Finite Group Databasebeta

← #10696150

Subgroups — #10696150

Order 768.

636 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3100C211yesmaximal, characteristic, complemented
2#83286C211yesmaximal, characteristic, complemented
3#82308C211yesmaximal, characteristic, complemented
4#71800C211yesmaximal, characteristic, complemented
5#71615C211yesmaximal, characteristic, complemented
6#70758C211yesmaximal, characteristic, complemented
7#67740C211yesmaximal, characteristic, complemented
8#10032—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×#950—22no—
12#5477C2211yescharacteristic
13C3×#480—21no—
14C3×#474—21no—
15C3×#459—21no—
16C3×#447—21no—
17C3×C4×D8—22no—
18C3×#431C2211yescharacteristic
19#5217—21no—
20C2×#830—22no—
21#5086—21no—
22#5074—21no—
23#5060—21no—
24#5058—21no—
25#5015—21no—
26C4×#213—22no—
27#5007—21no—
28#4967—21no—
29#4925C2211yescharacteristic
30C2×C4⋊(C3⋊C8)C2211yescharacteristic
31C2×#803C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4459—22no—
34#3150—31no—
35#3100—31no—
36#3100D311yescharacteristic, complemented
37#1955—31no—
38#1480—31no—
39#1480—31no—
40#1275—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×#235—21no—
45C2×C3×D8—22no—
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60#950—44no—
61#950—44no—
62C2×#216—22no—
63C2×#215—22no—
64C2×#214—22no—
65C2×#213—22no—
66C2×#213—22no—
67#930—21no—
68#930—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8D412yes—
71#875—21no—
72#874D411yescharacteristic
73C4×D12—21no—
74C4×D12—21no—
75#869D411yescharacteristic
76C3×#107—22no—
77C3×C4⋊C8—21no—
78C3×#102—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C4×C8D412yes—
82#832—22no—
83#830—21no—
84#830—44no—
85#809—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90#803—21no—
91#803—21no—
92C4×C3⋊C8D412yes—
93C2×#135—62no—
94#532—31no—
95#532C2×D311yescharacteristic
96C2×C4×D4C2×D311yescharacteristic
97#497—61no—
98#497—61no—
99#496—61no—
100#496—61no—
101#480—21no—
102#480—61no—
103#474—21no—
104#474—61no—
105#459—21no—
106#459—61no—
107#447—21no—
108#447—61no—
109C4×D8—22no—
110C4×D8—62no—
111#431—31no—
112#431C2×D311yescharacteristic
113C2×C4⋊C8—31no—
114C2×C22⋊C8—62no—
115C2×#97—31no—
116D4⋊C8—62no—
117D4⋊C8—62no—
118C24×C3—21no—
119C3×#41—42no—
120C3×#41—42no—
121C2×C3×Q8—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—41no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—41no—
139#235—42no—
140#235—42no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12—21no—
144C2×C4×D3—41no—
145C2×Dic6—21no—
146C3×D8—44no—
147C2×C3×C8—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160#216—44no—
161#215—44no—
162#214—44no—
163#213—44no—
164#213—44no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8D814yescomplemented
175C2×#41—21no—
176C2×#41—61no—
177C22×D4#6711yescharacteristic
178C23×C4—21no—
179#135—124no—
180#135—124no—
181C2×Q16—62no—
182C2×SD16—62no—
183C2×SD16—62no—
184C2×D8—22no—
185C2×D8—62no—
186C2×D8—62no—
187C22×C8—32no—
188C22×C8—61no—
189C4⋊Q8—31no—
190C4⋊Q8#6711yescharacteristic
191#127—31no—
192#127#6711yescharacteristic
193#124—21no—
194#124—61no—
195#121—21no—
196#121—21no—
197#121—61no—
198#121—61no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—61no—
206C4×D4—61no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C42C22×D311yescharacteristic
210#107—22no—
211#107—62no—
212C4⋊C8—21no—
213C4⋊C8—61no—
214C4⋊C8—61no—
215C4⋊C8—61no—
216#102—22no—
217#102—22no—
218#102—62no—
219#102—62no—
