Finite Group Databasebeta

← #10896451

Subgroups — #10896451

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2957C211yesmaximal, characteristic
2#78543C211yesmaximal, characteristic, complemented
3#72086C211yesmaximal, characteristic
4#72085C211yesmaximal, characteristic
5#72028C212yesmaximal, complemented
6#72026C212yesmaximal, complemented
7#70706C211yesmaximal, characteristic, complemented
8#70705C211yesmaximal, characteristic, complemented
9#70405C212yesmaximal, complemented
10#70394C212yesmaximal, complemented
11#69896C211yesmaximal, characteristic, complemented
12#29271—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C3×#568C2212yes—
15#5517C2211yescharacteristic
16C3×#444—22no—
17C3×#443—22no—
18C2×C3×#108C2211yescharacteristic
19C3×#421C2212yes—
20C3×#405C2211yescharacteristic
21#5240C2212yes—
22#5236C2212yes—
23C2×#837C2212yes—
24#5234C2212yes—
25#5124C2212yes—
26#5119C2211yescharacteristic
27#5119C2212yes—
28#5118C2211yescharacteristic
29C2×#819C2211yescharacteristic
30#5106C2211yescharacteristic
31#5101C2211yescharacteristic
32C2×#815C2211yescharacteristic
33#4999C2211yescharacteristic
34#4701—22no—
35#4679—22no—
36#4556C2212yes—
37#4556C2212yes—
38#4555C2212yes—
39#4531C2212yes—
40#2961—31no—
41#2957D311yescharacteristic
42#2098—32no—
43#2031—32no—
44#1976—31no—
45#1975—31no—
46#1953—31no—
47#1946—31no—
48#1868—32no—
49#1861—32no—
50#1833—31no—
51C2×C3×#41C2312yes—
52C3×#135—21no—
53C3×#135—21no—
54C3×#135—21no—
55C3×#135—21no—
56C2×C3×Q16Q811yescharacteristic, complemented
57C2×C3×SD16Q812yescomplemented
58C2×C3×D8Q811yescharacteristic, complemented
59C3×#131—24no—
60C3×#131—42no—
61C2×C3×M16—42no—
62C2×C3×M16C2312yes—
63C2×C3×M16C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#117C2311yescharacteristic
66#926C2312yes—
67#926C2311yescharacteristic
68C2×C4×Dic3C2311yescharacteristic
69C2×#208C2311yescharacteristic
70C2×#208C2312yes—
71C22×C3⋊C8C2311yescharacteristic
72#890—21no—
73#888—21no—
74#878—22no—
75C4×Dic6—22no—
76C3×#108—21no—
77C3×#108—21no—
78C3×#104—24no—
79C3×#97D412yes—
80C3×C4×C8D412yes—
81#837—22no—
82#837—22no—
83#837—42no—
84#834—24no—
85#830—42no—
86#822—42no—
87#819—21no—
88#819—21no—
89#818—21no—
90#817—21no—
91#815—21no—
92#815—21no—
93#815—21no—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96Dic3⋊C8—42no—
97C8×Dic3—21no—
98C4⋊(C3⋊C8)—22no—
99C2×#135C2×D311yescharacteristic
100#568C2×D312yes—
101#533—31no—
102#446—62no—
103#446—62no—
104#444—22no—
105#443—22no—
106#430—32no—
107C2×#108—31no—
108C2×#108C2×D311yescharacteristic
109#427—31no—
110#424—31no—
111#424—31no—
112#424—32no—
113#423—31no—
114#421—32no—
115#421C2×D312yes—
116C2×#104—32no—
117#410—32no—
118#408—32no—
119#405—31no—
120#405C2×D311yescharacteristic
121C2×#97—31no—
122#344—32no—
123#344—32no—
124#339—32no—
125H(3,ℤ/4)—32no—
126C3×#41—22no—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—22no—
130C2×C3×Q8C2×Q812yes—
