Finite Group Databasebeta

← #10799587

Subgroups — #10799587

Order 768.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79165C211yesmaximal, characteristic
2#79071C212yesmaximal, complemented
3C3×#1945C211yesmaximal, characteristic, complemented
4#70696C211yesmaximal, characteristic
5Dic3×#108C211yesmaximal, characteristic, complemented
6#70662C211yesmaximal, characteristic, complemented
7#70536C212yesmaximal, complemented
8#69795C211yesmaximal, characteristic
9#69749C211yesmaximal, characteristic, complemented
10#68746C212yesmaximal, complemented
11#68729C212yesmaximal, complemented
12#21294—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14#5522C2212yes—
15C2×C3×#108C2211yescharacteristic
16C3×#427C2211yescharacteristic
17C3×#424C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C2×#825C2212yes—
20#5129C2212yes—
21Dic3×M16C2212yes—
22#5110C2212yes—
23C2×#819C2211yescharacteristic
24C2×#815C2211yescharacteristic
25C2×#815C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#5002C2212yes—
29#4999C2212yes—
30#4932C2211yescharacteristic
31C2×#805C2212yes—
32C2×#803C2211yescharacteristic
33#4873C412yescomplemented
34#4872C412yescomplemented
35#4852C414yescomplemented
36#4840—24no—
37#4835—24no—
38C3×#339C2212yes—
39#4558C2212yes—
40#4532C2212yes—
41C4×#135—31no—
42#2923—32no—
43#1974—31no—
44#1974—31no—
45#1945—31no—
46#1945D311yescharacteristic, complemented
47#1938—32no—
48#1938—32no—
49C4×#108—31no—
50#1759—32no—
51#1726—31no—
52C2×#235C2312yes—
53C2×C3×M16C2312yes—
54C22×C3×C8C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×#117C2312yes—
57C2×C3×C42C2311yescharacteristic
58C2×C4×Dic3C2311yescharacteristic
59C2×C4×Dic3C2311yescharacteristic
60C2×#208C2312yes—
61C22×C3⋊C8C2311yescharacteristic
62C22×C3⋊C8C2311yescharacteristic
63C2×Dic12C2×C411yescharacteristic
64#904—22no—
65#904—22no—
66C2×D24C2×C411yescharacteristic
67C2×#204C2×C412yes—
68#893C2×C414yes—
69#887C2×C414yes—
70#67⋊C4—24no—
71#872—22no—
72C42×D3—22no—
73C3×#108—22no—
74C3×#107C2×C412yes—
75C3×#106C2×C412yes—
76C3×C4⋊C8—24no—
77C3×C4×C8—22no—
78C3⋊#95—22no—
79#825—24no—
80(C2×D3)⋊C8—24no—
81(C2×D3)⋊C8—24no—
82#819—22no—
83#815—22no—
84#815—22no—
85#815D412yes—
86#815D412yes—
87C8×Dic3—22no—
88C8×Dic3—22no—
89C8×Dic3C2×C412yes—
90C8×Dic3C2×C412yes—
91#810C2×C414yes—
92#809C2×C414yes—
93#805—24no—
94C4⋊(C3⋊C8)—24no—
95#803—22no—
96C2×#135—31no—
97C4×#41—32no—
98C4×Q16—32no—
99C4×SD16—34no—
100C4×D8—32no—
101#433—64no—
102#432—64no—
103C2×#108—31no—
104C2×#108C2×D311yescharacteristic
105#427C2×D311yescharacteristic
106#424—32no—
107#424C2×D312yes—
108C2×#104—32no—
109C2×#104—32no—
110#416—32no—
111#408—32no—
112#408—32no—
113C4×M16—32no—
114C2×#97—31no—
115C2×#97—31no—
116C2×#97—31no—
117C2×C4×C8—31no—
118C2×C4×C8—31no—
119C2×C4×C8C2×D311yescharacteristic
120#376—31no—
121#341—32no—
122#339—32no—
123#339C2×D312yes—
124C22×C3×C4—21no—
125C22×C3×C4C2411yescharacteristic
126C2×#67—22no—
127C22×Dic3—21no—
128C22×Dic3#4112yes—
129#235—22no—
130#235—22no—
131#235—22no—
132#235—22no—
133C2×D12C22×C412yes—
134C2×C4×D3—22no—
135C2×C4×D3—24no—
136C2×Dic6C22×C412yes—
137C3×M16—22no—
138C3×M16—22no—
139C2×C3×C8—21no—
140C2×C3×C8—21no—
141C2×C3×C8—22no—
142C2×C3×C8C22×C411yescharacteristic
143C2×C3×C8C22×C411yescharacteristic
144C2×C3×C8C2×D412yes—
145C2×C3×C8#4112yes—
146C2×C3×C8#4112yes—
147C3×C4⋊C4C22×C414yes—
148C3×C22⋊C4—22no—
149C3×C42—22no—
150C3×C42—21no—
151C3×C42#4112yes—
152(C2×D3)⋊C4—24no—
153Dic3⋊C4—24no—
154C4×Dic3—21no—
155C4×Dic3—24no—
156C4×Dic3C22×C412yes—
157C4×Dic3C2×D411yescharacteristic
158C4×Dic3C2×D411yescharacteristic
159C4×Dic3#4112yes—
160#208—22no—
161#208—22no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8C22×C412yes—
168C2×C3⋊C8C2×D412yes—
169Dic12—22no—
170D24—22no—
171#204—24no—
172C2×#41—32no—
173#135—62no—
174#135—62no—
175C2×Q16—31no—
176C2×SD16—32no—
177C2×D8—31no—
178C2×M16—32no—
179C2×M16C22×D312yes—
180C22×C8—31no—
181C22×C8—31no—
182C22×C8C22×D311yescharacteristic
183C22×C8C22×D311yescharacteristic
184C4×Q8—34no—
185C4×D4—34no—
186C4×D4—64no—
187#117—62no—
188#117C22×D312yes—