220C22⋊C8—21no—
221C22⋊C8—61no—
222C22⋊C8—124no—
223#97—61no—
224#97—61no—
225C4×C8—32no—
226C4×C8#6712yes—
227C23×C3—22no—
228C23×C3—41no—
229C23×C3—42no—
230C23×C3—41no—
231C23×C3—42no—
232C23×C3—41no—
233C23×C3#12111yescharacteristic
234C22×D3—41no—
235C3×Q8—42no—
236C3×D4—22no—
237C3×D4—22no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—82no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—82no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12711yescharacteristic
260C2×C3×C4C2×D812yes—
261C2×C3×C4AGL(1,ℤ/8)12yes—
262#67—82no—
263C2×Dic3—41no—
264D12—42no—
265D12—42no—
266D12—42no—
267C4×D3—81no—
268C4×D3—81no—
269Dic6—42no—
270C3×C8—42no—
271C3⋊C8—24no—
272C3⋊C8—42no—
273C3⋊C8—81no—
274C24—21no—
275#41—42no—
276#41—42no—
277#41—122no—
278#41—122no—
279C2×Q8—21no—
280C2×Q8—61no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—121no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—121no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—124no—
303SD16—124no—
304SD16—124no—
305D8—44no—
306D8—124no—
307D8—124no—
308C2×C8—22no—
309C2×C8—34no—
310C2×C8—62no—
311C2×C8—62no—
312C2×C8—62no—
313C2×C8—122no—
314C2×C8—121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42D3×D411yescharacteristic
328C42D3×D411yescharacteristic
329C42C2×#6711yescharacteristic
330C42C2×#6711yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—44no—
334C22×C3—42no—
335C22×C3—82no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3#39211yescharacteristic
342C22×C3#45912yes—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—81no—
351C3×C4—81no—
352C3×C4#44712yes—
353C3×C4#49311yescharacteristic
354C3×C4#49611yescharacteristic
355Dic3—81no—
356C23—22no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—121no—
363C23#93911yescharacteristic
364Q8—42no—
365Q8—122no—
366D4—22no—
367D4—22no—
368D4—41no—
369D4—41no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—82no—
374D4—122no—
375D4—122no—
376D4—122no—
377D4—242no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4D3×D812yes—
392C2×C4#91112yes—
393C2×C4#93711yescharacteristic
394C2×C4#93711yescharacteristic
395C2×C4#93811yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#94011yescharacteristic
398C2×C4#95311yescharacteristic
399C8—42no—
400C8—64no—
401C8—122no—
402C8—241no—
403C2×C3—22no—
404C2×C3—82no—
405C2×C3—161no—
406C2×C3#201011yescharacteristic
407C2×C3#201111yescharacteristic
408C2×C3#201711yescharacteristic
409D3—161no—
410C22—21no—
411C22—21no—
412C22—44no—
413C22—42no—
414C22—82no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—241no—
421C22—241no—
422C22—241no—
423C22#515012yes—
424C22#522311yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—81no—
430C4—81no—
431C4—241no—
432C4#515111yescharacteristic
433C4#515411yescharacteristic
434C4#515612yes—
435C3#1003211yescharacteristic, complemented, Sylow
436C2—22no—
437C2—82no—
438C2—161no—
439C2—481no—
440C2#7171511yescharacteristic
441C2#7204411yescharacteristic
442C2#7212111yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3100C2
socleC22×C3#392
hypercenter#3100D3
hypocenterC3#10032
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. C3×#3100C2semidirect
2. #83286C2semidirect
3. #82308C2semidirect
4. #71800C2semidirect
5. #71615C2semidirect
6. #70758C2semidirect
7. #67740C2semidirect
36. #3100D3semidirect
174. C2×C3⋊C8D8semidirect
435. C3#10032semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
#832862
#823082
#718002
#716152
#707582
#677402
#100323