Finite Group Databasebeta

← #10690676

Subgroups — #10690676

Order 768.

488 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79277C211yesmaximal, characteristic
2C4×#909C211yesmaximal, characteristic
3#70786C211yesmaximal, characteristic, complemented
4C2×#4484C211yesmaximal, characteristic, complemented
5#70425C211yesmaximal, characteristic
6C3×#1283C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#9098—31nomaximal, Sylow
9C2×#909C411yescharacteristic
10C4×#235C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5133C411yescharacteristic
16#5121C411yescharacteristic
17#5120C411yescharacteristic
18#4925C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20#4838—21no—
21#4831—21no—
22#4730—22no—
23#4716—21no—
24#4710—21no—
25C4×Dic12—21no—
26C4×#204—21no—
27C4×ΓL(1,𝔽25)—21no—
28#4685—21no—
29#4484—21no—
30#4484—21no—
31#4484—21no—
32#4483—21no—
33#4469—22no—
34#2965—31no—
35C4×#137—31no—
36#1477—31no—
37C2×Q8⋊C8—31no—
38#1461—31no—
39#1283D311yescharacteristic, complemented
40#1278—31no—
41C2×#235C2×C411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16C2×C411yescharacteristic
44C2×C3×M16C2×C411yescharacteristic
45C22×C3×C8—21no—
46C2×C3×C42C2311yescharacteristic
47C4×#67—21no—
48C4×#67—21no—
49#926C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C22×C3⋊C8—21no—
54#909—22no—
55#909—22no—
56C2×Dic12—21no—
57C2×#204—21no—
58C2×ΓL(1,𝔽25)—22no—
59C4×D12D411yescharacteristic
60C4×D12D411yescharacteristic
61#872—21no—
62C42×D3—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6D411yescharacteristic
68C4×Dic6D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#830—21no—
77#821—21no—
78(C2×D3)⋊C8—22no—
79#818—21no—
80#817—21no—
81#816—21no—
82#816—21no—
83#816—21no—
84#815—22no—
85Dic3⋊C8—22no—
86C4⋊(C3⋊C8)—21no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D411yescharacteristic
90C4×C3⋊C8D411yescharacteristic
91C2×#137—31no—
92C4×#41—31no—
93C2×C4×Q8—31no—
94#446—61no—
95#441—61no—
96#440—61no—
97C4×Q16—61no—
98C4×SD16—61no—
99#436—61no—
100#435—62no—
101#431—31no—
102#431—61no—
103#431C2×D311yescharacteristic
104#428—31no—
105C2×C4⋊C8C2×D311yescharacteristic
106#419—31no—
107#417—31no—
108C4×M16—61no—
109C4×M16C2×D311yescharacteristic
110C2×C4×C8—31no—
111Q8⋊C8—61no—
112Q8⋊C8—61no—
113Q8⋊C8—61no—
114D4⋊C8—61no—
115#324—62no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—21no—
123#235—21no—
124#235—21no—
125#235—21no—
126C2×D12C22⋊C411yescharacteristic
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6C22⋊C411yescharacteristic
136C2×Dic6M1612yescomplemented
137C3×M16—21no—
138C3×M16—21no—
139C3×M16—21no—
140C3×M16M1612yescomplemented
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C2×C3×C8—81no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—21no—
151(C2×D3)⋊C4—21no—
152(C2×D3)⋊C4—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3C22⋊C411yescharacteristic
156C4⋊Dic3C22⋊C411yescharacteristic
157Dic3⋊C4—21no—
158Dic3⋊C4—21no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170Dic12—24no—
171#204—24no—
172ΓL(1,𝔽25)—44no—
173C2×#41—31no—
174C22×Q8—31no—
175#137—62no—
176#137—62no—
177C2×Q16—61no—
178C2×SD16—61no—
179C2×M16—62no—
180C2×M16C4×D311yescharacteristic
181C2×M16C4×D311yescharacteristic
182C22×C8—21no—
183C22×C8—61no—
184C4×Q8—31no—
185C4×Q8—31no—
186C4×Q8—61no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×D4—31no—
191C4×D4—31no—
192C4×D4—61no—
193C4×D4—61no—
194#117—31no—
195#117—61no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—61no—
198C2×C42—61no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—61no—
202#106—61no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—62no—
207C4⋊C8—61no—
208#103—61no—
209#103—61no—
210#103—61no—
211#102—61no—
212C22⋊C8—62no—
213C22⋊C8—61no—
214C22⋊C8D1212yes—
215#97—21no—
216#97—62no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—31no—
220C4×C8—31no—
221C4×C8—61no—
222C4×C8—61no—
223C23×C3C4×D411yescharacteristic
