Finite Group Databasebeta

← #10826075

Subgroups — #10826075

Order 768.

616 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79257C211yesmaximal, characteristic
2#79076C211yesmaximal, characteristic, complemented
3C3×#2103C211yesmaximal, characteristic, complemented
4#70728C211yesmaximal, characteristic, complemented
5#70547C211yesmaximal, characteristic
6#69042C212yesmaximal, complemented
7#69014C212yesmaximal, complemented
8#68935C212yesmaximal, complemented
9#68926C212yesmaximal, complemented
10#68719C211yesmaximal, characteristic
11#68717C211yesmaximal, characteristic, complemented
12#23833—31nomaximal, Sylow
13#5748C2211yescharacteristic
14C2×#907C2212yes—
15#5517C2211yescharacteristic
16C3×#430C2211yescharacteristic
17C3×#424C2211yescharacteristic
18#5139C2212yes—
19C2×#825C2212yes—
20#5132C2212yes—
21#5122C2212yes—
22#5122C2211yescharacteristic
23#5118C2211yescharacteristic
24#5113C2212yes—
25#5102C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27#5006C2212yes—
28#4997C2211yescharacteristic
29#4895—22no—
30#4894—22no—
31#4830—22no—
32#4828—22no—
33C3×#344C2212yes—
34C3×#341C2212yes—
35C3×#338C2211yescharacteristic
36#4557C2212yes—
37#4557C2211yescharacteristic
38#4556C2212yes—
39#4555C2212yes—
40#2960—31no—
41#2955—31no—
42#2103D311yescharacteristic, complemented
43#2102—31no—
44#2080—32no—
45#2053—31no—
46#2050—31no—
47#2031—32no—
48#1764—32no—
49#1757—32no—
50#1747—31no—
51C2×#235C2312yes—
52C2×C3×M16C2312yes—
53C2×C3×M16C2312yes—
54C2×C3×M16C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×#117C2311yescharacteristic
57#926C2312yes—
58#926C2311yescharacteristic
59C2×C4×Dic3C2311yescharacteristic
60C2×#208C2311yescharacteristic
61C22×C3⋊C8C2311yescharacteristic
62#909—22no—
63#908—22no—
64#907—22no—
65#907—22no—
66#907—22no—
67#907—22no—
68#904—22no—
69C2×ΓL(1,𝔽25)—22no—
70C2×D3×C8—22no—
71#890—21no—
72#888—21no—
73#878—22no—
74C4×Dic6—22no—
75C3×#108—22no—
76C3×C4⋊C8—22no—
77#825—24no—
78#825—22no—
79#825—22no—
80#824—22no—
81#823—22no—
82#821—22no—
83#816—22no—
84#815—22no—
85#815D412yes—
86Dic3⋊C8—22no—
87Dic3⋊C8—22no—
88C8×Dic3—21no—
89C8×Dic3—21no—
90C8×Dic3—22no—
91C8×Dic3D412yes—
92#806—24no—
93#803—21no—
94C4×C3⋊C8—21no—
95#575—31no—
96C2×#131—32no—
97#533—31no—
98#446—62no—
99C8×Q8—62no—
100#444—62no—
101#443—62no—
102#430C2×D311yescharacteristic
103#424—31no—
104#424C2×D311yescharacteristic
105#423—31no—
106#421—32no—
107C2×#104—32no—
108#416—32no—
109#413—32no—
110#410—32no—
111#405—31no—
112#405—32no—
113#405—31no—
114C2×C4×C8—31no—
115#344—32no—
116#344C2×D312yes—
117#341C2×D312yes—
118#339—32no—
119#338C2×D311yescharacteristic
120H(3,ℤ/4)—31no—
121H(3,ℤ/4)—32no—
122C22×C3×C4C2411yescharacteristic
123C2×#67—22no—
124C22×Dic3—21no—
125#235—22no—
126#235—22no—
127#235—22no—
128#235—22no—
129C2×D12#4112yes—
130C2×C4×D3—22no—
131C2×Dic6—22no—
132C2×Dic6#4112yes—
133C3×M16—22no—
134C3×M16—22no—
135C3×M16—22no—
136C3×M16—22no—
137C3×M16—22no—
138C2×C3×C8—21no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8C2×D412yes—
143C3×C4⋊C4—21no—
144C3×C22⋊C4#4112yes—
145C3×C42—21no—
146#217—21no—
147#217—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—22no—
151Dic3⋊C4—22no—
152Dic3⋊C4—22no—
153C4×Dic3—21no—
154C4×Dic3—22no—
155C4×Dic3C2×D412yes—
156C4×Dic3#4111yescharacteristic
157C4×Dic3#4111yescharacteristic
158#208—22no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8C2×D411yescharacteristic
161C2×C3⋊C8C2×D411yescharacteristic
162C2×C3⋊C8#4112yes—
163C2×C3⋊C8#4112yes—
164Dic12—22no—
165D24—22no—
166#204—24no—
167ΓL(1,𝔽25)—42no—
168ΓL(1,𝔽25)—42no—
169D3×C8—42no—
170D3×C8—42no—
171C2×#41—32no—
172#137—62no—
173AGL(1,ℤ/8)—62no—
174#135—62no—
175#131—62no—
176#131—62no—
177#131—62no—
178#131—62no—
179C2×M16—31no—
