Finite Group Databasebeta

← #10896459

Subgroups — #10896459

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
1#79249C211yesmaximal, characteristic
2#78543C211yesmaximal, characteristic, complemented
3C3×#1953C211yesmaximal, characteristic, complemented
4#70738C212yesmaximal, complemented
5#70735C212yesmaximal, complemented
6#70705C211yesmaximal, characteristic, complemented
7#70698C211yesmaximal, characteristic, complemented
8#70697C211yesmaximal, characteristic
9#69897C211yesmaximal, characteristic
10#69042C212yesmaximal, complemented
11#69034C212yesmaximal, complemented
12#29271—31nomaximal, Sylow
13#5745C2212yes—
14C2×#904C2211yescharacteristic
15#5517C2211yescharacteristic
16C3×#430C2212yes—
17C2×C3×#108C2211yescharacteristic
18C2×C3×#108C2211yescharacteristic
19C3×#405C2211yescharacteristic
20#5139C2212yes—
21#5139C2212yes—
22C2×#825C2212yes—
23#5129C2212yes—
24#5119C2211yescharacteristic
25#5119C2212yes—
26#5118C2211yescharacteristic
27#5106C2211yescharacteristic
28#5101C2211yescharacteristic
29C2×#815C2211yescharacteristic
30#5006C2212yes—
31#4999C2211yescharacteristic
32#4895—22no—
33#4894—22no—
34#4701—22no—
35#4679—22no—
36C3×#344C2212yes—
37#4559C2212yes—
38#4557C2212yes—
39#4555C2212yes—
40#2961—31no—
41#2957—31no—
42#2098—32no—
43#2031—32no—
44#1976—31no—
45#1975—31no—
46#1953D311yescharacteristic, complemented
47#1946—31no—
48#1868—32no—
49#1861—32no—
50#1833—31no—
51C2×#235C2312yes—
52C2×C3×M16C2312yes—
53C2×C3×M16C2311yescharacteristic
54C2×C3×M16C2312yes—
55C22×C3×C8C2311yescharacteristic
56C3×#117C2311yescharacteristic
57#926C2312yes—
58#926C2311yescharacteristic
59C2×C4×Dic3C2311yescharacteristic
60C2×#208C2311yescharacteristic
61C22×C3⋊C8C2311yescharacteristic
62#907—24no—
63D3×M16—42no—
64C2×Dic12Q811yescharacteristic, complemented
65#904—21no—
66#904—21no—
67#904—21no—
68#904—21no—
69C2×D24Q811yescharacteristic, complemented
70C2×#204Q812yescomplemented
71#901—42no—
72#890—21no—
73#888—21no—
74#878—22no—
75C4×Dic6—22no—
76C3×#108—21no—
77C3×#108—21no—
78C3×#108—21no—
79C3×#108—21no—
80C3×#108—42no—
81C3×#97—21no—
82C3×C4×C8—21no—
83#825—24no—
84#825—22no—
85#825—22no—
86#825—42no—
87(C2×D3)⋊C8—42no—
88#818—21no—
89#817—21no—
90#815—21no—
91#815—21no—
92#815D412yes—
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95Dic3⋊C8—42no—
96C8×Dic3D412yes—
97#806—24no—
98C4⋊(C3⋊C8)—22no—
99C2×#135—31no—
100#568—32no—
101#533—31no—
102#446—62no—
103#446—62no—
104#444—62no—
105#443—62no—
106#430C2×D312yes—
107C2×#108C2×D311yescharacteristic
108C2×#108C2×D311yescharacteristic
109#427—31no—
110#424—31no—
111#424—31no—
112#424—32no—
113#423—31no—
114#421—32no—
115#421—32no—
116C2×#104—32no—
117#410—32no—
118#408—32no—
119#405—31no—
120#405C2×D311yescharacteristic
121C2×#97—31no—
122#344—32no—
123#344C2×D312yes—
124#339—32no—
125H(3,ℤ/4)—32no—
126C22×C3×C4C2411yescharacteristic
127C2×#67—22no—
128C22×Dic3—21no—
129#235—22no—
130#235—22no—
131#235—22no—
132#235—22no—
133C2×D12C2×Q812yes—
134C2×C4×D3—22no—
135C2×Dic6—22no—
136C2×Dic6C2×Q812yes—
137C3×M16—22no—
138C3×M16—22no—
139C3×M16—22no—
140C3×M16—21no—
141C3×M16—21no—
142C3×M16—22no—
143C2×C3×C8—21no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
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—21no—
153C3×C22⋊C4#4112yes—
154C3×C42—21no—
155#217—21no—
156#217—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—21no—
159C4⋊Dic3—22no—
160Dic3⋊C4—22no—
161Dic3⋊C4—22no—
162C4×Dic3—21no—
163C4×Dic3—22no—
164C4×Dic3C2×D412yes—
165C4×Dic3#4111yescharacteristic
166C4×Dic3#4111yescharacteristic
167#208—21no—
168#208—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8C2×D412yes—
173Dic12—21no—
174Dic12—21no—
175D24—21no—
176D24—21no—
177#204—22no—
178#204—22no—
179ΓL(1,𝔽25)—44no—
180D3×C8—44no—
181C2×#41—32no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135—61no—
186C2×Q16—31no—
187C2×SD16—32no—
188C2×D8—31no—
189#131—64no—
190#131—122no—
191C2×M16—31no—
192C2×M16—122no—
193C2×M16C22×D312yes—
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—21no—
209#108—21no—
210#108—42no—
211#107—61no—
