Finite Group Databasebeta

← #86765

Subgroups — #86765

Order 384.

1022 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×Q8C211yesmaximal, characteristic, complemented
2#5817C212yesmaximal, complemented
3#5816C212yesmaximal, complemented
4#5733C212yesmaximal, complemented
5#5730C212yesmaximal, complemented
6#5725C211yesmaximal, characteristic, complemented
7#5721C211yesmaximal, characteristic, complemented
8#5647C214yesmaximal, complemented
9#5643C212yesmaximal, complemented
10#5638C214yesmaximal, complemented
11#5631C212yesmaximal, complemented
12#5585C211yesmaximal, characteristic, complemented
13#5578C211yesmaximal, characteristic, complemented
14#5562C212yesmaximal, complemented
15#5559C212yesmaximal, complemented
16#5557C211yesmaximal, characteristic, complemented
17#5553C211yesmaximal, characteristic, complemented
18#3554—31nomaximal, Sylow
19C22×C3×Q8C2212yes—
20C2×#239C2212yes—
21C2×#235C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×C4⋊Q8C2212yes—
24C3×C22⋊Q8C2214yes—
25C3×C22⋊Q8C2212yes—
26C3×C4×Q8C2211yescharacteristic
27C3×C4×D4C2211yescharacteristic
28C3×C4×D4C2212yes—
29#947C2212yes—
30#947C2214yes—
31#947C2212yes—
32#946C2214yes—
33#946C2214yes—
34#946C2214yes—
35#946C2211yescharacteristic
36Q8×Dic3C2212yes—
37#944C2212yes—
38#944C2211yescharacteristic
39#940C2212yes—
40#938C2211yescharacteristic
41#936C2211yescharacteristic
42#930C2212yes—
43#930C2214yes—
44#929C2214yes—
45#929C2214yes—
46C4×#67C2214yes—
47C4×#67C2212yes—
48#926C2212yes—
49#898C2214yes—
50#897C2214yes—
51#897C2211yescharacteristic
52#897C2211yescharacteristic
53#896C2212yes—
54#895C2212yes—
55#895C2212yes—
56#894C2214yes—
57#894C2214yes—
58#893C2212yes—
59#893C2212yes—
60#893C2212yes—
61#893C2212yes—
62D3×C4⋊C4C2214yes—
63#890C2212yes—
64#890C2211yescharacteristic
65#889C2212yes—
66#889C2214yes—
67#888C2211yescharacteristic
68#887C2212yes—
69#885C2214yes—
70#884C2214yes—
71#884C2214yes—
72#883C2214yes—
73#883C2214yes—
74#883C2212yes—
75#879C2212yes—
76#875C2212yes—
77#874C2211yescharacteristic
78C4×D12C2212yes—
79#872C2212yes—
80C42×D3C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#562—34no—
85#559—31no—
86#559—32no—
87#557—31no—
88#554—32no—
89#550—31no—
90D4×Q8D311yescharacteristic, complemented
91#548—32no—
92#547—32no—
93#542—32no—
94#542—32no—
95#540—32no—
96#539—34no—
97#536—32no—
98#533—31no—
99#532—31no—
100#529—31no—
101C2×C3×Q8—44no—
102C2×C3×Q8C2312yes—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×Q8C2314yes—
105C2×C3×D4C2311yescharacteristic
106C22×C3×C4C2312yes—
107C22×C3×C4C2314yes—
108C2×#67C2312yes—
109C2×#67C2314yes—
110C2×#67C2314yes—
111#239—44no—
112#235—44no—
113C2×D12C2312yes—
114C2×D12C2312yes—
115C2×D12C2312yes—
116C2×D12C2314yes—
117C2×C4×D3C2311yescharacteristic
118C2×C4×D3C2311yescharacteristic
119C2×C4×D3C2311yescharacteristic
120C2×C4×D3C2314yes—
121C2×C4×D3C2314yes—
122C2×C4×D3C2314yes—
123C2×Dic6C2311yescharacteristic
124C2×Dic6C2311yescharacteristic
125C2×Dic6C2312yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2312yes—
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2311yescharacteristic
131C3×C4⋊C4C2314yes—
132C3×C22⋊C4C2312yes—
133C3×C22⋊C4C2314yes—
134C3×C42C2311yescharacteristic
135C3×C42C2312yes—
136#217C2312yes—
137#217C2312yes—
138#217C2312yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2314yes—
153C4⋊Dic3C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163C4×Dic3C2311yescharacteristic
164C4×Dic3C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2314yes—
169C2×#41—32no—
170C2×#41—32no—
171C22×Q8C2×D312yes—
172C4⋊Q8—31no—
173C4⋊Q8—32no—
174C4⋊Q8—31no—
175C4⋊Q8C2×D311yescharacteristic
176C4⋊Q8C2×D312yes—
177#127—32no—
178#127—31no—
179#126—34no—
180#126—32no—
181#125—31no—
182#125—32no—
183#125—34no—
184#125—32no—
185#124—32no—
186#124—32no—
187#124—34no—
188#124—31no—
189#124—34no—
190#124—32no—
191#123—34no—
192#123—34no—
193#123—34no—
194#123—34no—
195#123—32no—
196#123—34no—
197#123—34no—
198C22⋊Q8—31no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8—34no—
202C22⋊Q8—31no—
203C22⋊Q8—31no—
204C22⋊Q8—32no—
205C22⋊Q8—34no—
206C22⋊Q8C2×D314yes—
207C22⋊Q8C2×D312yes—
208#121—31no—
209#121—34no—
210#121—32no—
211#121—32no—
212#121—32no—
213#121—34no—
214#121—34no—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—31no—
218C4×Q8—31no—
219C4×Q8—32no—
220C4×Q8C2×D311yescharacteristic
221C4×D4—32no—
