Finite Group Databasebeta

← #13061717

Subgroups — #13061717

Order 1344.

432 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2#10621430C211yesmaximal, characteristic, complemented
3#10621427C211yesmaximal, characteristic, complemented
4#10621412C211yesmaximal, characteristic, complemented
5C4×#4268C211yesmaximal, characteristic, complemented
6#10621381C211yesmaximal, characteristic, complemented
7#10621379C211yesmaximal, characteristic, complemented
8#10621379C211yesmaximal, characteristic, complemented
9#10621377C211yesmaximal, characteristic, complemented
10#10621375C211yesmaximal, characteristic, complemented
11#10621370C211yesmaximal, characteristic, complemented
12#10621368C211yesmaximal, characteristic, complemented
13#10621368C211yesmaximal, characteristic, complemented
14#10621367C211yesmaximal, characteristic
15#10621362C211yesmaximal, characteristic, complemented
16#90331—31nomaximal, Hall
17C2×C3×C7×Q8C2211yescharacteristic
18C22×C3×C4×C7C2211yescharacteristic
19C2×#4268C2211yescharacteristic
20C2×#4268C2211yescharacteristic
21C2×D84C2211yescharacteristic
22C2×C4×D21C2211yescharacteristic
23C2×C4×D21C2211yescharacteristic
24C2×Dic42C2211yescharacteristic
25C2×Dic42C2211yescharacteristic
26C3×C7×C4⋊C4C2211yescharacteristic
27C3×C7×C4⋊C4C2211yescharacteristic
28C3×C7×C4⋊C4C2211yescharacteristic
29C3×C7×C22⋊C4C2211yescharacteristic
30C3×C7×C22⋊C4C2211yescharacteristic
31#66636C2211yescharacteristic
32#66636C2211yescharacteristic
33#66631C2211yescharacteristic
34#66631C2211yescharacteristic
35#66631C2211yescharacteristic
36#66631C2211yescharacteristic
37#66631C2211yescharacteristic
38#66631C2211yescharacteristic
39#66631C2211yescharacteristic
40#66631C2211yescharacteristic
41C4⋊Dic21C2211yescharacteristic
42#66629C2211yescharacteristic
43#66629C2211yescharacteristic
44#66629C2211yescharacteristic
45#66629C2211yescharacteristic
46#66629C2211yescharacteristic
47#66629C2211yescharacteristic
48C4×Dic21C2211yescharacteristic
49C4×Dic21C2211yescharacteristic
50C4×Dic21C2211yescharacteristic
51C4×Dic21C2211yescharacteristic
52C7×C22⋊Q8D311yescharacteristic, complemented
53#6510—31no—
54#6507—31no—
55#6492—31no—
56C4×C22⋊D7—31no—
57#6461—31no—
58#6459—31no—
59#6459—31no—
60#6457—31no—
61#6455—31no—
62#6450—31no—
63#6448—31no—
64#6448—31no—
65#6447—31no—
66#6442—31no—
67#5640—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C22×D21C2311yescharacteristic
71C3×C7×Q8—41no—
72C2×C3×C4×C7—41no—
73C2×C3×C4×C7C2311yescharacteristic
74C2×C3×C4×C7C2311yescharacteristic
75C2×C3×C4×C7C2311yescharacteristic
76C2×C3×C4×C7C2311yescharacteristic
77C2×C3×C4×C7C2311yescharacteristic
78C2×C3×C4×C7C2311yescharacteristic
79#4268—22no—
80#4268—41no—
81C2×Dic21C2311yescharacteristic
82C2×Dic21C2311yescharacteristic
83C2×Dic21C2311yescharacteristic
84C2×Dic21C2311yescharacteristic
85C2×Dic21C2311yescharacteristic
86C2×Dic21C2311yescharacteristic
87D84—41no—
88C4×D21—22no—
89C4×D21—41no—
90Dic42—22no—
91Dic42—41no—
92C2×C7×Q8C2×D311yescharacteristic
93C22×C4×C7C2×D311yescharacteristic
94C2×C22⋊D7—31no—
95C2×C22⋊D7—31no—
96C2×D28—31no—
97C2×C4×D7—31no—
98C2×C4×D7—31no—
99C2×Dic14—31no—
100C2×Dic14—31no—
101C7×C4⋊C4C2×D311yescharacteristic
102C7×C4⋊C4C2×D311yescharacteristic
103C7×C4⋊C4C2×D311yescharacteristic
104C7×C22⋊C4C2×D311yescharacteristic
105C7×C22⋊C4C2×D311yescharacteristic
106C22⋊Dic7—31no—
107C22⋊Dic7—31no—