189C2×C42—31no—
190C2×C42—31no—
191C2×C42—62no—
192C2×C42C22×D311yescharacteristic
193#108—22no—
194#108—62no—
195#107C4×D312yes—
196#106C4×D312yes—
197C4⋊C8—24no—
198C4⋊C8—64no—
199#104—64no—
200#104—64no—
201#103—34no—
202#102—34no—
203C22⋊C8—64no—
204C22⋊C8—64no—
205#97—32no—
206#97—32no—
207#97—62no—
208#97—62no—
209#97—62no—
210C4×C8—22no—
211C4×C8—32no—
212C4×C8—32no—
213C4×C8—62no—
214C4×C8—62no—
215#95—62no—
216C23×C3C2×#4111yescharacteristic
217C22×D3—22no—
218C2×C3×C4—24no—
219C2×C3×C4—22no—
220C2×C3×C4—42no—
221C2×C3×C4C23×C411yescharacteristic
222C2×C3×C4C22×D411yescharacteristic
223C2×C3×C4C2×#4111yescharacteristic
224C2×C3×C4C2×#4111yescharacteristic
225C2×C3×C4C2×#4111yescharacteristic
226C2×C3×C4C2×#4111yescharacteristic
227#67—42no—
228#67—42no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3C4×D412yes—
234C2×Dic3C4×D412yes—
235D12—22no—
236D12—22no—
237C4×D3—42no—
238C4×D3—42no—
239C4×D3—84no—
240Dic6—22no—
241Dic6—22no—
242C3×C8—22no—
243C3×C8—42no—
244C3×C8#11712yes—
245C3×C8#11712yes—
246C3⋊C8—22no—
247C3⋊C8—22no—
248C3⋊C8—42no—
249#41—62no—
250#41—62no—
251#41—62no—
252#41—62no—
253C2×Q8—32no—
254C2×D4—32no—
255C2×D4—62no—
256C22×C4—21no—
257C22×C4—32no—
258C22×C4—64no—
259C22×C4—61no—
260C22×C4—62no—
261C22×C4C23×D311yescharacteristic
262Q16—62no—
263SD16—64no—
264D8—62no—
265M16—22no—
266M16—22no—
267M16—62no—
268M16—62no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—32no—
273C2×C8—32no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—62no—
279C2×C8C2×C4×D311yescharacteristic
280C2×C8C2×C4×D311yescharacteristic
281C2×C8D3×D412yes—
282C2×C8#23712yes—
283C2×C8#23912yes—
284C4⋊C4—64no—
285C4⋊C4C2×C4×D314yes—
286C22⋊C4—22no—
287C22⋊C4—64no—
288C42—22no—
289C42—21no—
290C42—32no—
291C42—31no—
292C42—31no—
293C42—32no—
294C42—64no—
295C42—61no—
296C42#23912yes—
297C22×C3—22no—
298C22×C3C2×C4×D411yescharacteristic
299C22×C3#52911yescharacteristic
300C22×C3#52911yescharacteristic
301C2×D3—44no—
302C2×D3—42no—
303C3×C4—42no—
304C3×C4—44no—
305C3×C4C2×#11711yescharacteristic
306C3×C4C4×#4111yescharacteristic
307C3×C4#53211yescharacteristic
308C3×C4#53211yescharacteristic
309Dic3—22no—
310Dic3—22no—
311Dic3—42no—
312Dic3—42no—
313C23—62no—
314C23D3×#4111yescharacteristic
315Q8—62no—
316Q8—62no—
317D4—62no—
318D4—62no—
319D4—122no—
320D4—122no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—42no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—244no—
333C2×C4C22×C4×D311yescharacteristic
334C2×C4C2×D3×D411yescharacteristic
335C2×C4C2×#23711yescharacteristic
336C2×C4C2×#23911yescharacteristic
337C2×C4C2×#23911yescharacteristic
338C2×C4D3×#4111yescharacteristic
339C8—22no—
340C8—42no—
341C8—62no—
342C8—62no—
343C8—122no—
344C8#89212yes—
345C8#89212yes—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#229311yescharacteristic
349C2×C3#295712yes—
350D3—82no—
351C22—22no—
352C22—124no—
353C22—122no—
354C22C4×D3×D411yescharacteristic
355C22#562411yescharacteristic
356C22#569111yescharacteristic
357C4—42no—
358C4—44no—
359C4—62no—
360C4—62no—
361C4—122no—
362C4—122no—
363C4C2×#89211yescharacteristic
364C4C4×#23911yescharacteristic
365C4#559611yescharacteristic
366C4#572511yescharacteristic
367C3#2129411yescharacteristic, complemented, Sylow
368C2—21no—
369C2—21no—
370C2—242no—
371C2#7596111yescharacteristic
372C2#7989012yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1945C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1945D3
hypocenterC3#21294
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1945

Complements

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

NH ≅ G/Ntype
2. #79071C2semidirect
3. C3×#1945C2semidirect
5. Dic3×#108C2semidirect
6. #70662C2semidirect
7. #70536C2semidirect
9. #69749C2semidirect
10. #68746C2semidirect
11. #68729C2semidirect
33. #4873C4semidirect
34. #4872C4semidirect
35. #4852C4semidirect
46. #1945D3semidirect
367. C3#21294semidirect

Maximal subgroups

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

subgroupindex
#791652
#790712
C3×#19452
#706962
Dic3×#1082
#706622
#705362
#697952
#697492
#687462
#687292
#212943