Finite Group Databasebeta

← #10798416

Subgroups — #10798416

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2922C211yesmaximal, characteristic, complemented
2#82506C211yesmaximal, characteristic
3C4×#901C211yesmaximal, characteristic, complemented
4#71643C211yesmaximal, characteristic, complemented
5#70545C212yesmaximal, complemented
6#70534C212yesmaximal, complemented
7#70011C212yesmaximal
8#69996C212yesmaximal, complemented
9#69873C211yesmaximal, characteristic, complemented
10#69866C211yesmaximal, characteristic, complemented
11#69751C211yesmaximal, characteristic, complemented
12#21138—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C2×#901C2212yes—
16C4×#235C2211yescharacteristic
17C3×#433—21no—
18C3×#433—21no—
19C3×#431—22no—
20C3×#424C2212yes—
21C3×#408C2212yes—
22C3×C4×M16C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C2×#837C2212yes—
25#5233C2211yescharacteristic
26C2×#825C2212yes—
27C2×#822C2212yes—
28#5101C2212yes—
29#5030C412yescomplemented
30#5026C412yescomplemented
31#5018C412yescomplemented
32#5014C412yescomplemented
33#5002C2211yescharacteristic
34#4998C2211yescharacteristic
35#4932C2211yescharacteristic
36C4×#208C2211yescharacteristic
37C2×#803C2211yescharacteristic
38#4689—22no—
39C4×ΓL(1,𝔽25)—21no—
40C4×D3×C8—21no—
41#4558C2212yes—
42#4555C2212yes—
43#4550C2212yes—
44#4550C2212yes—
45#4532C2212yes—
46#2943—31no—
47#2922D311yescharacteristic, complemented
48C4×#131—31no—
49#1929—32no—
50#1922—32no—
51#1876—32no—
52#1866—32no—
53#1778—31no—
54#1759—31no—
55#1759—31no—
56#1746—31no—
57C2×C3×#41C2311yescharacteristic
58C2×#235C2311yescharacteristic
59C2×C3×M16C2312yes—
60C22×C3×C8C2312yes—
61C3×C4×Q8C2×C412yes—
62C3×C4×D4—22no—
63C3×C4×D4C2×C412yes—
64C3×#117—21no—
65C3×#117C2311yescharacteristic
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68#950—24no—
69C2×#216C2×C411yescharacteristic
70C2×#215C2×C411yescharacteristic
71C2×#214C2×C411yescharacteristic
72C2×#213C2×C411yescharacteristic
73C4×#67—22no—
74#926C2311yescharacteristic
75C2×C4×Dic3C2312yes—
76C2×#208C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79#901—24no—
80#901—24no—
81C2×ΓL(1,𝔽25)—22no—
82C2×D3×C8—22no—
83C4×D12C2×C412yes—
84#872—21no—
85C42×D3—21no—
86C4×Dic6C2×C412yes—
87C3×C4⋊C8—22no—
88C3×C4⋊C8—22no—
89C3×C22⋊C8—22no—
90C3×C22⋊C8—22no—
91C3×#97—21no—
92C3×C4×C8—21no—
93C3×C4×C8D411yescharacteristic
94C3×C4×C8D411yescharacteristic
95C3×C4×C8D412yes—
96#837—24no—
97#835C2×C412yes—
98#832C2×C412yes—
99C3⋊#95—22no—
100#825—24no—
101#822—24no—
102#815—22no—
103C8×Dic3—22no—
104#810C2×C412yes—
105#809C2×C412yes—
106#808C2×C412yes—
107#807C2×C412yes—
108#803—21no—
109#803C2×C412yes—
110#803C2×C412yes—
111C4×C3⋊C8—21no—
112C2×#135—31no—
113C2×#131—32no—
114C4×#41—31no—
115C4×#41C2×D311yescharacteristic
116#442—32no—
117#441—32no—
118#440—32no—
119#440—32no—
120#433—21no—
121#433—21no—
122#431—22no—
123C2×#108—32no—
124#427—31no—
125#424C2×D312yes—
126C2×#104—32no—
127C2×#104—32no—
128#416—31no—
129#416—31no—
130#408C2×D312yes—
131#405—32no—
132#405—62no—
133C4×M16—31no—
134C4×M16—61no—
135C4×M16C2×D311yescharacteristic
