Finite Group Databasebeta

← #10713598

Subgroups — #10713598

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×#2145C211yesmaximal, characteristic, complemented
3#72086C211yesmaximal, characteristic, complemented
4#71450C211yesmaximal, characteristic, complemented
5#71419C211yesmaximal, characteristic, complemented
6#70658C211yesmaximal, characteristic, complemented
7#70648C211yesmaximal, characteristic, complemented
8#13243—31nomaximal, Sylow
9C2×C3×#135C2211yescharacteristic
10C2×#904C2211yescharacteristic
11C2×C3×M32C2211yescharacteristic
12C3×#427C2211yescharacteristic
13#5240—21no—
14#5236—21no—
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#5106C2211yescharacteristic
24#5101C2211yescharacteristic
25C22×C3⋊C16C2211yescharacteristic
26#5006—21no—
27#5005—21no—
28C3×#360—22no—
29#4518—22no—
30#4502—21no—
31Dic3×C16—21no—
32#4490—21no—
33#4489—21no—
34C2×#508—31no—
35#2158—31no—
36#2145—31no—
37#2145D311yescharacteristic, complemented
38#2108—31no—
39#1976—31no—
40#1976—31no—
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
50C22×C3×C8C2311yescharacteristic
51C3×#117—21no—
52#926—21no—
53#926D411yescharacteristic
54C2×#208D411yescharacteristic
55C2×Dic12C2×C411yescharacteristic
56#904—21no—
57#904—21no—
58#904—21no—
59#904—21no—
60C2×D24C2×C411yescharacteristic
61C2×#204—21no—
62C3×M32—21no—
63C3×M32D412yes—
64C2×C3×C16—21no—
65C3×#107—21no—
66C3×#106—21no—
67#837—42no—
68#834—42no—
69#829—21no—
70#829—21no—
71#828—21no—
72#828—21no—
73#827—21no—
74#827—21no—
75#826—21no—
76#826—21no—
77#818—21no—
78#817—21no—
79#815—21no—
80C8×Dic3—21no—
81C2×C3⋊C16C2×C411yescharacteristic
82C2×C3⋊C16C2×C411yescharacteristic
83C2×C3⋊C16D411yescharacteristic
84C2×C3⋊C16D411yescharacteristic
85C2×C3⋊C16D411yescharacteristic
86C2×C3⋊C16D411yescharacteristic
87#806—42no—
88#805—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×M32C2×D311yescharacteristic
100C22×C16—31no—
101#427—31no—
102#427C2×D311yescharacteristic
103#421—61no—
104#421—61no—
105#410—61no—
106#410—61no—
107#405—31no—
108#367—61no—
109#366—61no—
110#360—22no—
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—
137C2×C3×C8C22×C411yescharacteristic
138C2×C3×C8C2×D411yescharacteristic
139C2×C3×C8C2×D411yescharacteristic
140C2×C3×C8C2×D411yescharacteristic
141C2×C3×C8#4111yescharacteristic
142C2×C3×C8#4111yescharacteristic
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146#217—41no—
147#217D812yes—
148C4⋊Dic3—21no—
149C4⋊Dic3—41no—
150C4×Dic3—21no—
151C4×Dic3—41no—
152#208—21no—
153#208D812yes—
154C2×C3⋊C8—21no—
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—31no—
183C22×C8C22×D311yescharacteristic
184#117—21no—
185#117—31no—
186#117—61no—
187Q32—61no—
188Q32—61no—
189SD32—61no—
190SD32—61no—
191SD32—61no—
192SD32—61no—
193D16—61no—
194D16—61no—
195M32—21no—
196M32#6712yes—
197C2×C16—21no—
198C2×C16—31no—
199C2×C16—31no—
200C2×C16—31no—
201C2×C16—31no—
202C2×C16—31no—
203C2×C16—31no—
204#107—21no—
205#107—61no—
206#106—21no—
207#106—61no—
208#104—122no—
209#104—122no—
210#99—122no—
211#99—122no—
212#97—61no—
213C4×C8—61no—
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—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×C3×C4C2×D811yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4#13511yescharacteristic
233#67—81no—
234#67—81no—
235C2×Dic3—22no—
236C2×Dic3—41no—
237C2×Dic3—81no—
238D12—41no—
239D12—41no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—41no—
243Dic6—41no—
244C3×C8C2×C22⋊C411yescharacteristic
245C3×C8C4×D411yescharacteristic
246C3×C8#12111yescharacteristic
247C3×C8#12411yescharacteristic
248C3⋊C8—22no—
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×C4D3×D411yescharacteristic
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—32no—
279M16—61no—
280C2×C8—61no—
281C2×C8C2×C4×D311yescharacteristic
282C2×C8#23511yescharacteristic
283C2×C8D3×D411yescharacteristic
284C2×C8D3×D411yescharacteristic
285C2×C8#23711yescharacteristic
286C2×C8C2×#6711yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C22⋊C4—32no—
291C22⋊C4—41no—
292C22⋊C4—121no—
293C42—41no—
294C42—61no—
295C42—121no—
296C16—22no—
297C16—31no—
298C16—31no—
299C16—31no—
300C16—31no—
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—81no—
310C3×C4—161no—
311C3×C4#39011yescharacteristic
312C3×C4C4×D811yescharacteristic
313C3×C4#46611yescharacteristic
314C3×C4#49511yescharacteristic
315Dic3—42no—
316Dic3—81no—
317Dic3—161no—
318C23—41no—
319C23—121no—
320C23D3×D811yescharacteristic
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—41no—
334C2×C4—62no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—121no—
339C2×C4—241no—
340C2×C4—241no—
341C2×C4—241no—
342C2×C4#67⋊C411yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4D3×D811yescharacteristic
345C2×C4#91611yescharacteristic
346C2×C4#91611yescharacteristic
347C2×C4#94011yescharacteristic
348C8—62no—
349C8D3×C22⋊C411yescharacteristic
350C8#88511yescharacteristic
351C8C4×#6711yescharacteristic
352C8#93911yescharacteristic
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—
364C22#475611yescharacteristic
365C22#477511yescharacteristic
366C22#515111yescharacteristic
367C4—81no—
368C4—122no—
369C4—161no—
370C4—241no—
371C4—481no—
372C4#475611yescharacteristic
373C4#477511yescharacteristic
374C4#495711yescharacteristic
375C4#516811yescharacteristic
376C3#1324311yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—21no—
380C2—161no—
381C2—481no—
382C2#7019011yescharacteristic
subgroupG/H
centerC4#4775
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2145C2
socleC2×C3#1970
prime radicalC2×#5194C2
cosocleC22×C3×C8C23
hypercenter#2145D3
hypocenterC3#13243
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
2. C3×#2145C2semidirect
3. #72086C2semidirect
4. #71450C2semidirect
5. #71419C2semidirect
6. #70658C2semidirect
7. #70648C2semidirect
19. C2×#829C4semidirect
20. C2×#828C4semidirect
21. C2×#827C4semidirect
22. C2×#826C4semidirect
37. #2145D3semidirect
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×#21452
#720862
#714502
#714192
#706582
#706482
#132433