Finite Group Databasebeta

← #10691298

Subgroups — #10691298

Order 768.

536 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#131C211yesmaximal, characteristic, complemented
2#82529C211yesmaximal, characteristic
3#71641C211yesmaximal, characteristic, complemented
4#69859C211yesmaximal, characteristic, complemented
5#69735C211yesmaximal, characteristic, complemented
6#67669C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9211—31nomaximal, Sylow
9C2×C3×#131—21no—
10C3×C4×#41C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#405—21no—
13C3×#405—22no—
14C3×C4×M16—21no—
15C3×C4×M16C2211yescharacteristic
16C2×C3×C4×C8—21no—
17C2×C3×C4×C8C2211yescharacteristic
18#5238—21no—
19#5027C411yescharacteristic
20C4×#215—21no—
21#5021C411yescharacteristic
22#5015C411yescharacteristic
23C4×#213—21no—
24#5008C411yescharacteristic
25#5003—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31#4464—22no—
32#4459—21no—
33#4459—21no—
34#3106—31no—
35C4×#131D311yescharacteristic, complemented
36#1525—31no—
37#1492—31no—
38#1489—31no—
39#1299—31no—
40#1278—31no—
41C2×C3×#41D411yescharacteristic
42C2×#235—21no—
43C3×#131—24no—
44C3×#131—24no—
45C2×C3×M16—22no—
46C2×C3×M16—21no—
47C22×C3×C8—22no—
48C22×C3×C8—21no—
49C3×C4×Q8C2×C411yescharacteristic
50C3×C4×Q8D411yescharacteristic
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4C2×C411yescharacteristic
54C3×C4×D4D411yescharacteristic
55C3×#117—21no—
56C3×#117D411yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#215—21no—
60C2×#213—21no—
61#930—21no—
62#930—21no—
63C2×#208—21no—
64C2×#208—21no—
65#875—21no—
66#874C2×C411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#869C2×C411yescharacteristic
70C3×#97—22no—
71C3×#97D412yes—
72C3×C4×C8—22no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75C3×C4×C8D411yescharacteristic
76C3×C4×C8D411yescharacteristic
77#835—21no—
78#832—21no—
79#830—21no—
80#809—21no—
81#809—21no—
82#808—21no—
83#807—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88#803—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×#131—21no—
92#532—31no—
93C4×#41C2×D311yescharacteristic
94#482—61no—
95#477—31no—
96#476—31no—
97#460—31no—
98#459—31no—
99#449—61no—
100C4×SD16—61no—
101C4×D8—61no—
102#431—31no—
103#423—31no—
104#405—21no—
105#405—22no—
106C4×M16—21no—
107C4×M16—31no—
108C4×M16C2×D311yescharacteristic
109C2×C4×C8—21no—
110C2×C4×C8C2×D311yescharacteristic
111#329—62no—
112Q8⋊C8—61no—
113D4⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—61no—
116C3×#41—22no—
117C3×#41—22no—
118C2×C3×Q8C22⋊C411yescharacteristic
119C2×C3×D4—21no—
120C2×C3×D4C22⋊C411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—41no—
128#235—81no—
129#235—81no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C3×M16—28no—
136C3×M16—22no—
137C3×M16—22no—
138C2×C3×C8—28no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—21no—
142C2×C3×C8—22no—
143C2×C3×C8—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42C22×C411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158#215—42no—
159#213—42no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164#208—22no—
165#208—41no—
166#208—81no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C2×#41—61no—
172C2×#41#6711yescharacteristic
173C2×SD16—61no—
174C2×D8—61no—
175#131—24no—
176#131—24no—
177C2×M16—22no—
178C2×M16—21no—
179C2×M16—61no—
180C2×M16—61no—
181C22×C8—22no—
182C22×C8—21no—
183C4⋊Q8—31no—
184#127—31no—
185#124—61no—
186#121—61no—
187#121—61no—
188C4×Q8C4×D311yescharacteristic
189C4×Q8D1211yescharacteristic
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—61no—
193C4×D4—61no—
194C4×D4C4×D311yescharacteristic
195C4×D4D1211yescharacteristic
196#117—21no—
197#117#6711yescharacteristic
198C2×C42—21no—
199C2×C42C22×D311yescharacteristic
200#107—61no—
201#106—61no—
202C4⋊C8—31no—