108(C2×D7)⋊C4—31no—
109(C2×D7)⋊C4—31no—
110(C2×D7)⋊C4—31no—
111(C2×D7)⋊C4—31no—
112(C2×D7)⋊C4—31no—
113(C2×D7)⋊C4—31no—
114(C2×D7)⋊C4—31no—
115(C2×D7)⋊C4—31no—
116C4⋊Dic7—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119Dic7⋊C4—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123C4×Dic7—31no—
124C4×Dic7—31no—
125C4×Dic7—31no—
126C4×Dic7—31no—
127C3×C22⋊Q8D711yescharacteristic, complemented
128#947—71no—
129#944—71no—
130#929—71no—
131C4×#67—71no—
132#898—71no—
133#896—71no—
134#896—71no—
135#894—71no—
136#892—71no—
137#887—71no—
138#885—71no—
139#885—71no—
140#884—71no—
141#879—71no—
142C22×C3×C7—21no—
143C22×C3×C7—21no—
144C22×C3×C7—21no—
145C22×C3×C7C2411yescharacteristic
146C2×D21—21no—
147C2×D21—21no—
148C2×D21—21no—
149C2×D21—21no—
150C2×D21—21no—
151C2×D21—21no—
152C3×C4×C7—21no—
153C3×C4×C7—21no—
154C3×C4×C7—21no—
155C3×C4×C7—21no—
156C3×C4×C7—21no—
157C3×C4×C7—21no—
158Dic21—21no—
159Dic21—21no—
160Dic21—21no—
161Dic21—21no—
162Dic21—21no—
163Dic21#4112yes—
164#542—211noSylow
165C23×C7C22×D311yescharacteristic
166C22×D7—31no—
167C22×D7—31no—
168C7×Q8—41no—
169C2×C4×C7—41no—
170C2×C4×C7C22×D311yescharacteristic
171C2×C4×C7C22×D311yescharacteristic
172C2×C4×C7C22×D311yescharacteristic
173C2×C4×C7C22×D311yescharacteristic
174C2×C4×C7C22×D311yescharacteristic
175C2×C4×C7C22×D311yescharacteristic
176C22⋊D7—62no—
177C22⋊D7—121no—
178C2×Dic7—31no—
179C2×Dic7—31no—
180C2×Dic7—31no—
181C2×Dic7—31no—
182C2×Dic7—31no—
183C2×Dic7—31no—
184D28—121no—
185C4×D7—62no—
186C4×D7—121no—
187Dic14—62no—
188Dic14—121no—
189C2×C3×Q8C2×D711yescharacteristic
190C22×C3×C4C2×D711yescharacteristic
191C2×#67—71no—
192C2×#67—71no—
193C2×D12—71no—
194C2×C4×D3—71no—
195C2×C4×D3—71no—
196C2×Dic6—71no—
197C2×Dic6—71no—
198C3×C4⋊C4C2×D711yescharacteristic
199C3×C4⋊C4C2×D711yescharacteristic
200C3×C4⋊C4C2×D711yescharacteristic
201C3×C22⋊C4C2×D711yescharacteristic
202C3×C22⋊C4C2×D711yescharacteristic
203#217—71no—
204#217—71no—
205(C2×D3)⋊C4—71no—
206(C2×D3)⋊C4—71no—
207(C2×D3)⋊C4—71no—
208(C2×D3)⋊C4—71no—
209(C2×D3)⋊C4—71no—
210(C2×D3)⋊C4—71no—
211(C2×D3)⋊C4—71no—
212(C2×D3)⋊C4—71no—
213C4⋊Dic3—71no—
214Dic3⋊C4—71no—
215Dic3⋊C4—71no—
216Dic3⋊C4—71no—
217Dic3⋊C4—71no—
218Dic3⋊C4—71no—
219Dic3⋊C4—71no—
220C4×Dic3—71no—
221C4×Dic3—71no—
222C4×Dic3—71no—
223C4×Dic3—71no—
224C2×C3×C7—41no—
225C2×C3×C7C2×#4111yescharacteristic
226C2×C3×C7H(5,𝔽2)11yescharacteristic
227C2×C3×C72−1+411yescharacteristic
228D21—41no—
229D21—41no—
230C4⋊Q8—211no—
231#126—211no—
232#126—211no—
233#124—211no—
234#124—211no—
235#124—211no—
236#123—211no—
237#123—211no—
238C22⋊Q8—211no—
239C22⋊Q8—211no—
240C22⋊Q8D2111yescharacteristic, complemented
241#121—211no—
242C4×Q8—211no—
243C4×D4—211no—
244#117—211no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7C23×D311yescharacteristic
249C2×D7—61no—
250C2×D7—61no—
251C2×D7—61no—
252C2×D7—61no—
253C2×D7—61no—
254C2×D7—61no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—21no—
261Dic7—32no—
262Dic7—61no—
263Dic7—61no—
264Dic7—61no—
265Dic7—61no—
266Dic7—61no—
267C23×C3C22×D711yescharacteristic
268C22×D3—71no—
269C22×D3—71no—
270C3×Q8—41no—
271C2×C3×C4—41no—
272C2×C3×C4C22×D711yescharacteristic
273C2×C3×C4C22×D711yescharacteristic
274C2×C3×C4C22×D711yescharacteristic
275C2×C3×C4C22×D711yescharacteristic
276C2×C3×C4C22×D711yescharacteristic
277C2×C3×C4C22×D711yescharacteristic
278#67—142no—
279#67—281no—