131C2×C3×D4—22no—
132C2×C3×D4C2×Q812yes—
133C22×C3×C4—22no—
134C22×C3×C4C2411yescharacteristic
135C22×Dic3—21no—
136C2×Dic6—22no—
137C3×Q16—21no—
138C3×Q16—21no—
139C3×SD16—22no—
140C3×SD16—22no—
141C3×D8—21no—
142C3×D8—21no—
143C3×M16—21no—
144C3×M16—21no—
145C3×M16—22no—
146C3×M16—22no—
147C3×M16—44no—
148C2×C3×C8—22no—
149C2×C3×C8—44no—
150C2×C3×C8C2×D412yes—
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8C2×D411yescharacteristic
153C2×C3×C8C2×Q811yescharacteristic
154C2×C3×C8C2×Q811yescharacteristic
155C2×C3×C8#4111yescharacteristic
156C2×C3×C8#4111yescharacteristic
157C3×C4⋊C4—21no—
158C3×C22⋊C4—21no—
159C3×C42C2×D412yes—
160#217—22no—
161#217#4112yes—
162C4⋊Dic3—22no—
163C4⋊Dic3—21no—
164C4⋊Dic3—22no—
165Dic3⋊C4—22no—
166Dic3⋊C4—22no—
167C4×Dic3—21no—
168C4×Dic3—21no—
169C4×Dic3—22no—
170C4×Dic3#4111yescharacteristic
171C4×Dic3#4111yescharacteristic
172#208—22no—
173#208—22no—
174#208—21no—
175#208—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—21no—
181C2×#41C22×D312yes—
182#135—21no—
183#135—21no—
184#135—21no—
185#135—21no—
186C2×Q16Dic611yescharacteristic, complemented
187C2×SD16Dic612yescomplemented
188C2×D8Dic611yescharacteristic, complemented
189#131—24no—
190#131—42no—
191C2×M16—32no—
192C2×M16—31no—
193C2×M16—42no—
194C2×M16C22×D312yes—
195C2×M16C22×D311yescharacteristic
196C22×C8—31no—
197C22×C8C22×D311yescharacteristic
198C4⋊Q8—61no—
199#125—61no—
200C22⋊Q8—62no—
201C4×Q8—62no—
202#117—31no—
203#117—32no—
204#117C22×D311yescharacteristic
205C2×C42—31no—
206#108—21no—
207#108—21no—
208#108—61no—
209#108—61no—
210#108—122no—
211#107—61no—
212#106—61no—
213C4⋊C8—62no—
214C4⋊C8—62no—
215C4⋊C8—62no—
216C4⋊C8—122no—
217#104—24no—
218#104—62no—
219#104—62no—
220#104—122no—
221#99—64no—
222C22⋊C8—122no—
223#97—61no—
224#97—61no—
225#97—61no—
226#97#6712yes—
227C4×C8—61no—
228C4×C8#6712yes—
229C23×C3—22no—
230C23×C3C2×#4111yescharacteristic
231C3×Q8—22no—
232C3×Q8—22no—
233C3×D4—22no—
234C3×D4—22no—
235C3×D4—42no—
236C3×D4—42no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4C22×D411yescharacteristic
242C2×C3×C4C22×Q811yescharacteristic
243C2×C3×C4C2×#4111yescharacteristic
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4H(5,𝔽2)11yescharacteristic
246C2×C3×C4H(5,𝔽2)11yescharacteristic
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—22no—
250C2×Dic3—42no—
251C2×Dic3—81no—
252Dic6—82no—
253C3×C8—22no—
254C3×C8—42no—
255C3×C8C22⋊Q811yescharacteristic
256C3×C8C22⋊Q811yescharacteristic
257C3×C8C22⋊Q811yescharacteristic
258C3×C8C22⋊Q811yescharacteristic
259C3⋊C8—41no—
260C3⋊C8—41no—
261C3⋊C8—42no—
262#41—22no—
263#41—22no—
264#41—22no—
265#41—22no—
266C2×Q8—62no—
267C2×Q8C2×Dic612yes—
268C2×D4—22no—
269C2×D4C2×Dic612yes—
270C22×C4—22no—
271C22×C4—61no—
272C22×C4C23×D311yescharacteristic
273Q16—21no—
274Q16—21no—
275SD16—22no—
276SD16—22no—
277D8—21no—
278D8—21no—
279M16—21no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—44no—