224C22×D3—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C2×M1611yescharacteristic
235C2×C3×C4C2×M1611yescharacteristic
236C2×C3×C4#13111yescharacteristic
237C2×C3×C4#13111yescharacteristic
238#67—41no—
239#67—41no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—22no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—81no—
248D12—21no—
249D12—21no—
250C4×D3—42no—
251C4×D3—41no—
252C4×D3—41no—
253C4×D3—41no—
254C4×D3—41no—
255Dic6—22no—
256Dic6—21no—
257Dic6—21no—
258Dic6—22no—
259Dic6—41no—
260Dic6—41no—
261C3×C8—22no—
262C3×C8—42no—
263C3×C8—41no—
264C3×C8—81no—
265C3⋊C8—22no—
266C3⋊C8—41no—
267C3⋊C8—81no—
268#41—61no—
269#41—61no—
270#41—61no—
271#41—61no—
272C2×Q8—31no—
273C2×Q8—32no—
274C2×Q8—61no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—61no—
279C2×D4—31no—
280C2×D4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4C2×C4×D311yescharacteristic
287C22×C4C2×D1211yescharacteristic
288C22×C4#23511yescharacteristic
289Q16—64no—
290SD16—64no—
291M16—21no—
292M16—21no—
293M16—21no—
294M16—124no—
295M16ΓL(1,𝔽25)12yescomplemented
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—81no—
303C2×C8—122no—
304C2×C8—121no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C42—61no—
317C42—61no—
318C42—61no—
319C42—121no—
320C42—121no—
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C42D3×D411yescharacteristic
324C42#23911yescharacteristic
325C22×C3—21no—
326C22×C3—21no—
327C22×C3#38611yescharacteristic
328C22×C3#43511yescharacteristic
329C22×C3#43611yescharacteristic
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4#40711yescharacteristic
337C3×C4#40811yescharacteristic
338C3×C4#43511yescharacteristic
339C3×C4#43611yescharacteristic
340C3×C4#47111yescharacteristic
341C3×C4#47311yescharacteristic
342Dic3—42no—
343Dic3—41no—
344Dic3—41no—
345Dic3—41no—
346Dic3—81no—
347C23—61no—
348C23C4×D1211yescharacteristic
349Q8—62no—
350Q8—61no—
351Q8—61no—
352Q8—62no—
353Q8—121no—
354Q8—121no—
355D4—61no—
356D4—61no—
357D4—121no—
358D4—121no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—62no—
366C2×C4—122no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4D3×C22⋊C411yescharacteristic
376C2×C4#88211yescharacteristic
377C2×C4#88411yescharacteristic
378C2×C4#89311yescharacteristic
379C2×C4#89411yescharacteristic
380C2×C4#89511yescharacteristic
381C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
382C2×C4#90111yescharacteristic
383C2×C4D3×M1611yescharacteristic
384C2×C4#90711yescharacteristic
385C8—22no—
386C8—42no—
387C8—41no—
388C8—62no—
389C8—81no—
390C8—121no—
391C8—241no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3#187011yescharacteristic
395C2×C3#190611yescharacteristic
396C2×C3#190811yescharacteristic
397D3—81no—
398C22—21no—
399C22—21no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22#466511yescharacteristic
404C22#468811yescharacteristic
405C22#470611yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—121no—
410C4—122no—
411C4—121no—
412C4—121no—
413C4—241no—
414C4#472611yescharacteristic
415C4#472711yescharacteristic
416C4#474811yescharacteristic
417C4#475511yescharacteristic
418C4#483611yescharacteristic
419C4#483811yescharacteristic
420C3#909811yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—241no—
424C2#6866911yescharacteristic
425C2#6900711yescharacteristic
426C2#6903911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1283C2
socleC22×C3#386
prime radicalC2×#235C2×C4
cosocleC2×C3×C42C23
hypercenter#1283D3
hypocenterC3#9098
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
3. #70786C2semidirect
4. C2×#4484C2semidirect
6. C3×#1283C2semidirect
7. #67667C2semidirect
39. #1283D3semidirect
136. C2×Dic6M16semidirect
140. C3×M16M16semidirect
295. M16ΓL(1,𝔽25)semidirect
420. C3#9098semidirect

Maximal subgroups

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

subgroupindex
#792772
C4×#9092
#707862
C2×#44842
#704252
C3×#12832
#676672
#90983