222C4×D4—34no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—32no—
226C4×D4—32no—
227C4×D4—32no—
228C4×D4C2×D311yescharacteristic
229C4×D4C2×D312yes—
230#117—32no—
231#117—32no—
232C2×C4⋊C4—34no—
233C2×C42—31no—
234C23×C3C2412yes—
235C22×D3C2411yescharacteristic
236C22×D3C2414yes—
237C3×Q8—24no—
238C3×Q8—22no—
239C3×Q8—44no—
240C3×D4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—44no—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2414yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2412yes—
247C2×C3×C4C2412yes—
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250#67—44no—
251#67—44no—
252#67—42no—
253C2×Dic3C2412yes—
254C2×Dic3C2412yes—
255C2×Dic3C2411yescharacteristic
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2411yescharacteristic
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260D12—24no—
261D12—24no—
262D12—42no—
263D12—44no—
264C4×D3—22no—
265C4×D3—22no—
266C4×D3—41no—
267C4×D3—44no—
268C4×D3—44no—
269C4×D3—44no—
270Dic6—22no—
271Dic6—22no—
272Dic6—42no—
273#41—124no—
274#41—124no—
275C2×Q8—31no—
276C2×Q8—31no—
277C2×Q8—32no—
278C2×Q8—44no—
279C2×Q8C22×D312yes—
280C2×Q8C22×D311yescharacteristic
281C2×Q8C22×D314yes—
282C2×D4—32no—
283C2×D4—34no—
284C2×D4—34no—
285C2×D4—32no—
286C2×D4—32no—
287C2×D4—32no—
288C2×D4—34no—
289C2×D4C22×D311yescharacteristic
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—34no—
294C22×C4—34no—
295C22×C4—34no—
296C22×C4C22×D312yes—
297C22×C4C22×D314yes—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—34no—
301C4⋊C4—32no—
302C4⋊C4—32no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—32no—
307C4⋊C4—31no—
308C4⋊C4—32no—
309C4⋊C4—34no—
310C4⋊C4—32no—
311C4⋊C4—34no—
312C4⋊C4—32no—
313C4⋊C4C22×D314yes—
314C4⋊C4C22×D311yescharacteristic
315C4⋊C4C22×D311yescharacteristic
316C4⋊C4C22×D312yes—
317C4⋊C4C22×D312yes—
318C4⋊C4C22×D312yes—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—34no—
331C22⋊C4C22×D312yes—
332C22⋊C4C22×D314yes—
333C42—31no—
334C42—31no—
335C42—31no—
336C42—32no—
337C42—32no—
338C42—34no—
339C42C22×D311yescharacteristic
340C42C22×D312yes—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3C2511yescharacteristic
345C2×D3—21no—
346C2×D3—24no—
347C2×D3—21no—
348C2×D3—24no—
349C2×D3—24no—
350C2×D3—21no—
351C3×C4—22no—
352C3×C4—24no—
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—21no—
356C3×C4H(5,𝔽2)12yes—
357C3×C42−1+412yes—
358Dic3—22no—
359Dic3—21no—
360Dic3—22no—
361Dic3—21no—
362Dic3—21no—
363Dic3—22no—
364Dic3—22no—
365C23—31no—
366C23—34no—
367C23C23×D312yes—
368Q8—22no—
369Q8—24no—
370Q8—44no—
371Q8—62no—
372Q8—62no—
373Q8—122no—
374D4—22no—
375D4—64no—
376D4—64no—
377D4—124no—
378D4—122no—
379D4—124no—
380D4—122no—
381D4—124no—
382C2×C4—24no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—44no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—124no—
394C2×C4—124no—
395C2×C4—124no—
396C2×C4—121no—
397C2×C4C23×D311yescharacteristic
398C2×C4C23×D314yes—
399C2×C4C23×D312yes—
400C2×C4C23×D312yes—
401C2×C4C23×D312yes—
402C2×C4C23×D311yescharacteristic
403C2×C4C23×D311yescharacteristic
404C2×C3—42no—
405C2×C3C2×H(5,𝔽2)11yescharacteristic
406C2×C3C2×2−1+411yescharacteristic
407C2×C3#58511yescharacteristic
408D3—41no—
409D3—44no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—61no—
414C22—64no—
415C22—61no—
416C22—64no—
417C22—64no—
418C22—61no—
419C22C24×D311yescharacteristic
420C4—22no—
421C4—22no—
422C4—24no—
423C4—22no—
424C4—21no—
425C4—61no—
426C4—62no—
427C4—61no—
428C4—62no—
429C4—62no—
430C4—61no—
431C4—62no—
432C4#100412yes—
433C4#100712yes—
434C3#355411yescharacteristic, complemented, Sylow
435C2—42no—
436C2—121no—
437C2—124no—
438C2C2×#100411yescharacteristic
439C2C2×#100711yescharacteristic
440C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×D4×Q8C2
hypercenterD4×Q8D3
hypocenterC3#3554
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×D4×Q8C2semidirect
2. #5817C2semidirect
3. #5816C2semidirect
4. #5733C2semidirect
5. #5730C2semidirect
6. #5725C2semidirect
7. #5721C2semidirect
8. #5647C2semidirect
9. #5643C2semidirect
10. #5638C2semidirect
11. #5631C2semidirect
12. #5585C2semidirect
13. #5578C2semidirect
14. #5562C2semidirect
15. #5559C2semidirect
16. #5557C2semidirect
17. #5553C2semidirect
90. D4×Q8D3semidirect
434. C3#3554semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Q82
#58172
#58162
#57332
#57302
#57252
#57212
#56472
#56432
#56382
#56312
#55852
#55782
#55622
#55592
#55572
#55532
#35543