284M16—62no—
285M16—61no—
286M16—61no—
287M16—62no—
288C2×C8—22no—
289C2×C8—44no—
290C2×C8—61no—
291C2×C8—62no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8C2×Dic611yescharacteristic
296C2×C8C2×Dic611yescharacteristic
297C2×C8#23511yescharacteristic
298C2×C8#23511yescharacteristic
299C2×C8C2×#6712yes—
300C2×C8C2×#6711yescharacteristic
301C2×C8C2×#6711yescharacteristic
302C4⋊C4—21no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—62no—
308C22⋊C4—21no—
309C22⋊C4—32no—
310C22⋊C4—62no—
311C42—31no—
312C42—31no—
313C42—61no—
314C42—61no—
315C42—62no—
316C42C2×#6712yes—
317C22×C3—41no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3#52511yescharacteristic
321C22×C3#55411yescharacteristic
322C22×C3#55911yescharacteristic
323C3×C4—42no—
324C3×C4—42no—
325C3×C4C2×C22⋊Q811yescharacteristic
326C3×C4#53511yescharacteristic
327C3×C4#55411yescharacteristic
328C3×C4#55911yescharacteristic
329Dic3—41no—
330Dic3—42no—
331Dic3—41no—
332Dic3—82no—
333C23—22no—
334C23D3×#4111yescharacteristic
335Q8—22no—
336Q8—22no—
337Q8—242no—
338D4—22no—
339D4—22no—
340D4—42no—
341D4—42no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—62no—
349C2×C4—122no—
350C2×C4—241no—
351C2×C4C22×Dic611yescharacteristic
352C2×C4C2×#23511yescharacteristic
353C2×C4#100411yescharacteristic
354C2×C4#100411yescharacteristic
355C2×C4D3×#4111yescharacteristic
356C2×C4C22×#6711yescharacteristic
357C8—22no—
358C8—42no—
359C8—121no—
360C8—121no—
361C8—122no—
362C8#92411yescharacteristic
363C8#92411yescharacteristic
364C8#92411yescharacteristic
365C8#92411yescharacteristic
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—82no—
370C2×C3#258011yescharacteristic
371C22—41no—
372C22—42no—
373C22—82no—
374C22#553911yescharacteristic
375C22#555911yescharacteristic
376C22#582811yescharacteristic
377C4—42no—
378C4—42no—
379C4—121no—
380C4—121no—
381C4—122no—
382C4—242no—
383C4#553911yescharacteristic
384C4#555911yescharacteristic
385C4C2×#92411yescharacteristic
386C4#580511yescharacteristic
387C3#2927111yescharacteristic, complemented, Sylow
388C2—21no—
389C2—21no—
390C2—21no—
391C2—82no—
392C2#8238811yescharacteristic
subgroupG/H
centerC4#5805
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2957C2
socleC2×C3#2580
prime radicalC2×C3×#135C22
hypercenter#2957D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. #78543C2semidirect
5. #72028C2semidirect
6. #72026C2semidirect
7. #70706C2semidirect
8. #70705C2semidirect
9. #70405C2semidirect
10. #70394C2semidirect
11. #69896C2semidirect
56. C2×C3×Q16Q8semidirect
57. C2×C3×SD16Q8semidirect
58. C2×C3×D8Q8semidirect
186. C2×Q16Dic6semidirect
187. C2×SD16Dic6semidirect
188. C2×D8Dic6semidirect
387. C3#29271semidirect

Maximal subgroups

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

subgroupindex
C3×#29572
#785432
#720862
#720852
#720282
#720262
#707062
#707052
#704052
#703942
#698962
#292713