Finite Group Databasebeta

← #10896458

Subgroups — #10896458

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#78580C211yesmaximal, characteristic
2#78543C211yesmaximal, characteristic, complemented
3C3×#1833C211yesmaximal, characteristic, complemented
4#70707C211yesmaximal, characteristic, complemented
5#70697C211yesmaximal, characteristic
6#70663C211yesmaximal, characteristic, complemented
7#70658C211yesmaximal, characteristic
8#69034C212yesmaximal, complemented
9#69033C212yesmaximal, complemented
10#68715C212yesmaximal, complemented
11#68714C212yesmaximal, complemented
12#29271—31nomaximal, Sylow
13#5745C2212yes—
14C2×#904C2211yescharacteristic
15#5517C2211yescharacteristic
16C2×C3×#108C2211yescharacteristic
17C3×#427C2211yescharacteristic
18C3×#405C2211yescharacteristic
19C2×#825C2212yes—
20#5129C2212yes—
21#5125C2212yes—
22#5119C2211yescharacteristic
23#5118C2211yescharacteristic
24C2×#819C2211yescharacteristic
25#5101C2211yescharacteristic
26C2×#815C2211yescharacteristic
27#5006C2212yes—
28#5005C2212yes—
29#5005C2212yes—
30#4999C2211yescharacteristic
31#4999C2212yes—
32#4717—22no—
33#4701—22no—
34#4700—22no—
35#4679—22no—
36C3×#344C2212yes—
37C3×H(3,ℤ/4)C2212yes—
38#4559C2212yes—
39#4532C2212yes—
40#2961—31no—
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#1833D311yescharacteristic, complemented
51C2×#235C2312yes—
52C2×C3×M16C2312yes—
53C2×C3×M16C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C3×#117C2312yes—
56C3×#117C2311yescharacteristic
57#926C2311yescharacteristic
58C2×C4×Dic3C2311yescharacteristic
59C2×#208C2311yescharacteristic
60C2×#208C2312yes—
61C22×C3⋊C8C2311yescharacteristic
62#907—42no—
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—24no—
72#890—21no—
73#888—21no—
74#878—22no—
75C4×Dic6—22no—
76C3×#108—21no—
77C3×#108—21no—
78C3×#107—21no—
79C3×#106—21no—
80C3×#97D412yes—
81C3×C4×C8D412yes—
82#825—22no—
83#825—22no—
84(C2×D3)⋊C8—42no—
85#819—21no—
86#819—21no—
87#819—42no—
88#815—21no—
89#815—21no—
90#815—21no—
91Dic3⋊C8—22no—
92Dic3⋊C8—22no—
93C8×Dic3—21no—
94#806—24no—
95#805—24no—
96#805—42no—
97C4⋊(C3⋊C8)—22no—
98C4⋊(C3⋊C8)—42no—
99C2×#135—31no—
100#568—32no—
101#533—31no—
102#446—62no—
103#446—62no—
104#444—62no—
105#443—62no—
106#430—32no—
107C2×#108—31no—
108C2×#108C2×D311yescharacteristic
109#427C2×D311yescharacteristic
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)C2×D312yes—
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—21no—
138C3×M16—21no—
139C3×M16—22no—
140C3×M16—22no—
141C2×C3×C8—22no—
142C2×C3×C8C2×D412yes—
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8C2×Q811yescharacteristic
146C2×C3×C8C2×Q811yescharacteristic
147C2×C3×C8#4111yescharacteristic
148C2×C3×C8#4111yescharacteristic
149C3×C4⋊C4—21no—
150C3×C4⋊C4—22no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—22no—
153C3×C42—22no—
154C3×C42C2×D412yes—
155#217#4112yes—
156C4⋊Dic3—22no—
157C4⋊Dic3—21no—
158Dic3⋊C4—22no—
159Dic3⋊C4—22no—
160C4×Dic3—21no—
161C4×Dic3—21no—
162C4×Dic3#4111yescharacteristic
163C4×Dic3#4111yescharacteristic
164#208—22no—
165#208—21no—
166#208—21no—
167#208—22no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—21no—
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—32no—
192C2×M16—31no—
193C2×M16—122no—
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#117C22×D312yes—
204#117C22×D311yescharacteristic
205C2×C42—31no—
206#108—21no—
207#108—21no—
208#108—61no—
209#108—61no—
210#108—122no—
211#107—21no—
212#106—21no—
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—61no—
224#97—61no—
225#97—61no—
226#97D1212yes—
227C4×C8—61no—
228C4×C8D1212yes—
