Finite Group Databasebeta

← #10795560

Subgroups — #10795560

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78486C211yesmaximal, characteristic
2C3×#1773C211yesmaximal, characteristic, complemented
3#70718C211yesmaximal, characteristic
4#70698C211yesmaximal, characteristic, complemented
5#70664C211yesmaximal, characteristic
6#70657C211yesmaximal, characteristic, complemented
7#69898C212yesmaximal
8#69895C212yesmaximal, complemented
9#69829C211yesmaximal, characteristic, complemented
10#68749C212yesmaximal, complemented
11#68737C212yesmaximal
12#20788—31nomaximal, Sylow
13C2×#901C2211yescharacteristic
14C2×#901C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×C3×#108C2211yescharacteristic
17C3×#427C2211yescharacteristic
18C3×#405C2212yes—
19C2×C3×#97C2211yescharacteristic
20#5132C2212yes—
21C2×#819C2211yescharacteristic
22#5106C2211yescharacteristic
23#5101C2212yes—
24#5101C2211yescharacteristic
25#5101C2211yescharacteristic
26#5006C2212yes—
27C4×#208C2211yescharacteristic
28#4708C412yescomplemented
29#4708C412yescomplemented
30#4707C412yescomplemented
31D3×#97C412yescomplemented
32C3×#339C2212yes—
33C3⋊#338C2212yes—
34#4559C2212yes—
35#4557C2212yes—
36#4533C2212yes—
37#4531C2212yes—
38#4531C2212yes—
39#4523C2212yes—
40#2875—31no—
41#1841—32no—
42#1833—31no—
43#1833—31no—
44#1827—32no—
45#1813—31no—
46#1812—31no—
47#1773—31no—
48#1773D311yescharacteristic, complemented
49#1757—32no—
50#1747—32no—
51C2×#235C2311yescharacteristic
52C2×C3×M16C2312yes—
53C22×C3×C8C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C3×#117C2312yes—
56C2×C3×C42C2311yescharacteristic
57C4×#67—22no—
58#926C2312yes—
59#926C2311yescharacteristic
60C2×#208C2312yes—
61C2×#208C2311yescharacteristic
62C2×#208C2311yescharacteristic
63#901—21no—
64#901—21no—
65#901—21no—
66#901—21no—
67#901—22no—
68#901—22no—
69C2×ΓL(1,𝔽25)C2×C412yes—
70C2×ΓL(1,𝔽25)C2×C412yes—
71C2×D3×C8C2×C412yes—
72C2×D3×C8C2×C412yes—
73C4×D12—21no—
74#872C2×C412yes—
75C42×D3C2×C412yes—
76C4×Dic6—21no—
77C3×#108—21no—
78C3×#108—21no—
79C3×#107—21no—
80C3×#106—21no—
81C3×#97—22no—
82C3×#97C2×C411yescharacteristic
83C3×#97C2×C411yescharacteristic
84C3×#97C2×C411yescharacteristic
85C3×#97C2×C411yescharacteristic
86C3×C4×C8—22no—
87#824—22no—
88#823—22no—
89#819—21no—
90#819—21no—
91#818—21no—
92#817—21no—
93#815—22no—
94#815C2×C412yes—
95#815C2×C412yes—
96C8×Dic3—22no—
97C8×Dic3C2×C412yes—
98C8×Dic3C2×C412yes—
99#806—24no—
100#803C2×C412yes—
101C4×C3⋊C8C2×C412yes—
102C2×#131—31no—
103C2×#131—31no—
104C4×#41—31no—
105C2×#108—31no—
106C2×#108C2×D311yescharacteristic
107#427—31no—
108#427C2×D311yescharacteristic
109#413—32no—
110#410—32no—
111#405—31no—
112#405—31no—
113#405—32no—
114#405—32no—
115#405—32no—
116#405C2×D312yes—
117C4×M16—31no—
118C4×M16—32no—
119C2×#97—32no—
120C2×#97C2×D311yescharacteristic
121#344—32no—
122#344—32no—
123#339—32no—
124#339C2×D312yes—
125#338—32no—
126H(3,ℤ/4)—32no—
127H(3,ℤ/4)—32no—
128H(3,ℤ/4)—32no—
129C22×C3×C4—21no—
130C22×C3×C4C2411yescharacteristic
131C2×#67#4112yes—
132#235—21no—