180C2×M16—62no—
181C2×M16C22×D312yes—
182C2×M16C22×D312yes—
183C2×M16C22×D311yescharacteristic
184C22×C8—31no—
185C22×C8—62no—
186C22×C8C22×D311yescharacteristic
187C4⋊Q8—61no—
188#125—61no—
189C22⋊Q8—62no—
190C4×Q8—62no—
191#117—31no—
192#117—32no—
193#117C22×D311yescharacteristic
194C2×C42—31no—
195#108—22no—
196C4⋊C8—22no—
197C4⋊C8—62no—
198C4⋊C8—62no—
199#104—62no—
200#104—62no—
201#104—64no—
202#103—62no—
203#102—62no—
204#101—62no—
205#100—62no—
206#99—64no—
207#97—32no—
208#97—61no—
209#97—62no—
210C4×C8—32no—
211C4×C8—61no—
212C4×C8—61no—
213C4×C8—61no—
214C4×C8—62no—
215C23×C3C2×#4111yescharacteristic
216C22×D3—22no—
217C2×C3×C4—22no—
218C2×C3×C4C22×D411yescharacteristic
219C2×C3×C4C2×#4112yes—
220C2×C3×C4C2×#4112yes—
221C2×C3×C4C2×#4111yescharacteristic
222#67—42no—
223#67—42no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230D12—22no—
231D12—22no—
232C4×D3—42no—
233C4×D3—42no—
234Dic6—22no—
235Dic6—22no—
236Dic6—44no—
237C3×C8—22no—
238C3×C8—41no—
239C3×C8—42no—
240C3×C8—42no—
241C3⋊C8—41no—
242C3⋊C8#12411yescharacteristic
243C3⋊C8#12412yes—
244C3⋊C8#12411yescharacteristic
245#41—62no—
246#41—62no—
247#41—62no—
248#41—62no—
249C2×Q8—32no—
250C2×Q8—62no—
251C2×D4—32no—
252C2×D4—62no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4C23×D311yescharacteristic
256Q16—62no—
257SD16—64no—
258D8—62no—
259M16—22no—
260M16—22no—
261M16—22no—
262M16—22no—
263M16—22no—
264M16—62no—
265M16—122no—
266M16—122no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—31no—
272C2×C8—31no—
273C2×C8—32no—
274C2×C8—32no—
275C2×C8—61no—
276C2×C8—122no—
277C2×C8—122no—
278C2×C8D3×D412yes—
279C4⋊C4—21no—
280C4⋊C4—62no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—61no—
284C4⋊C4—62no—
285C22⋊C4—61no—
286C22⋊C4—62no—
287C22⋊C4#23912yes—
288C42—21no—
289C42—32no—
290C42—31no—
291C42—31no—
292C42—61no—
293C42—62no—
294C22×C3—41no—
295C22×C3#52511yescharacteristic
296C22×C3#52912yes—
297C2×D3—42no—
298C2×D3—82no—
299C3×C4—42no—
300C3×C4#52511yescharacteristic
301C3×C4C2×#12411yescharacteristic
302C3×C4#52912yes—
303Dic3—22no—
304Dic3—42no—
305Dic3—41no—
306Dic3—42no—
307Dic3—82no—
308C23—62no—
309C23C2×#23911yescharacteristic
310Q8—62no—
311Q8—62no—
312Q8—124no—
313D4—62no—
314D4—62no—
315D4—122no—
316D4—122no—
317C2×C4—22no—
318C2×C4—61no—
319C2×C4—62no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4C2×D3×D411yescharacteristic
327C2×C4D3×#4112yes—
328C2×C4D3×#4112yes—
329C2×C4D3×#4111yescharacteristic
330C8—22no—
331C8—31no—
332C8—32no—
333C8—31no—
334C8—41no—
335C8—42no—
336C8—42no—
337C8—121no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3#239811yescharacteristic
341D3—82no—
342C22—41no—
343C22—122no—
344C22—242no—
345C22#562711yescharacteristic
346C22#569112yes—
347C4—42no—
348C4—62no—
349C4—121no—
350C4—122no—
351C4—122no—
352C4—242no—
353C4#565611yescharacteristic
354C4D3×#12411yescharacteristic
355C4#570612yes—
356C3#2383311yescharacteristic, complemented, Sylow
357C2—21no—
358C2—22no—
359C2—242no—
360C2#7813111yescharacteristic
subgroupG/H
centerC4D3×#124
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2103C2
socleC2×C3#2398
prime radical#5748C22
hypercenter#2103D3
hypocenterC3#23833
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2103

Complements

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

NH ≅ G/Ntype
2. #79076C2semidirect
3. C3×#2103C2semidirect
4. #70728C2semidirect
6. #69042C2semidirect
7. #69014C2semidirect
8. #68935C2semidirect
9. #68926C2semidirect
11. #68717C2semidirect
42. #2103D3semidirect
356. C3#23833semidirect

Maximal subgroups

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

subgroupindex
#792572
#790762
C3×#21032
#707282
#705472
#690422
#690142
#689352
#689262
#687192
#687172
#238333