136C2×#97—31no—
137C2×C4×C8—61no—
138C2×C4×C8C2×D311yescharacteristic
139#376—31no—
140#341—32no—
141#339—32no—
142#339—32no—
143#336—32no—
144#336—32no—
145C3×#41—22no—
146C3×#41—22no—
147C2×C3×Q8C22×C411yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4C22×C411yescharacteristic
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4#4112yes—
153C22×C3×C4C2411yescharacteristic
154C2×#67—21no—
155C22×Dic3—22no—
156#235—22no—
157#235#4114yes—
158C2×D12C22×C411yescharacteristic
159C2×C4×D3—21no—
160C2×C4×D3—22no—
161C2×Dic6C22×C411yescharacteristic
162C3×M16—22no—
163C3×M16—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C2×D412yes—
168C2×C3×C8C2×D412yes—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C22×C412yes—
171C3×C22⋊C4—21no—
172C3×C22⋊C4—22no—
173C3×C42—22no—
174C3×C42—21no—
175C3×C42C22×C412yes—
176C3×C42C2×D411yescharacteristic
177C3×C42C2×D411yescharacteristic
178#217—21no—
179#216—22no—
180#215—22no—
181#214—22no—
182#213—22no—
183(C2×D3)⋊C4—22no—
184C4⋊Dic3C22×C412yes—
185Dic3⋊C4—22no—
186C4×Dic3—22no—
187C4×Dic3—21no—
188C4×Dic3—22no—
189C4×Dic3—22no—
190#208—22no—
191#208#4114yes—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8C22×C411yescharacteristic
196C2×C3⋊C8C22×C411yescharacteristic
197C2×C3⋊C8C22×C412yes—
198ΓL(1,𝔽25)—28no—
199D3×C8—28no—
200C2×#41—31no—
201C2×#41C22×D311yescharacteristic
202#135—64no—
203C2×Q16—31no—
204C2×SD16—31no—
205C2×SD16—31no—
206C2×D8—31no—
207#131—64no—
208#131—64no—
209C2×M16—32no—
210C2×M16—62no—
211C2×M16C22×D312yes—
212C22×C8—31no—
213C22×C8—31no—
214C22×C8—62no—
215C22×C8C22×D312yes—
216C4×Q8—32no—
217C4×Q8C4×D312yes—
218C4×D4—22no—
219C4×D4—32no—
220C4×D4—62no—
221C4×D4C4×D312yes—
222#117—21no—
223#117—31no—
224#117—61no—
225#117C22×D311yescharacteristic
226C2×C42—21no—
227C2×C42—32no—
228C2×C42—61no—
229C2×C42C22×D311yescharacteristic
230#108—64no—
231#107—32no—
232#106—32no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235#104—64no—
236#104—64no—
237#103—32no—
238#103—32no—
239#102—32no—
240#102—32no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243#97—21no—
244#97—32no—
245#97—32no—
246#97—62no—
247#97—61no—
248C4×C8—21no—
249C4×C8—62no—
250C4×C8—61no—
251C4×C8#6711yescharacteristic
252C4×C8#6711yescharacteristic
253C4×C8#6712yes—
254#95—62no—
255C23×C3—21no—
256C23×C3C2×#4111yescharacteristic
257C22×D3—21no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×D4—21no—
261C3×D4—21no—
262C3×D4—42no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—44no—
267C2×C3×C4—42no—
268C2×C3×C4C4×D412yes—
269C2×C3×C4C4×D412yes—
270C2×C3×C4C23×C411yescharacteristic
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C2×#4112yes—
273C2×C3×C4C2×#4112yes—
274#67—24no—
275C2×Dic3—21no—
276C2×Dic3—22no—
277C2×Dic3—44no—
278C2×Dic3—42no—
279D12—21no—
280D12#11712yes—
281C4×D3—24no—
282C4×D3—44no—
283Dic6—21no—
284Dic6#11712yes—
285C3×C8—22no—
286C3×C8—22no—
287C3×C8—42no—
288C3⋊C8—21no—
289C3⋊C8—21no—
290C3⋊C8—22no—
291C3⋊C8—24no—
292#41—22no—
293#41—22no—
294#41—34no—
295#41—62no—
296C2×Q8—31no—
297C2×Q8C2×C4×D311yescharacteristic
298C2×D4—21no—
