Finite Group Databasebeta

← #10713600

Subgroups — #10713600

Order 768.

410 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5194C211yesmaximal, characteristic
2C3×#1976C211yesmaximal, characteristic, complemented
3#72076C211yesmaximal, characteristic, complemented
4#70656C211yesmaximal, characteristic, complemented
5#70648C211yesmaximal, characteristic, complemented
6#69897C211yesmaximal, characteristic, complemented
7#69881C211yesmaximal, characteristic, complemented
8#13243—31nomaximal, Sylow
9C2×C3×#135C2211yescharacteristic
10C2×#904C2211yescharacteristic
11C3×#427C2211yescharacteristic
12C3×#421—21no—
13C3×#410—21no—
14C3×#405C2211yescharacteristic
15#5194—21no—
16#5194—21no—
17#5194—21no—
18#5194—21no—
19C2×#829C411yescharacteristic, complemented
20C2×#828C411yescharacteristic, complemented
21C2×#827C411yescharacteristic, complemented
22C2×#826C411yescharacteristic, complemented
23#5139—21no—
24#5106C2211yescharacteristic
25C2×#812C2211yescharacteristic
26C22×C3⋊C16C2211yescharacteristic
27#5006—21no—
28#4566—22no—
29#4490—21no—
30#4489—21no—
31#4488—22no—
32#4461—21no—
33C4×C3⋊C16—21no—
34C2×#508—31no—
35#2158—31no—
36#2145—31no—
37#2145—31no—
38#2108—31no—
39#1976—31no—
40#1976D311yescharacteristic, complemented
41C2×C3×#41—21no—
42C2×#235—21no—
43C3×#135—21no—
44C3×#135—21no—
45C3×#135—21no—
46C3×#135—21no—
47C2×C3×Q16C2×C411yescharacteristic
48C2×C3×SD16—21no—
49C2×C3×D8C2×C411yescharacteristic
50C2×C3×M16D411yescharacteristic
51C22×C3×C8C2311yescharacteristic
52C3×#117—21no—
53C3×#117D411yescharacteristic
54#926—21no—
55C2×Dic12C2×C411yescharacteristic
56#904—21no—
57#904—21no—
58#904—21no—
59#904—21no—
60C2×D24C2×C411yescharacteristic
61C2×#204—21no—
62C3×#107—21no—
63C3×#106—21no—
64C3×#104—42no—
65C3×#99—42no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#829—21no—
69#829—21no—
70#828—21no—
71#828—21no—
72#827—21no—
73#827—21no—
74#826—21no—
75#826—21no—
76#825—42no—
77#818—21no—
78#817—21no—
79#812—21no—
80#812D412yes—
81C2×C3⋊C16—21no—
82C2×C3⋊C16C2×C411yescharacteristic
83C2×C3⋊C16C2×C411yescharacteristic
84C2×C3⋊C16D411yescharacteristic
85C2×C3⋊C16D411yescharacteristic
86C2×C3⋊C16D411yescharacteristic
87C2×C3⋊C16D411yescharacteristic
88#806—42no—
89C2×#135—31no—
90C2×#135C2×D311yescharacteristic
91#508—61no—
92#508—61no—
93#508—61no—
94#508—61no—
95C2×Q32—31no—
96C2×SD32—31no—
97C2×SD32—31no—
98C2×D16—31no—
99C2×M32—31no—
100C22×C16—31no—
101#427—31no—
102#427C2×D311yescharacteristic
103#421—21no—
104#421—61no—
105#410—21no—
106#410—61no—
107#405C2×D311yescharacteristic
108#367—61no—
109#366—61no—
110#360—62no—
111#360—62no—
112#346—61no—
113C4×C16—61no—
114C3×#41—41no—
115C3×#41—41no—
116C3×#41—41no—
117C3×#41—41no—
118C2×C3×Q8—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—41no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×C4×D3—41no—
130C2×Dic6—21no—
131C3×Q16—21no—
132C3×Q16—21no—
133C3×SD16—41no—
134C3×SD16—41no—
135C3×D8—21no—
136C3×D8—21no—
137C3×M16—21no—
138C3×M16D812yes—
139C2×C3×C8—21no—
140C2×C3×C8C22×C411yescharacteristic
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8C2×D411yescharacteristic
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8#4111yescharacteristic
145C2×C3×C8#4111yescharacteristic
146C3×C4⋊C4—21no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4D812yes—
150C3×C42—21no—
151C3×C42—41no—
152#217—41no—
153C4⋊Dic3—41no—
154C4×Dic3—41no—
155Dic12—21no—
156Dic12—21no—
157D24—21no—
158D24—21no—
159#204—41no—
160#204—41no—
161C3⋊C16—22no—
162C3⋊C16C22⋊C411yescharacteristic
163C3⋊C16C22⋊C411yescharacteristic
164C3⋊C16C22⋊C411yescharacteristic
165C3⋊C16C22⋊C411yescharacteristic
166C2×#41—21no—
167C2×#41—61no—
168#135—21no—
169#135—21no—
170#135—21no—
171#135—21no—
172#135—61no—
173#135—61no—
174#135—61no—
175#135—61no—
176C2×Q16—31no—
177C2×Q16C4×D311yescharacteristic
178C2×SD16—21no—
179C2×SD16—61no—
180C2×D8—31no—
181C2×D8C4×D311yescharacteristic
182C2×M16#6711yescharacteristic
183C22×C8C22×D311yescharacteristic
184#117—21no—
185#117—61no—
186#117#6711yescharacteristic
187Q32—61no—
188Q32—61no—
189SD32—61no—
190SD32—61no—
191SD32—61no—