133#235—21no—
134#235—21no—
135#235—21no—
136C2×D12#4111yescharacteristic
137C2×C4×D3—22no—
138C2×C4×D3C22×C412yes—
139C2×Dic6#4111yescharacteristic
140C3×M16—22no—
141C3×M16—22no—
142C2×C3×C8—22no—
143C2×C3×C8C22×C411yescharacteristic
144C2×C3×C8C22×C411yescharacteristic
145C2×C3×C8C22×C411yescharacteristic
146C2×C3×C8C22×C411yescharacteristic
147C2×C3×C8#4111yescharacteristic
148C2×C3×C8#4111yescharacteristic
149C2×C3×C8#4111yescharacteristic
150C2×C3×C8#4111yescharacteristic
151C2×C3×C8#4112yes—
152C2×C3×C8#4112yes—
153C3×C4⋊C4—22no—
154C3×C22⋊C4—22no—
155C3×C42—21no—
156C3×C42—22no—
157C3×C42C22×C411yescharacteristic
158C3×C42C22×C411yescharacteristic
159#217—21no—
160#217—22no—
161(C2×D3)⋊C4—22no—
162C4⋊Dic3—21no—
163C4⋊Dic3—22no—
164Dic3⋊C4—22no—
165C4×Dic3—22no—
166C4×Dic3—22no—
167C4×Dic3C22×C412yes—
168#208—21no—
169#208—21no—
170#208—21no—
171#208—21no—
172#208—22no—
173#208—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8C22×C412yes—
176C2×C3⋊C8C22×C412yes—
177ΓL(1,𝔽25)—22no—
178ΓL(1,𝔽25)—22no—
179ΓL(1,𝔽25)—24no—
180D3×C8—22no—
181D3×C8—22no—
182D3×C8—24no—
183C2×#41—31no—
184#131—61no—
185#131—61no—
186#131—61no—
187#131—61no—
188#131—62no—
189#131—62no—
190C2×M16—31no—
191C2×M16—31no—
192C2×M16—32no—
193C2×M16—32no—
194C2×M16—32no—
195C2×M16C22×D312yes—
196C22×C8—32no—
197C22×C8—32no—
198C22×C8C22×D311yescharacteristic
199C22×C8C22×D311yescharacteristic
200C4×Q8—61no—
201C4×D4—61no—
202C4×D4—62no—
203#117—31no—
204#117—32no—
205#117—32no—
206#117C22×D312yes—
207C2×C42—32no—
208C2×C42C22×D311yescharacteristic
209#108—21no—
210#108—21no—
211#108—61no—
212#108—61no—
213#107—21no—
214#107—61no—
215#106—21no—
216#106—61no—
217#101—62no—
218#100—62no—
219#99—64no—
220#97—22no—
221#97—32no—
222#97—32no—
223#97—32no—
224#97—62no—
225#97C4×D311yescharacteristic
226#97C4×D311yescharacteristic
227#97C4×D311yescharacteristic
228#97C4×D311yescharacteristic
229C4×C8—22no—
230C4×C8—32no—
231C4×C8—32no—
232C4×C8—32no—
233C4×C8—62no—
234C23×C3C2×#4111yescharacteristic
235C22×D3#11712yes—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—42no—
239C2×C3×C4—81no—
240C2×C3×C4C23×C411yescharacteristic
241C2×C3×C4C2×#4111yescharacteristic
242C2×C3×C4C2×#4111yescharacteristic
243C2×C3×C4C2×#4111yescharacteristic
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4C2×#4111yescharacteristic
246#67—22no—
247#67—22no—
248C2×Dic3—22no—
249C2×Dic3—42no—
250C2×Dic3#11712yes—
251D12—21no—
252D12—21no—
253C4×D3—22no—
254C4×D3—22no—
255C4×D3—82no—
256Dic6—21no—
257Dic6—21no—
258C3×C8—42no—
259C3×C8#11711yescharacteristic
260C3×C8#11711yescharacteristic
261C3×C8#11711yescharacteristic
262C3×C8#11711yescharacteristic
263C3×C8#11712yes—
264C3×C8#11712yes—
265C3⋊C8—22no—
266C3⋊C8—22no—
267C3⋊C8—42no—
268#41—61no—
269#41—61no—
270#41—61no—
271#41—61no—
272C2×Q8—31no—
273C2×D4—32no—
274C2×D4—31no—
275C22×C4—21no—
276C22×C4—32no—
277C22×C4—62no—
278C22×C4C23×D311yescharacteristic
279M16—22no—
280M16—22no—
281M16—61no—
282M16—61no—
283M16—61no—
284M16—61no—
285M16—62no—