229C23×C3C2×#4111yescharacteristic
230C22×D3—22no—
231C2×C3×C4—22no—
232C2×C3×C4—42no—
233C2×C3×C4C22×D411yescharacteristic
234C2×C3×C4C22×Q811yescharacteristic
235C2×C3×C4C2×#4111yescharacteristic
236C2×C3×C4C2×#4111yescharacteristic
237C2×C3×C4H(5,𝔽2)11yescharacteristic
238C2×C3×C4H(5,𝔽2)11yescharacteristic
239#67—42no—
240#67—42no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—81no—
246D12—22no—
247D12—22no—
248C4×D3—42no—
249C4×D3—42no—
250Dic6—22no—
251Dic6—22no—
252Dic6—82no—
253C3×C8—22no—
254C3×C8—42no—
255C3×C8C22⋊Q811yescharacteristic
256C3×C8C22⋊Q811yescharacteristic
257C3×C8C22⋊Q811yescharacteristic
258C3×C8C22⋊Q811yescharacteristic
259C3⋊C8—42no—
260C3⋊C8—41no—
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—21no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—62no—
284M16—61no—
285M16—61no—
286M16—62no—
287M16—124no—
288C2×C8—22no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—62no—
294C2×C8—124no—
295C2×C8C2×D1212yes—
296C2×C8C2×D1211yescharacteristic
297C2×C8C2×D1211yescharacteristic
298C2×C8#23711yescharacteristic
299C2×C8#23711yescharacteristic
300C2×C8D3×Q811yescharacteristic
301C2×C8D3×Q811yescharacteristic
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—61no—
308C22⋊C4—21no—
309C22⋊C4—22no—
310C22⋊C4—32no—
311C42—22no—
312C42—31no—
313C42—31no—
314C42—61no—
315C42—61no—
316C42C2×D1212yes—
317C22×C3—41no—
318C22×C3#52511yescharacteristic
319C22×C3#55411yescharacteristic
320C22×C3#55911yescharacteristic
321C2×D3—42no—
322C2×D3—82no—
323C3×C4—42no—
324C3×C4—82no—
325C3×C4C2×C22⋊Q811yescharacteristic
326C3×C4#53511yescharacteristic
327C3×C4#55411yescharacteristic
328C3×C4#55911yescharacteristic
329Dic3—41no—
330Dic3—42no—
331Dic3—41no—
332Dic3—42no—
333C23—62no—
334C23D3×#4111yescharacteristic
335Q8—62no—
336Q8—62no—
337Q8—242no—
338D4—62no—
339D4—62no—
340D4—122no—
341D4—122no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—62no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—62no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—241no—
351C2×C4C22×D1211yescharacteristic
352C2×C4C2×#23711yescharacteristic
353C2×C4C2×D3×Q811yescharacteristic
354C2×C4D3×#4111yescharacteristic
355C2×C4#100911yescharacteristic
356C2×C4#100911yescharacteristic
357C8—22no—
358C8—42no—
359C8—121no—
360C8—122no—
361C8—121no—
362C8#89711yescharacteristic
363C8#89711yescharacteristic
364C8#89711yescharacteristic
365C8#89711yescharacteristic
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3#258011yescharacteristic
370D3—82no—
371C22—41no—
372C22—122no—
373C22—242no—
374C22#552411yescharacteristic
375C22#568111yescharacteristic
376C22#569011yescharacteristic
377C4—42no—
378C4—82no—
379C4—121no—
380C4—121no—
381C4—122no—
382C4—122no—
383C4C2×#89711yescharacteristic
384C4#552511yescharacteristic
385C4#568111yescharacteristic
386C4#569011yescharacteristic
387C3#2927111yescharacteristic, complemented, Sylow
388C2—21no—
389C2—21no—
390C2—21no—
391C2—242no—
392C2#7419011yescharacteristic
subgroupG/H
centerC4#5525
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1833C2
socleC2×C3#2580
prime radicalC2×#904C22
hypercenter#1833D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1833

Complements

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

NH ≅ G/Ntype
2. #78543C2semidirect
3. C3×#1833C2semidirect
4. #70707C2semidirect
6. #70663C2semidirect
8. #69034C2semidirect
9. #69033C2semidirect
10. #68715C2semidirect
11. #68714C2semidirect
50. #1833D3semidirect
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
#785802
#785432
C3×#18332
#707072
#706972
#706632
#706582
#690342
#690332
#687152
#687142
#292713