203C4⋊C8—31no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206#103—61no—
207#102—61no—
208#102—61no—
209#102—61no—
210C22⋊C8—61no—
211#97—22no—
212#97—61no—
213#97#6712yes—
214C4×C8—22no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—61no—
218C4×C8—61no—
219C4×C8#6711yescharacteristic
220C4×C8#6711yescharacteristic
221C23×C3—21no—
222C23×C3#12311yescharacteristic
223C22×D3—41no—
224C3×Q8—21no—
225C3×Q8#10412yes—
226C3×D4—24no—
227C3×D4—21no—
228C3×D4#10412yes—
229C2×C3×C4—24no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249#67—82no—
250C2×Dic3—41no—
251D12—41no—
252D12—41no—
253D12—41no—
254D12—41no—
255D12—81no—
256C4×D3—81no—
257C4×D3—81no—
258Dic6—81no—
259C3×C8—22no—
260C3×C8—24no—
261C3×C8—22no—
262C3⋊C8—42no—
263C3⋊C8—41no—
264C3⋊C8—41no—
265C3⋊C8—81no—
266#41—22no—
267#41—22no—
268#41—241no—
269#41—241no—
270C2×Q8—61no—
271C2×Q8(C2×D3)⋊C411yescharacteristic
272C2×D4—21no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—121no—
277C2×D4(C2×D3)⋊C411yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—121no—
282C22×C4#23511yescharacteristic
283C22×C4#23511yescharacteristic
284C22×C4C2×#6711yescharacteristic
285SD16—122no—
286D8—122no—
287M16—28no—
288M16—22no—
289M16—22no—
290M16—62no—
291M16—121no—
292M16—241no—
293C2×C8—28no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—61no—
300C2×C8—61no—
301C2×C8—121no—
302C2×C8—121no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4(C2×D3)⋊C411yescharacteristic
308C4⋊C4(C2×D3)⋊C411yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42C2×C4×D311yescharacteristic
318C42C2×D1211yescharacteristic
319C42C2×#6711yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3#38611yescharacteristic
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—22no—
334C3×C4—21no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4C2×#10411yescharacteristic
338C3×C4C2×#10411yescharacteristic
339C3×C4#44311yescharacteristic
340C3×C4#44411yescharacteristic
341C3×C4#47111yescharacteristic
342C3×C4#47211yescharacteristic
343Dic3—81no—
344C23—21no—
345C23—121no—
346C23#92911yescharacteristic
347Q8—21no—
348Q8—241no—
349Q8#80512yes—
350D4—24no—
351D4—21no—
352D4—121no—
353D4—121no—
354D4—121no—
355D4—121no—
356D4—241no—
357D4—242no—
358D4#80512yes—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4C2×(C2×D3)⋊C411yescharacteristic
377C2×C4C4×#6711yescharacteristic
378C2×C4C4×#6711yescharacteristic
379C2×C4#93011yescharacteristic
380C2×C4#93011yescharacteristic
381C2×C4#95311yescharacteristic
382C8—22no—
383C8—24no—
384C8—22no—
385C8—122no—
386C8—121no—
387C8—121no—
388C8—241no—
389C2×C3—41no—
390C2×C3—42no—
391C2×C3#187611yescharacteristic
392C2×C3#189611yescharacteristic
393C2×C3#190711yescharacteristic
394D3—161no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22#521311yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—241no—
412C4C2×#80511yescharacteristic
413C4C2×#80511yescharacteristic
414C4#514111yescharacteristic
415C4#514211yescharacteristic
416C4#514411yescharacteristic
417C4#514511yescharacteristic
418C3#921111yescharacteristic, complemented, Sylow
419C2—41no—
420C2—42no—
421C2—481no—
422C2#7153211yescharacteristic
423C2#7201011yescharacteristic
424C2#7201811yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#131C2
socleC22×C3#386
prime radical#82529C2
cosocleC2×C3×C42C23
hypercenterC4×#131D3
hypocenterC3#9211
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131

Complements

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

NH ≅ G/Ntype
1. C3×C4×#131C2semidirect
3. #71641C2semidirect
4. #69859C2semidirect
5. #69735C2semidirect
6. #67669C2semidirect
7. #67656C2semidirect
35. C4×#131D3semidirect
418. C3#9211semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1312
#825292
#716412
#698592
#697352
#676692
#676562
#92113