286M16—62no—
287M16—64no—
288M16—62no—
289M16—62no—
290C2×C8—22no—
291C2×C8—32no—
292C2×C8—32no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—64no—
296C2×C8—62no—
297C2×C8C2×C4×D311yescharacteristic
298C2×C8C2×C4×D311yescharacteristic
299C2×C8C2×C4×D311yescharacteristic
300C2×C8C2×C4×D311yescharacteristic
301C2×C8#23511yescharacteristic
302C2×C8#23511yescharacteristic
303C2×C8#23511yescharacteristic
304C2×C8#23511yescharacteristic
305C2×C8#23712yes—
306C2×C8#23912yes—
307C4⋊C4—22no—
308C4⋊C4—62no—
309C4⋊C4—61no—
310C4⋊C4—62no—
311C22⋊C4—22no—
312C22⋊C4—62no—
313C22⋊C4—61no—
314C22⋊C4—62no—
315C42—21no—
316C42—22no—
317C42—32no—
318C42—62no—
319C42—62no—
320C42C2×C4×D311yescharacteristic
321C42C2×C4×D311yescharacteristic
322C22×C3—41no—
323C22×C3C2×#11711yescharacteristic
324C22×C3#52911yescharacteristic
325C22×C3#52911yescharacteristic
326C2×D3—22no—
327C2×D3—42no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—82no—
331C3×C4C2×#11711yescharacteristic
332C3×C4C2×#11711yescharacteristic
333C3×C4#52911yescharacteristic
334C3×C4#52911yescharacteristic
335Dic3—22no—
336Dic3—42no—
337Dic3—82no—
338C23—32no—
339C23D3×#4111yescharacteristic
340Q8—61no—
341Q8—61no—
342D4—62no—
343D4—62no—
344D4—61no—
345D4—61no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—32no—
349C2×C4—42no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—81no—
354C2×C4—122no—
355C2×C4—242no—
356C2×C4C22×C4×D311yescharacteristic
357C2×C4C2×#23511yescharacteristic
358C2×C4C2×#23511yescharacteristic
359C2×C4C2×#23711yescharacteristic
360C2×C4C2×#23911yescharacteristic
361C2×C4D3×#4111yescharacteristic
362C8—42no—
363C8—62no—
364C8—62no—
365C8—122no—
366C8#87211yescharacteristic
367C8#87211yescharacteristic
368C8#87211yescharacteristic
369C8#87211yescharacteristic
370C8#89212yes—
371C8#89212yes—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3#228311yescharacteristic
376D3—42no—
377C22—41no—
378C22—62no—
379C22—122no—
380C22D3×#11711yescharacteristic
381C22#555711yescharacteristic
382C22#558011yescharacteristic
383C4—41no—
384C4—41no—
385C4—62no—
386C4—82no—
387C4—122no—
388C4—242no—
389C4C2×#87211yescharacteristic
390C4C2×#89211yescharacteristic
391C4#555711yescharacteristic
392C4#558011yescharacteristic
393C3#2078811yescharacteristic, complemented, Sylow
394C2—21no—
395C2—21no—
396C2—21no—
397C2—122no—
398C2#7375211yescharacteristic
subgroupG/H
centerC4C2×#892
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC2×C3#2283
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#20788
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. C3×#1773C2semidirect
4. #70698C2semidirect
6. #70657C2semidirect
8. #69895C2semidirect
9. #69829C2semidirect
10. #68749C2semidirect
28. #4708C4semidirect
29. #4708C4semidirect
30. #4707C4semidirect
31. D3×#97C4semidirect
48. #1773D3semidirect
393. C3#20788semidirect

Maximal subgroups

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

subgroupindex
#784862
C3×#17732
#707182
#706982
#706642
#706572
#698982
#698952
#698292
#687492
#687372
#207883