299C2×D4—31no—
300C2×D4—61no—
301C2×D4C2×C4×D311yescharacteristic
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—61no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4#23512yes—
308C22×C4C23×D311yescharacteristic
309Q16—62no—
310SD16—62no—
311SD16—62no—
312D8—62no—
313M16—22no—
314M16—22no—
315M16—34no—
316M16—68no—
317M16—62no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—32no—
322C2×C8—31no—
323C2×C8—31no—
324C2×C8—68no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—62no—
328C2×C8C2×#6712yes—
329C2×C8C2×#6712yes—
330C4⋊C4—22no—
331C4⋊C4—32no—
332C4⋊C4—62no—
333C4⋊C4C2×C4×D312yes—
334C22⋊C4—21no—
335C22⋊C4—22no—
336C22⋊C4—61no—
337C22⋊C4—62no—
338C42—22no—
339C42—21no—
340C42—62no—
341C42—61no—
342C42—62no—
343C42—62no—
344C42C2×C4×D312yes—
345C42C2×#6711yescharacteristic
346C42C2×#6711yescharacteristic
347C22×C3—22no—
348C22×C3—42no—
349C22×C3—41no—
350C22×C3C2×C4×D411yescharacteristic
351C22×C3#52912yes—
352C2×D3—22no—
353C2×D3—42no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—42no—
357C3×C4—41no—
358C3×C4C2×#11711yescharacteristic
359C3×C4C4×#4111yescharacteristic
360C3×C4#52512yes—
361Dic3—22no—
362Dic3—42no—
363Dic3—82no—
364C23—21no—
365C23—61no—
366C23C2×#23511yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—32no—
370Q8—61no—
371D4—21no—
372D4—21no—
373D4—32no—
374D4—42no—
375D4—64no—
376D4—61no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—61no—
383C2×C4—64no—
384C2×C4—62no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—122no—
388C2×C4C4×#6712yes—
389C2×C4C4×#6712yes—
390C2×C4C22×C4×D311yescharacteristic
391C2×C4D3×#4112yes—
392C2×C4D3×#4112yes—
393C2×C4C22×#6711yescharacteristic
394C8—22no—
395C8—22no—
396C8—42no—
397C8—61no—
398C8—61no—
399C8—62no—
400C8—64no—
401C2×C3—22no—
402C2×C3—81no—
403C2×C3#229011yescharacteristic
404C2×C3#295512yes—
405D3—42no—
406C22—22no—
407C22—42no—
408C22—41no—
409C22—62no—
410C22—122no—
411C22#552812yes—
412C22C2×C4×#6711yescharacteristic
413C4—22no—
414C4—22no—
415C4—42no—
416C4—41no—
417C4—62no—
418C4—122no—
419C4—242no—
420C4D3×#11711yescharacteristic
421C4#552211yescharacteristic
422C4#582812yes—
423C3#2113811yescharacteristic, complemented, Sylow
424C2—22no—
425C2—81no—
426C2—122no—
427C2#8226811yescharacteristic
428C2#8379212yes—
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2922C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#2922D3
hypocenterC3#21138
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2922

Complements

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

NH ≅ G/Ntype
1. C3×#2922C2semidirect
3. C4×#901C2semidirect
4. #71643C2semidirect
5. #70545C2semidirect
6. #70534C2semidirect
8. #69996C2semidirect
9. #69873C2semidirect
10. #69866C2semidirect
11. #69751C2semidirect
29. #5030C4semidirect
30. #5026C4semidirect
31. #5018C4semidirect
32. #5014C4semidirect
47. #2922D3semidirect
423. C3#21138semidirect

Maximal subgroups

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

subgroupindex
C3×#29222
#825062
C4×#9012
#716432
#705452
#705342
#700112
#699962
#698732
#698662
#697512
#211383