212#106—61no—
213C4⋊C8—62no—
214C4⋊C8—62no—
215C4⋊C8—62no—
216C4⋊C8—122no—
217#104—62no—
218#104—62no—
219#104—64no—
220#104—122no—
221#99—64no—
222C22⋊C8—122no—
223#97—21no—
224#97—32no—
225#97—61no—
226#97—61no—
227C4×C8—21no—
228C4×C8—32no—
229C23×C3C2×#4111yescharacteristic
230C22×D3—22no—
231C2×C3×C4—22no—
232C2×C3×C4C22×D411yescharacteristic
233C2×C3×C4C22×Q811yescharacteristic
234C2×C3×C4C2×#4111yescharacteristic
235C2×C3×C4C2×#4111yescharacteristic
236C2×C3×C4H(5,𝔽2)11yescharacteristic
237C2×C3×C4H(5,𝔽2)11yescharacteristic
238#67—42no—
239#67—42no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—42no—
245C2×Dic3—81no—
246D12—22no—
247D12—22no—
248C4×D3—42no—
249C4×D3—42no—
250Dic6—22no—
251Dic6—22no—
252Dic6—82no—
253C3×C8—42no—
254C3×C8—41no—
255C3×C8—42no—
256C3×C8C22⋊Q811yescharacteristic
257C3×C8C22⋊Q811yescharacteristic
258C3×C8C22⋊Q811yescharacteristic
259C3×C8C22⋊Q811yescharacteristic
260C3⋊C8—22no—
261C3⋊C8—41no—
262#41—62no—
263#41—62no—
264#41—62no—
265#41—62no—
266C2×Q8—32no—
267C2×Q8—62no—
268C2×D4—32no—
269C2×D4—62no—
270C22×C4—61no—
271C22×C4—62no—
272C22×C4C23×D311yescharacteristic
273Q16—61no—
274Q16—61no—
275SD16—62no—
276SD16—62no—
277D8—61no—
278D8—61no—
279M16—22no—
280M16—22no—
281M16—21no—
282M16—22no—
283M16—22no—
284M16—21no—
285M16—61no—
286M16—61no—
287M16—124no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—32no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—124no—
296C2×C8#23511yescharacteristic
297C2×C8#23511yescharacteristic
298C2×C8D3×D411yescharacteristic
299C2×C8D3×D411yescharacteristic
300C2×C8D3×Q811yescharacteristic
301C2×C8D3×Q811yescharacteristic
302C4⋊C4—21no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—62no—
308C22⋊C4—61no—
309C22⋊C4—62no—
310C22⋊C4#23912yes—
311C42—21no—
312C42—32no—
313C42—31no—
314C42—31no—
315C42—61no—
316C42—62no—
317C22×C3—41no—
318C22×C3#52511yescharacteristic
319C22×C3#55411yescharacteristic
320C22×C3#55911yescharacteristic
321C2×D3—42no—
322C2×D3—82no—
323C3×C4—42no—
324C3×C4C2×C22⋊Q811yescharacteristic
325C3×C4#53511yescharacteristic
326C3×C4#55411yescharacteristic
327C3×C4#55911yescharacteristic
328Dic3—41no—
329Dic3—42no—
330Dic3—41no—
331Dic3—42no—
332Dic3—82no—
333C23—62no—
334C23C2×#23911yescharacteristic
335Q8—62no—
336Q8—62no—
337Q8—242no—
338D4—62no—
339D4—62no—
340D4—122no—
341D4—122no—
342C2×C4—22no—
343C2×C4—62no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—62no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—241no—
351C2×C4C2×#23511yescharacteristic
352C2×C4C2×D3×D411yescharacteristic
353C2×C4#100411yescharacteristic
354C2×C4C2×D3×Q811yescharacteristic
355C2×C4D3×#4111yescharacteristic
356C2×C4#100911yescharacteristic
357C8—42no—
358C8—42no—
359C8—41no—
360C8—62no—
361C8—121no—
362C8#89611yescharacteristic
363C8#89611yescharacteristic
364C8#89611yescharacteristic
365C8#89611yescharacteristic
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3#258011yescharacteristic
370D3—82no—
371C22—41no—
372C22—122no—
373C22—242no—
374C22#558511yescharacteristic
375C22#562711yescharacteristic
376C22#573011yescharacteristic
377C4—42no—
378C4—121no—
379C4—121no—
380C4—122no—
381C4—122no—
382C4—242no—
383C4C2×#89611yescharacteristic
384C4#555911yescharacteristic
385C4#560711yescharacteristic
386C4#569011yescharacteristic
387C3#2927111yescharacteristic, complemented, Sylow
388C2—21no—
389C2—21no—
390C2—21no—
391C2—242no—
392C2#7607511yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1953C2
socleC2×C3#2580
prime radicalC2×#904C22
hypercenter#1953D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
2. #78543C2semidirect
3. C3×#1953C2semidirect
4. #70738C2semidirect
5. #70735C2semidirect
6. #70705C2semidirect
7. #70698C2semidirect
10. #69042C2semidirect
11. #69034C2semidirect
46. #1953D3semidirect
64. C2×Dic12Q8semidirect
69. C2×D24Q8semidirect
70. C2×#204Q8semidirect
387. C3#29271semidirect

Maximal subgroups

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

subgroupindex
#792492
#785432
C3×#19532
#707382
#707352
#707052
#706982
#706972
#698972
#690422
#690342
#292713