Finite Group Databasebeta

← #10712372

Subgroups — #10712372

Order 768.

476 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3414C211yesmaximal, characteristic
2#83899C211yesmaximal, characteristic, complemented
3#72086C211yesmaximal, characteristic, complemented
4#72076C211yesmaximal, characteristic, complemented
5C2×#4566C211yesmaximal, characteristic, complemented
6#67819C211yesmaximal, characteristic, complemented
7#67814C211yesmaximal, characteristic, complemented
8#13030—31nomaximal, Sylow
9C2×C3×#135C2211yescharacteristic
10C2×C3×#135C2211yescharacteristic
11#5827—21no—
12C3×#510—21no—
13C3×#510C412yescomplemented
14C3×#509—21no—
15C3×#509C412yescomplemented
16C3×#508—22no—
17C2×C3×M32C2211yescharacteristic
18#5240—21no—
19#5236—21no—
20#5232—21no—
21#5184—21no—
22Dic3×SD16—22no—
23#5152—21no—
24#5106C2211yescharacteristic
25C2×#817C2211yescharacteristic
26#5101C2211yescharacteristic
27C2×#812C2211yescharacteristic
28#4566—22no—
29#4566—22no—
30#3414D311yescharacteristic
31#2952—31no—
32#2145—31no—
33C2×#360—31no—
34#1976—31no—
35#1386—31no—
36#1378—31no—
37C2×C3×#41—21no—
38C2×C3×#41—21no—
39C3×#135—21no—
40C3×#135—21no—
41C3×#135—21no—
42C3×#135—21no—
43C3×#135—21no—
44C3×#135—21no—
45C3×#135C2×C412yes—
46C3×#135D412yes—
47C2×C3×Q16C2×C411yescharacteristic, complemented
48C2×C3×Q16D411yescharacteristic
49C2×C3×SD16—21no—
50C2×C3×SD16—21no—
51C2×C3×D8C2×C411yescharacteristic, complemented
52C2×C3×D8D411yescharacteristic
53C22×C3×C8C2311yescharacteristic
54Q8×Dic3—22no—
55D4×Dic3—22no—
56D4×Dic3—42no—
57#926—21no—
58#926—41no—
59#926D411yescharacteristic
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—41no—
62C2×#208D411yescharacteristic
63C3×Q32—22no—
64C3×SD32—22no—
65C3×SD32—22no—
66C3×D16—22no—
67C3×M32—21no—
68C3×M32C2×C412yescomplemented
69C2×C3×C16—21no—
70#837—42no—
71#835—22no—
72#834—42no—
73#832—22no—
74#818—21no—
75#817—21no—
76#817—21no—
77#817D412yes—
78#815—21no—
79C8×Dic3—21no—
80#812—21no—
81#812—21no—
82C2×C3⋊C16—21no—
83C2×#135C2×D311yescharacteristic
84C2×#135C2×D311yescharacteristic
85#520—61no—
86#510—21no—
87#510Dic312yescomplemented
88#509—21no—
89#509Dic312yescomplemented
90#508—22no—
91C2×M32—31no—
92C2×M32C2×D311yescharacteristic
93#442—61no—
94#441—61no—
95C4×SD16—62no—
96#427—31no—
97C2×#106—31no—
98#421—61no—
99#417—61no—
100#410—61no—
101#405—31no—
102#360—62no—
103#360—62no—
104C3×#41—22no—
105C3×#41—22no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—42no—
122C22×Dic3—41no—
123C3×Q16—21no—
124C3×Q16—21no—
125C3×Q16—21no—
126C3×Q16C22⋊C412yes—
127C3×SD16—22no—
128C3×SD16—22no—
129C3×SD16—41no—
130C3×SD16—41no—
131C3×D8—21no—
132C3×D8—21no—
133C3×D8—21no—
134C3×D8C22⋊C412yes—
135C2×C3×C8C22×C411yescharacteristic
136C2×C3×C8C2×D411yescharacteristic
137C2×C3×C8C2×D411yescharacteristic
138C2×C3×C8C2×D411yescharacteristic
139C2×C3×C8#4111yescharacteristic
140C2×C3×C8#4111yescharacteristic
141#217—42no—
142#217—41no—
143#217D812yes—
144C4⋊Dic3—21no—
145C4⋊Dic3—22no—
146C4⋊Dic3—42no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4×Dic3—21no—
150C4×Dic3—42no—
151C4×Dic3—41no—
152#208—21no—
153#208—21no—
154C2×C3⋊C8—21no—
155C3×C16—22no—
156C3⋊C16—41no—
157C2×#41—21no—
158C2×#41—21no—
159#135—21no—
160#135—21no—
161#135—21no—
162#135—21no—
163#135—21no—
164#135—21no—
165#135C2×Dic312yes—
166#135#6712yes—
167C2×Q16C2×Dic311yescharacteristic, complemented
168C2×Q16#6711yescharacteristic
169C2×SD16—21no—
170C2×SD16—21no—
171C2×D8C2×Dic311yescharacteristic, complemented
172C2×D8#6711yescharacteristic
173C2×M16—31no—
174C22×C8C22×D311yescharacteristic
175C4×Q8—62no—
176C4×D4—62no—
177C4×D4—122no—
178#117—31no—
179#117—61no—
180#117—121no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—121no—
183Q32—22no—
184SD32—22no—
185SD32—22no—
186D16—22no—
187M32—21no—
188M32—61no—
189M32—61no—
190M32C2×Dic312yescomplemented
191C2×C16—21no—
192C2×C16—61no—
193#107—61no—
194#106—32no—
195#106—61no—
196#106—61no—
