292C4⋊C4—62no—
293C4⋊C4—61no—
294C4⋊C4—62no—
295C4⋊C4—128no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4#21514yescomplemented
299C22⋊C4—128no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—41no—
303C42—41no—
304C42C2×D1211yescharacteristic
305C42#23511yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C22×C3#39211yescharacteristic
309C22×C3#44811yescharacteristic
310C22×C3#44911yescharacteristic
311C22×C3#45012yes—
312C22×C3#45612yes—
313C2×D3—44no—
314C2×D3—42no—
315C2×D3—82no—
316C2×D3—81no—
317C2×D3—81no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—44no—
321C3×C4—82no—
322C3×C4#45011yescharacteristic
323C3×C4#45211yescharacteristic
324C3×C4#46012yes—
325C3×C4#46511yescharacteristic
326C3×C4#46811yescharacteristic
327C3×C4#49211yescharacteristic
328C3×C4#49611yescharacteristic
329Dic3—82no—
330Dic3—81no—
331C23—62no—
332C23—121no—
333C23—122no—
334C23—121no—
335C23—122no—
336C23#93711yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—81no—
340Q8—81no—
341Q8—242no—
342Q8—241no—
343Q8—241no—
344D4—62no—
345D4—62no—
346D4—121no—
347D4—121no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—124no—
358C2×C4—122no—
359C2×C4—242no—
360C2×C4—241no—
361C2×C4—241no—
362C2×C4#88211yescharacteristic
363C2×C4#88211yescharacteristic
364C2×C4#88411yescharacteristic
365C2×C4#88411yescharacteristic
366C2×C4#91111yescharacteristic
367C2×C4#91211yescharacteristic
368C2×C4D3×SD1611yescharacteristic
369C2×C4#91511yescharacteristic
370C2×C4#93011yescharacteristic
371C2×C4#94011yescharacteristic
372C2×C4C2×#21512yes—
373C2×C4#94912yes—
374C8—64no—
375C8—81no—
376C8—122no—
377C2×C3#165112yes—
378C2×C3#165412yes—
379C2×C3#200011yescharacteristic
380C2×C3#200511yescharacteristic
381C2×C3#201411yescharacteristic
382D3—82no—
383C22—122no—
384C22—124no—
385C22—241no—
386C22—242no—
387C22—241no—
388C22#496011yescharacteristic
389C22#504612yes—
390C22#515811yescharacteristic
391C22#517011yescharacteristic
392C22#520512yes—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—82no—
397C4—241no—
398C4—242no—
399C4#477611yescharacteristic
400C4#477811yescharacteristic
401C4#478011yescharacteristic
402C4#478311yescharacteristic
403C4#502712yes—
404C4#505811yescharacteristic
405C4#508511yescharacteristic
406C3#1006811yescharacteristic, complemented, Sylow
407C2—242no—
408C2#6968811yescharacteristic
409C2#7020511yescharacteristic
410C2#7034611yescharacteristic
411C2#7111112yes—
412C2#7112712yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1988D3
hypocenterC3#10068
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
2. C2×#5080C2semidirect
4. C3×#1988C2semidirect
5. C2×#4482C2semidirect
7. #67619C2semidirect
33. #1988D3semidirect
137. C3×C4⋊C4SD16semidirect
298. C4⋊C4#215semidirect
406. C3#10068semidirect

Maximal subgroups

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

subgroupindex
C2×#50832
C2×#50802
#738002
C3×#19882
C2×#44822
C2×#44832
#676192
#100683