280C2×Dic3—71no—
281C2×Dic3—71no—
282C2×Dic3—71no—
283C2×Dic3—71no—
284C2×Dic3—71no—
285C2×Dic3—71no—
286D12—281no—
287C4×D3—142no—
288C4×D3—281no—
289Dic6—142no—
290Dic6—281no—
291C3×C7#54211yescharacteristic, complemented, Hall
292C2×Q8—211no—
293C2×Q8—211no—
294C2×Q8C2×D2111yescharacteristic
295C2×D4—211no—
296C2×D4—211no—
297C2×D4—211no—
298C22×C4—211no—
299C22×C4—211no—
300C22×C4C2×D2111yescharacteristic
301C4⋊C4—211no—
302C4⋊C4—211no—
303C4⋊C4—211no—
304C4⋊C4—211no—
305C4⋊C4—211no—
306C4⋊C4—211no—
307C4⋊C4—211no—
308C4⋊C4C2×D2111yescharacteristic
309C4⋊C4C2×D2111yescharacteristic
310C4⋊C4C2×D2111yescharacteristic
311C22⋊C4—211no—
312C22⋊C4—211no—
313C22⋊C4—211no—
314C22⋊C4—211no—
315C22⋊C4—211no—
316C22⋊C4—211no—
317C22⋊C4—211no—
318C22⋊C4—211no—
319C22⋊C4—211no—
320C22⋊C4—211no—
321C22⋊C4C2×D2111yescharacteristic
322C22⋊C4C2×D2111yescharacteristic
323C42—211no—
324C42—211no—
325C42—211no—
326C42—211no—
327C2×C7—41no—
328C2×C7#100411yescharacteristic
329C2×C7#100711yescharacteristic
330C2×C7D3×#4111yescharacteristic
331D7—121no—
332D7—121no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3C23×D711yescharacteristic
337C2×D3—141no—
338C2×D3—141no—
339C2×D3—141no—
340C2×D3—141no—
341C2×D3—141no—
342C2×D3—141no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349Dic3—72no—
350Dic3—141no—
351Dic3—141no—
352Dic3—141no—
353Dic3—141no—
354Dic3—141no—
355C23—211no—
356C23—211no—
357C23C22×D2111yescharacteristic
358Q8—41no—
359Q8—422no—
360Q8—841no—
361D4—422no—
362D4—841no—
363D4—841no—
364C2×C4—41no—
365C2×C4—211no—
366C2×C4—211no—
367C2×C4—211no—
368C2×C4—211no—
369C2×C4—211no—
370C2×C4—211no—
371C2×C4—422no—
372C2×C4—841no—
373C2×C4C22×D2111yescharacteristic
374C2×C4C22×D2111yescharacteristic
375C2×C4C22×D2111yescharacteristic
376C2×C4C22×D2111yescharacteristic
377C2×C4C22×D2111yescharacteristic
378C2×C4C22×D2111yescharacteristic
379C7#564011yescharacteristic, complemented, Sylow
380C2×C3—41no—
381C2×C3#654811yescharacteristic
382C2×C3#655111yescharacteristic
383C2×C3D7×#4111yescharacteristic
384D3—281no—
385D3—281no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—421no—
390C22—421no—
391C22—421no—
392C22—421no—
393C22—421no—
394C22—421no—
395C22C23×D2111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—212no—
403C4—421no—
404C4—421no—
405C4—421no—
406C4—421no—
407C4—421no—
408C3#9033111yescharacteristic, complemented, Sylow
409C2—41no—
410C2—841no—
411C2—841no—
412C2#1062165711yescharacteristic
413C2#1062166011yescharacteristic
414C2#41×D2111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊Q8C2
prime radical#10621367C2
hypercenterC22⋊Q8D21
hypocenterC3×C7#542
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C7×C22⋊Q8C2semidirect
2. #10621430C2semidirect
3. #10621427C2semidirect
4. #10621412C2semidirect
5. C4×#4268C2semidirect
6. #10621381C2semidirect
7. #10621379C2semidirect
8. #10621379C2semidirect
9. #10621377C2semidirect
10. #10621375C2semidirect
11. #10621370C2semidirect
12. #10621368C2semidirect
13. #10621368C2semidirect
15. #10621362C2semidirect
52. C7×C22⋊Q8D3semidirect
127. C3×C22⋊Q8D7semidirect
240. C22⋊Q8D21semidirect
291. C3×C7#542semidirect
379. C7#5640semidirect
408. C3#90331semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
#106214302
#106214272
#106214122
C4×#42682
#106213812
#106213792
#106213772
#106213752
#106213702
#106213682
#106213672
#106213622
#903313
#56407