Finite Group Databasebeta

← #10896457

Subgroups — #10896457

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2961C211yesmaximal, characteristic, complemented
2#78580C211yesmaximal, characteristic
3#71842C212yesmaximal, complemented
4#70549C212yesmaximal, complemented
5#70546C212yesmaximal, complemented
6#70530C212yesmaximal, complemented
7#69901C211yesmaximal, characteristic, complemented
8#69897C211yesmaximal, characteristic
9#69896C211yesmaximal, characteristic, complemented
10#69845C211yesmaximal, characteristic
11#69844C211yesmaximal, characteristic, complemented
12#29271—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14#5745C2212yes—
15C2×#904C2211yescharacteristic
16C3×#446—22no—
17C3×#446—22no—
18C3×#424C2211yescharacteristic
19C3×#424C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#405C2211yescharacteristic
22C3×#405C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24#5139C2212yes—
25#5124C2212yes—
26#5110C2212yes—
27C2×#819C2211yescharacteristic
28C2×#819C2211yescharacteristic
29#5106C2211yescharacteristic
30#5102C2212yes—
31#5006C2212yes—
32#5005C2212yes—
33C2×#805C2212yes—
34#4717—22no—
35#4700—22no—
36#4556C2212yes—
37#4533C2212yes—
38#4531C2212yes—
39#4523C2212yes—
40#2961D311yescharacteristic, complemented
41#2957—31no—
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×#235C2312yes—
52C2×C3×M16C2311yescharacteristic
53C2×C3×M16C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×C4⋊Q8—21no—
57C3×#125—21no—
58C3×C22⋊Q8—22no—
59C3×C4×Q8—22no—
60C3×#117C2311yescharacteristic
61C3×#117C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63#926C2312yes—
64C2×#208C2312yes—
65C2×#208C2312yes—
66#907—42no—
67D3×M16—42no—
68C2×Dic12Q811yescharacteristic, complemented
69#904—21no—
70#904—21no—
71#904—21no—
72#904—21no—
73C2×D24Q811yescharacteristic, complemented
74C2×#204Q812yescomplemented
75#901—24no—
76C3×C4⋊C8—22no—
77C3×C4⋊C8—22no—
78C3×C4⋊C8—22no—
79C3×#97—21no—
80C3×#97—21no—
81C3×#97—21no—
82C3×#97D412yes—
83C3×C4×C8—21no—
84C3×C4×C8D412yes—
85#825—42no—
86#822—42no—
87(C2×D3)⋊C8—42no—
88#819—21no—
89#819—21no—
90#819—21no—
91#819—21no—
92#818—21no—
93#817—21no—
94Dic3⋊C8—42no—
95#806—24no—
96#805—24no—
97#805—22no—
98#805—22no—
99C2×#135—31no—
100#568—32no—
101#533C2×D311yescharacteristic
102#446—22no—
103#446—22no—
104#444—62no—
105#443—62no—
106#430—32no—
107C2×#108—31no—
108C2×#108—31no—
109#427—31no—
110#424—32no—
111#424C2×D311yescharacteristic
112#424C2×D311yescharacteristic
113#423C2×D311yescharacteristic
114#421—32no—
115#421—32no—
116C2×#104—32no—
117#410—32no—
118#408—32no—
119#405C2×D311yescharacteristic
120#405C2×D311yescharacteristic
121C2×#97C2×D311yescharacteristic
122#344—32no—
123#344—32no—
124#339—32no—
125H(3,ℤ/4)—32no—
126C2×C3×Q8—22no—
127C22×C3×C4—21no—
128C22×C3×C4C2411yescharacteristic
129C2×#67—22no—
130#235—22no—
131#235—22no—
132#235—22no—
133#235—22no—
134C2×D12C2×Q812yes—
135C2×C4×D3—22no—
136C2×Dic6C2×Q812yes—
137C3×M16—21no—
138C3×M16—21no—
139C3×M16—21no—
140C3×M16—21no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8C2×D412yes—
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8C2×Q811yescharacteristic
149C2×C3×C8C2×Q811yescharacteristic
150C2×C3×C8#4111yescharacteristic
151C2×C3×C8#4111yescharacteristic
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4#4112yes—
158C3×C42—21no—
159C3×C42—21no—
160C3×C42C2×D412yes—
161C3×C42#4111yescharacteristic
162C3×C42#4111yescharacteristic
163#217—22no—
164C4⋊Dic3—22no—
165C4×Dic3—22no—
166#208—22no—
167#208—22no—
168#208—22no—
169#208—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172Dic12—21no—
173Dic12—21no—
174D24—21no—
175D24—21no—
176#204—22no—
177#204—22no—
178ΓL(1,𝔽25)—44no—
179D3×C8—44no—
180C2×#41—32no—
181#135—61no—
182#135—61no—
183#135—61no—
184#135—61no—
185C2×Q16—31no—
186C2×SD16—32no—
187C2×D8—31no—
188#131—64no—
189#131—122no—
190C2×M16—32no—
191C2×M16—32no—
192C2×M16—122no—
193C2×M16C22×D311yescharacteristic
194C2×M16C22×D311yescharacteristic
195C22×C8C22×D311yescharacteristic
196C22×C8C22×D311yescharacteristic
197C4⋊Q8—21no—
198#125—21no—
199C22⋊Q8—22no—
200C4×Q8—22no—
201#117—32no—
202#117C22×D311yescharacteristic