197#104—122no—
198#103—62no—
199#102—62no—
200#99—122no—
201#97—61no—
202C4×C8—61no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3C2×D811yescharacteristic
206C3×Q8—22no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×Q8—41no—
210C3×D4—22no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—41no—
216C3×D4—81no—
217C3×D4—81no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×Dic3—22no—
231C2×Dic3—42no—
232C2×Dic3—81no—
233C2×Dic3—162no—
234C2×Dic3—161no—
235C3×C8C2×C22⋊C411yescharacteristic
236C3×C8C4×D411yescharacteristic
237C3×C8#12111yescharacteristic
238C3×C8#12311yescharacteristic
239C3⋊C8—41no—
240#41—22no—
241#41—22no—
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247#41—41no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—122no—
257C22×C4—121no—
258C22×C4D3×D411yescharacteristic
259Q16—21no—
260Q16—21no—
261Q16—21no—
262Q16#21712yes—
263SD16—22no—
264SD16—22no—
265SD16—41no—
266SD16—41no—
267D8—21no—
268D8—21no—
269D8—21no—
270D8#21712yes—
271M16—61no—
272M16—61no—
273C2×C8—61no—
274C2×C8D3×D411yescharacteristic
275C2×C8#23711yescharacteristic
276C2×C8#23711yescharacteristic
277C2×C8C22×Dic311yescharacteristic
278C2×C8C2×#6711yescharacteristic
279C2×C8C2×#6711yescharacteristic
280C4⋊C4—61no—
281C4⋊C4—62no—
282C4⋊C4—122no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—32no—
286C22⋊C4—122no—
287C22⋊C4—121no—
288C42—61no—
289C42—122no—
290C42—121no—
291C16—22no—
292C16—121no—
293C22×C3—41no—
294C22×C3—42no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—161no—
299C22×C3C4×D811yescharacteristic
300C22×C3#44711yescharacteristic
301C22×C3#45911yescharacteristic
302C3×C4—42no—
303C3×C4—81no—
304C3×C4#38611yescharacteristic
305C3×C4#44211yescharacteristic
306C3×C4#44911yescharacteristic
307C3×C4#46311yescharacteristic
308Dic3—42no—
309Dic3—82no—
310Dic3—161no—
311C23—41no—
312C23—41no—
313C23D3×D811yescharacteristic
314Q8—22no—
315Q8—41no—
316Q8—41no—
317Q8—41no—
318D4—22no—
319D4—42no—
320D4—42no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—81no—
325D4—81no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—62no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—122no—
334C2×C4—241no—
335C2×C4—482no—
336C2×C4—481no—
337C2×C4#91111yescharacteristic
338C2×C4#91111yescharacteristic
339C2×C4#91211yescharacteristic
340C2×C4D4×Dic311yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93911yescharacteristic
343C8—121no—
344C8D4×Dic311yescharacteristic
345C8#93511yescharacteristic
346C8#93911yescharacteristic
347C8C2×#21711yescharacteristic
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—82no—
352C2×C3—161no—
353C2×C3#187911yescharacteristic
354C22—41no—
355C22—42no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—161no—
360C22Dic3×D811yescharacteristic
361C22#515011yescharacteristic
362C22#515611yescharacteristic
363C4—42no—
364C4—81no—
365C4—122no—
366C4—242no—
367C4—481no—
368C4#515211yescharacteristic
369C4#515311yescharacteristic
370C4#515811yescharacteristic
371C4#522011yescharacteristic
372C3#1303011yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2—161no—
378C2#7203711yescharacteristic
subgroupG/H
centerC2#72037
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
Fitting, prime radicalC3×#3414C2
socleC2×C3#1879
cosocleC22×C3×C8C23
hypercenter#3414D3
hypocenterC3#13030
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
2. #83899C2semidirect
3. #72086C2semidirect
4. #72076C2semidirect
5. C2×#4566C2semidirect
6. #67819C2semidirect
7. #67814C2semidirect
13. C3×#510C4semidirect
15. C3×#509C4semidirect
47. C2×C3×Q16C2×C4semidirect
51. C2×C3×D8C2×C4semidirect
68. C3×M32C2×C4semidirect
87. #510Dic3semidirect
89. #509Dic3semidirect
167. C2×Q16C2×Dic3semidirect
171. C2×D8C2×Dic3semidirect
190. M32C2×Dic3semidirect
372. C3#13030semidirect

Maximal subgroups

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

subgroupindex
C3×#34142
#838992
#720862
#720762
C2×#45662
#678192
#678142
#130303