192SD32—61no—
193D16—61no—
194D16—61no—
195M32—32no—
196M32—61no—
197C2×C16—31no—
198C2×C16—31no—
199C2×C16—31no—
200C2×C16—31no—
201C2×C16—31no—
202C2×C16—31no—
203C2×C16—61no—
204#107—21no—
205#107—61no—
206#106—21no—
207#106—61no—
208#104—42no—
209#104—122no—
210#99—42no—
211#99—122no—
212#97—21no—
213C4×C8—21no—
214C23×C3—41no—
215C23×C3C2×D811yescharacteristic
216C22×D3—41no—
217C3×Q8—41no—
218C3×Q8—41no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—81no—
222C3×D4—81no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×C3×C4C2×D811yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4#13511yescharacteristic
234#67—81no—
235#67—81no—
236C2×Dic3—41no—
237C2×Dic3—81no—
238D12—41no—
239D12—41no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—41no—
243Dic6—41no—
244C3×C8—22no—
245C3×C8C2×C22⋊C411yescharacteristic
246C3×C8C4×D411yescharacteristic
247C3×C8#12111yescharacteristic
248C3×C8#12411yescharacteristic
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—121no—
254#41—121no—
255#41—121no—
256#41—121no—
257C2×Q8—21no—
258C2×Q8—61no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—61no—
262C2×D4—121no—
263C22×C4—41no—
264C22×C4—121no—
265C22×C4C2×#6711yescharacteristic
266Q16—21no—
267Q16—21no—
268Q16—61no—
269Q16—61no—
270SD16—41no—
271SD16—41no—
272SD16—121no—
273SD16—121no—
274D8—21no—
275D8—21no—
276D8—61no—
277D8—61no—
278M16—21no—
279M16#21312yes—
280C2×C8—21no—
281C2×C8C2×C4×D311yescharacteristic
282C2×C8#23511yescharacteristic
283C2×C8D3×D411yescharacteristic
284C2×C8D3×D411yescharacteristic
285C2×C8D3×D411yescharacteristic
286C2×C8#23711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—121no—
290C22⋊C4—41no—
291C22⋊C4—121no—
292C22⋊C4#21312yes—
293C42—21no—
294C42—41no—
295C42—121no—
296C16—31no—
297C16—31no—
298C16—31no—
299C16—31no—
300C16—62no—
301C22×C3—41no—
302C22×C3—81no—
303C22×C3—161no—
304C22×C3C4×D811yescharacteristic
305C22×C3#46611yescharacteristic
306C22×C3#49311yescharacteristic
307C2×D3—81no—
308C2×D3—161no—
309C3×C4—42no—
310C3×C4—81no—
311C3×C4—161no—
312C3×C4#39011yescharacteristic
313C3×C4C4×D811yescharacteristic
314C3×C4#46611yescharacteristic
315C3×C4#49511yescharacteristic
316Dic3—81no—
317Dic3—161no—
318C23—41no—
319C23—121no—
320C23C2×#21311yescharacteristic
321Q8—41no—
322Q8—41no—
323Q8—121no—
324Q8—121no—
325D4—41no—
326D4—41no—
327D4—81no—
328D4—81no—
329D4—121no—
330D4—121no—
331D4—241no—
332D4—241no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—121no—
339C2×C4—241no—
340C2×C4—241no—
341C2×C4—241no—
342C2×C4C4×#6711yescharacteristic
343C2×C4C2×#21311yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#94011yescharacteristic
346C2×C4#95011yescharacteristic
347C2×C4#95011yescharacteristic
348C8—22no—
349C8D3×C22⋊C411yescharacteristic
350C8#67⋊C411yescharacteristic
351C8#88411yescharacteristic
352C8#88511yescharacteristic
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—161no—
357C2×C3#197011yescharacteristic
358D3—161no—
359C22—41no—
360C22—81no—
361C22—161no—
362C22—241no—
363C22—481no—
364C22C4×#21311yescharacteristic
365C22#503811yescharacteristic
366C22#507311yescharacteristic
367C4—42no—
368C4—81no—
369C4—161no—
370C4—241no—
371C4—481no—
372C4#495711yescharacteristic
373C4C4×#21311yescharacteristic
374C4#503811yescharacteristic
375C4#505911yescharacteristic
376C3#1324311yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—21no—
380C2—161no—
381C2—481no—
382C2#6964711yescharacteristic
subgroupG/H
centerC4#5038
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#1976C2
socleC2×C3#1970
prime radicalC2×#5194C2
cosocleC22×C3×C8C23
hypercenter#1976D3
hypocenterC3#13243
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. C3×#1976C2semidirect
3. #72076C2semidirect
4. #70656C2semidirect
5. #70648C2semidirect
6. #69897C2semidirect
7. #69881C2semidirect
19. C2×#829C4semidirect
20. C2×#828C4semidirect
21. C2×#827C4semidirect
22. C2×#826C4semidirect
40. #1976D3semidirect
376. C3#13243semidirect

Maximal subgroups

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

subgroupindex
C2×#51942
C3×#19762
#720762
#706562
#706482
#698972
#698812
#132433