203#117C22×D311yescharacteristic
204C2×C42C22×D311yescharacteristic
205#108—61no—
206#108—61no—
207#108—61no—
208#108—61no—
209#108—122no—
210#107—61no—
211#106—61no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—122no—
216#104—62no—
217#104—62no—
218#104—64no—
219#104—122no—
220#99—64no—
221C22⋊C8—122no—
222#97—21no—
223#97—21no—
224#97—21no—
225#97#6712yes—
226C4×C8—21no—
227C4×C8#6712yes—
228C23×C3C2×#4111yescharacteristic
229C22×D3—22no—
230C3×Q8—82no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—81no—
236C2×C3×C4C22×D411yescharacteristic
237C2×C3×C4C22×Q811yescharacteristic
238C2×C3×C4C2×#4111yescharacteristic
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4H(5,𝔽2)11yescharacteristic
241C2×C3×C4H(5,𝔽2)11yescharacteristic
242#67—42no—
243#67—42no—
244C2×Dic3—22no—
245C2×Dic3—42no—
246D12—22no—
247D12—22no—
248C4×D3—42no—
249C4×D3—42no—
250Dic6—22no—
251Dic6—22no—
252C3×C8—22no—
253C3×C8—41no—
254C3×C8—41no—
255C3×C8C22⋊Q811yescharacteristic
256C3×C8C22⋊Q811yescharacteristic
257C3×C8C22⋊Q811yescharacteristic
258C3×C8C22⋊Q811yescharacteristic
259C3⋊C8—42no—
260C3⋊C8—42no—
261#41—62no—
262#41—62no—
263#41—62no—
264#41—62no—
265C2×Q8—22no—
266C2×Q8—32no—
267C2×D4—32no—
268C2×D4—62no—
269C22×C4—21no—
270C22×C4—62no—
271C22×C4C23×D311yescharacteristic
272Q16—61no—
273Q16—61no—
274SD16—62no—
275SD16—62no—
276D8—61no—
277D8—61no—
278M16—21no—
279M16—21no—
280M16—21no—
281M16—21no—
282M16—62no—
283M16—62no—
284M16—62no—
285M16—62no—
286M16—124no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—124no—
294C2×C8D3×Q811yescharacteristic
295C2×C8D3×Q811yescharacteristic
296C2×C8#23911yescharacteristic
297C2×C8#23911yescharacteristic
298C2×C8C2×#6712yes—
299C2×C8C2×#6711yescharacteristic
300C2×C8C2×#6711yescharacteristic
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—62no—
306C22⋊C4—21no—
307C22⋊C4—62no—
308C22⋊C4#23512yes—
309C42—21no—
310C42—21no—
311C42—62no—
312C42#23511yescharacteristic
313C42#23511yescharacteristic
314C42C2×#6712yes—
315C22×C3—41no—
316C22×C3#52511yescharacteristic
317C22×C3#55411yescharacteristic
318C22×C3#55911yescharacteristic
319C2×D3—42no—
320C2×D3—82no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4C2×C22⋊Q811yescharacteristic
326C3×C4#53511yescharacteristic
327C3×C4#55411yescharacteristic
328C3×C4#55911yescharacteristic
329Dic3—42no—
330Dic3—82no—
331C23—62no—
332C23C2×#23511yescharacteristic
333Q8—62no—
334Q8—62no—
335Q8—82no—
336D4—62no—
337D4—62no—
338D4—122no—
339D4—122no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—62no—
345C2×C4—81no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4C2×#23511yescharacteristic
350C2×C4C2×D3×Q811yescharacteristic
351C2×C4C2×#23911yescharacteristic
352C2×C4#100911yescharacteristic
353C2×C4#100911yescharacteristic
354C2×C4C22×#6711yescharacteristic
355C8—22no—
356C8—41no—
357C8—41no—
358C8—122no—
359C8—122no—
360C8#94611yescharacteristic
361C8#94611yescharacteristic
362C8#94611yescharacteristic
363C8#94611yescharacteristic
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3#258011yescharacteristic
368D3—82no—
369C22—41no—
370C22—122no—
371C22—242no—
372C22#557911yescharacteristic
373C22#558511yescharacteristic
374C22#579111yescharacteristic
375C4—41no—
376C4—41no—
377C4—42no—
378C4—42no—
379C4—122no—
380C4—242no—
381C4#557911yescharacteristic
382C4#558511yescharacteristic
383C4C2×#94611yescharacteristic
384C4#583011yescharacteristic
385C3#2927111yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—21no—
389C2—242no—
390C2#8250111yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2961C2
socleC2×C3#2580
prime radicalC2×#904C22
hypercenter#2961D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
1. C3×#2961C2semidirect
3. #71842C2semidirect
4. #70549C2semidirect
5. #70546C2semidirect
6. #70530C2semidirect
7. #69901C2semidirect
9. #69896C2semidirect
11. #69844C2semidirect
40. #2961D3semidirect
68. C2×Dic12Q8semidirect
73. C2×D24Q8semidirect
74. C2×#204Q8semidirect
385. C3#29271semidirect

Maximal subgroups

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

subgroupindex
C3×#29612
#785802
#718422
#705492
#705462
#705302
#699012
#698972
#698962
#698452
#698442
#292713